applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

optimal control of the free boundary in a two phase stefan problem we present an optimal control approach for the solidification process of a melt in a container the process is described by a two phase stefan problem with the free boundary interface between the two phases modelled as a graph we control the evolution of the free boundary using the temperature on the container wall the control goal consists in tracking a prescribed evolution of the free boundary we achieve this goal by minimizing a appropriate cost functional the resulting minimization problem is solved numerically by a steepest descent method with step size control where the gradient of the cost functional is expressed in terms of the adjoint variables several numerical examples are presented which illustrate the performance of the method the novelty of the approach presented consists in using a sharp interface model for the control of the free boundary this guarantees direct access to the free boundary as optimization variable in terms of its parametrization as a graph moreover at any stage of the optimization process the physical laws constituted by the mathematical model are conserved and the heat flux into the free boundary is variable and thus does not need to be specified a priori in the optimization process this guarantees more flexibility than in the optimization approaches to two phase stefan problems taken so far

a petrov galerkin finite element method for variable coefficient fractional diffusion equations abstract fractional diffusion equations have found increasingly more applications in recent years but introduce new mathematical and numerical difficulties galerkin formulation which was proved to be coercive and well posed for fractional diffusion equations with a constant diffusivity coefficient may lose its coercivity for variable coefficient problems the corresponding finite element method fails to converge we utilize the discontinuous petrov galerkin dpg framework to develop a petrov galerkin finite element method for variable coefficient fractional diffusion equations we prove the well posedness and optimal order convergence of the petrov galerkin finite element method numerical examples are presented to verify the theoretical results

finite element method for nonlinear riesz space fractional diffusion equations on irregular domains in this paper we consider two dimensional riesz space fractional diffusion equations with nonlinear source term on convex domains applying galerkin finite element method in space and backward difference method in time we present a fully discrete scheme to solve riesz space fractional diffusion equations our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes which can help us to deal with space fractional terms on any convex domain the stability and convergence of the scheme are also discussed numerical examples are given to verify accuracy and stability of our scheme a finite element method is developed to solve 2d space fractional diffusion equations with nonlinear source term we develop a finite element method for fpdes on irregular domain we obtain explicit expressions for fractional derivatives of shape functions we give the details on how to compute fractional stiffness matrix fractional fitzhugh nagumo model is solved on circular domain

theta point behavior of diluted polymer solutions can one observe the universal logarithmic corrections predicted by field theory in recent large scale monte carlo simulations of various models of theta point polymers in three dimensions grassberger and hegger found logarithmic corrections to mean field theory with amplitudes much larger than the universal amplitudes of the leading logarithmic corrections calculated by duplantier in the framework of tricritical o n field theory to resolve this issue we calculate the universal subleading correction of field theory which turns out to be of the same order of magnitude as the leading correction for all chain lengths available in present day simulations borel resummation of the renormalization group flow equations also shows the presence of such large corrections this suggests that the published simulations did not reach the asymptotic regime to further support this view we present results of monte carlo simulations on a domb joyce like model of weakly interacting random walks again the results cannot be explained by keeping only the leading corrections but are in fair accordance with our full theoretical result the corrections found for the domb joyce model are much smaller than those for other models which clearly shows that the effective corrections of are not yet in the asymptotic regime altogether our findings show that the existing simulations of theta polymers are compatible with tricritical field theory since the crossover to the asymptotic regime is very slow similar results were found earlier for self avoiding walks at their upper critical dimension d c 4

an adjoint method for the inverse design of solidification processes with natural convection this paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes in particular the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved this is the first time that an infinite dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse design problems with moving boundaries and boussinesq incompressible flow the present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary while overspecified boundary conditions are supplied in another part of the boundary here the freezing interface the solidification design problem is mathematically posed as a whole time domain optimization problem the gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem the minimization process is realized by the conjugate gradient method via the solutions of the direct adjoint and sensitivity sub problems the proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold the non uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized the inverse design problem is then posed as finding the appropriate spatial temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity the numerical example demonstrates the accuracy and convergence of the adjoint formulation finally open related research design problems are discussed 1998 john wiley sons ltd

optimal error estimates of spectral petrov galerkin and collocation methods for initial value problems of fractional differential equations we present optimal error estimates for spectral petrov galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval we also develop laguerre spectral petrov galerkin methods and collocation methods for fractional equations on the half line numerical results confirm the error estimates

error estimates of a spectral petrov galerkin method for two sided fractional reaction diffusion equations abstract we study regularity and the spectral method for two sided fractional diffusion equations with a reaction term we show that the regularity of the solution in weighted sobolev spaces can be greatly improved compared to that in standard sobolev spaces with this regularity we prove an optimal error estimate for the spectral petrov galerkin method numerical results are presented to verify our theoretical convergence orders

a fast finite difference method for distributed order space fractional partial differential equations on convex domains abstract fractional partial differential equations pdes provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long range spatial interactions which cannot be modeled accurately by classical second order diffusion equations however numerical methods for space fractional pdes usually generate dense or full stiffness matrices for which a direct solver requires o n 3 computations per time step and o n 2 memory where n is the number of unknowns the significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three dimensional space fractional diffusion equations computationally intractable fast numerical methods were previously developed for space fractional pdes on multidimensional rectangular domains without resorting to lossy compression but rather via the exploration of the tensor product form of the toeplitz like decompositions of the stiffness matrices in this paper we develop a fast finite difference method for distributed order space fractional pdes on a general convex domain in multiple space dimensions the fast method has an optimal order storage requirement and almost linear computational complexity without any lossy compression numerical experiments show the utility of the method

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

a divide and conquer fast finite difference method for spacetime fractional partial differential equation fractional partial differential equations fpdes provide better modeling capabilities for challenging phenomena with long range time memory and spatial interaction than integer order pdes do a conventional numerical discretization of spacetime fpdes requires o n2 mn memory and o mn3 m2n computational work where n is the number of spatial freedoms per time step and m is the number of time steps we develop a fast finite difference method fdm for spacetime fpde i we utilize the toeplitz like structure of the coefficient matrix to develop a matrix free preconditioned fast krylov subspace iterative solver to invert the coefficient matrix at each time step ii we utilize a divide and conquer strategy a recursive direct solver to handle the temporal coupling of the numerical scheme the fast method has an optimal memory requirement of o mn and an approximately linear computational complexity of o nm logn log2m without resorting to any lossy compression numerical experiments show the utility of the method

optimal control of the classical two phase stefan problem in level set formulation optimal control motion planning of the free interface in classical two phase stefan problems is considered the evolution of the free interface is modeled by a level set function the first order optimality system is derived on a formal basis it provides gradient information based on the adjoint temperature and adjoint level set function suitable discretization schemes for the forward and adjoint systems are described numerical examples verify the correctness and flexibility of the proposed scheme

preconditioned iterative methods for fractional diffusion equation in this paper we are concerned with numerical methods for the solution of initial boundary value problems of anomalous diffusion equations of order 1 2 the classical crank nicholson method is used to discretize the fractional diffusion equation and then the spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates two preconditioned iterative methods namely the preconditioned generalized minimal residual preconditioned gmres method and the preconditioned conjugate gradient for normal residual preconditioned cgnr method are proposed to solve relevant linear systems numerical experiments are given to illustrate the efficiency of the methods

the general epidemic process in a finite environment the general epidemic process gep a stochastic multiparticle process which exhibits a critical point near an absorbing state and leads to percolating clusters is studied in a finite environment using renormalisation group techniques the authors calculate the linear relaxation time in a cubic geometry of finite size l with periodic boundary conditions imposed the corresponding scaling behaviour to o epsilon epsilon 6 d d being the spatial dimension is presented in universal form

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

self avoiding walks in four dimensions logarithmic corrections we present simulation results for long n or 4000 self avoiding walks in four dimensions we find definite indications of logarithmic corrections but the data are poorly described by the asymptotically leading terms detailed comparisons are presented with renormalization group flow equations derived by direct renormalization and with the results of a field theoretic calculation

a preconditioned fast quadratic spline collocation method for two sided space fractional partial differential equations abstract quadratic spline collocation methods for the one and two dimensional fractional diffusion equations are proposed by carefully exploring the mathematical structure of the coefficient matrix we propose a matrix free fast krylov subspace iterative solver for the corresponding quadratic spline collocation scheme and then preconditioning technique is applied to further accelerate the convergence of the fast krylov subspace iterative method it shows that the fast quadratic spline collocation scheme has greatly reduced the computational cost from o n 2 to o n log n per iteration and memory requirement from o n 2 to o n while still well approximates the fractional diffusion equations without any accuracy lost numerical experiments are given to demonstrate the efficiency and effectiveness of the fast method

equation of state for directed percolation using field theoretic renormalization group methods we calculate the equation of state for non equilibrium systems belonging to the universality class of directed percolation to second order in by introducing a parametric representation the result can be written to this order in a very simple form we use our result to obtain a universal amplitude ratio to second order in

a finite difference scheme for semilinear space fractional diffusion equations with time delay a linearized quasi compact finite difference scheme is proposed for semilinear space fractional diffusion equations with a fixed time delay the nonlinear source term is discretized and linearized by taylor s expansion to obtain a second order discretization in time the space fractional derivatives are approximated by a weighted shifted grunwald letnikov formula which is of fourth order approximation under some smoothness assumptions of the exact solution under the local lipschitz conditions the solvability and convergence of the scheme are proved in the discrete maximum norm by the energy method numerical examples verify the theoretical predictions and illustrate the validity of the proposed scheme

a functional optimization approach to an inverse magneto convection problem a formulation and an fem implementation for the solution of an ill posed inverse design magneto convection problem is proposed in particular an incompressible viscous electrically conducting liquid material occupying a given domain is considered convection is driven by buoyancy effects as well as a lorentz force generated due to the applied magnetic field thermal boundary conditions are only prescribed in the part h0 of the boundary in addition the temperature distribution is also prescribed in the part i of the boundary h1 where the heat flux is known i e i is a boundary with overspecified thermal boundary conditions the inverse magneto convection problem is posed as an optimization problem in l2 i 0 tmax for the calculation of the boundary heat flux qo x t with x t h0 0 tmax the optimization scheme minimizes the discrepancy m x t x t qo l2 i 0 tmax between the temperature x t qo calculated from the solution of a direct problem for each flux qo and the desired or measured temperature m x t on the boundary i the exact l2 gradient of the cost functional is obtained from appropriately defined adjoint fields the adjoint problem is defined from the sensitivity operators obtained by linearization of the equations governing the direct problem the standard supg pspg stabilized fem formulation for incompressible fluid flow simulation is here extended to thermo magnetically driven flows the proposed fem formulation is used for the calculation of the direct adjoint and sensitivity thermal fluid flow and electric potential fields the entire optimization algorithm is solved using the conjugate gradient method finally the method is demonstrated through the solution of a few inverse problems with known results the need for regularization is identified in one of the example problems with uniformly distributed random errors in the temperature data m x t and a h1 regularized formulation is introduced to obtain stable solutions finally the numerical results are elucidated and potential applications are addressed

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

a finite volume method for two dimensional riemann liouville space fractional diffusion equation and its efficient implementation abstract we develop a finite volume method based on crank nicolson time discretization for the two dimensional nonsymmetric riemann liouville space fractional diffusion equation stability and convergence are then carefully discussed we prove that the finite volume scheme is unconditionally stable and convergent with second order accuracy in time and min 1 1 order in space with respect to a weighted discrete norm here 0 1 are the space fractional order indexes in x and y directions respectively furthermore we rewrite the finite volume scheme into a matrix form and develop a matrix free preconditioned fast krylov subspace iterative method which only requires storage of o n and computational cost of o n log n per iteration without losing any accuracy compared to the direct gaussian elimination method here n is the total number of spatial unknowns consequently the fast finite volume method is particularly suitable for large scale modeling and simulation numerical experiments verify the theoretical results and show strong potential of the fast method

finite size scaling of directed percolation in the steady state recently considerable progress has been made in understanding finite size scaling in equilibrium systems here we study finite size scaling in non equilibrium systems at the instance of directed percolation dp which has become the paradigm of non equilibrium phase transitions into absorbing states above at and below the upper critical dimension we investigate the finite size scaling behavior of dp analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d dimensional hypercube of finite edge length l with periodic boundary conditions near the bulk critical point in particular we study the order parameter and its higher moments using renormalized field theory we derive finite size scaling forms of the moments in a one loop calculation moreover we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous binder cumulant in equilibrium systems and that in particular provides a new signature of the dp universality class to complement our analytical work we perform monte carlo simulations which confirm our analytical results

direct and inverse one phase stefan problem solved by the variational iteration method in this paper solutions of one phase direct and inverse stefan problems are presented the direct problem consists in a calculation of temperature distribution and of a function which describes the position of the moving interface whilst the inverse problem consists in a calculation of temperature distribution as well as in the reconstruction of the function which describes the temperature distribution on the boundary when the position of the moving interface is known the proposed solution is based on the variational iteration method after the application of which we obtain the solution in the form of continuous functions

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

universal scaling behaviour of directed percolation and the pair contact process in an external field we consider the scaling behaviour of directed percolation and of the pair contact process with a conjugated field in particular we determine numerically the equation of state and show that both models are characterized by the same universal scaling function furthermore we derive the equation of state for the pair contact process within a mean field approach which again agrees with the mean field equation of state of the directed percolation universality class

numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation in this article we adopt the saddle point theoretical framework to analyze the conservative fractional diffusion equations by introducing a fractional order flux as auxiliary variable we establish the well posedness of the saddle point variational formulation based on the formulation we propose a locally conservative expanded mixed finite element procedure to approximate the unknown its derivative and the fractional flux directly and prove the existence and uniqueness of the mixed finite element solution by proving that the negative fractional derivative spaces are equivalent to the negative fractional order sobolev spaces we derive the approximation capability of the standard projection operator in fractional sobolev spaces for non smooth functions this leads an optimal order error estimates in terms of the right hand side both for sufficiently smooth solution and non smooth solution numerical experiments are included to confirm our theoretical findings a locally conservative expanded mixed finite element scheme is developed based on the saddle point variational formulation the well posedness of the saddle point variational formulation and the expanded mixed formulation is established the equivalence of the negative fractional spaces and the negative sobolev spaces is established optimal order error estimates are derived for sufficiently smooth solution and non smooth solution

universal scaling behavior of directed percolation around the upper critical dimension in this work we consider the steady state scaling behavior of directed percolation around the upper critical dimension in particular we determine numerically the order parameter its fluctuations as well as the susceptibility as a function of the control parameter and the conjugated field additionally to the universal scaling functions several universal amplitude combinations are considered we compare our results with those of a renormalization group approach

spectral approximation of a variable coefficient fractional diffusion equation in one space dimension abstract in this article we consider the approximation of a variable coefficient two sided fractional diffusion equation fde having unknown u by introducing an intermediate unknown q the variable coefficient fde is rewritten as a lower order constant coefficient fde a spectral approximation scheme using jacobi polynomials is presented for the approximation of q q n the approximate solution to u u n is obtained by post processing q n an a priori error analysis is given for q q n and u u n two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates

finite element method for the space and time fractional fokker planck equation we develop the finite element method for the numerical resolution of the space and time fractional fokker planck equation which is an effective tool for describing a process with both traps and flights the time fractional derivative of the equation is used to characterize the traps and the flights are depicted by the space fractional derivative the stability and error estimates are rigorously established and we prove that the convergent order is o k 2 alpha h mu where k is the time step size and h the space step size numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims

a simple finite element method for boundary value problems with a riemann liouville derivative we consider a boundary value problem involving a riemann liouville fractional derivative of order 3 2 2 on the unit interval 0 1 the standard galerkin finite element approximation converges slowly due to the presence of singularity term x 1 in the solution representation in this work we develop a simple technique by transforming it into a second order two point boundary value problem with nonlocal low order terms whose solution can reconstruct directly the solution to the original problem the stability of the variational formulation and the optimal regularity pickup of the solution are analyzed a novel galerkin finite element method with piecewise linear or quadratic finite elements is developed and l 2 d error estimates are provided the approach is then applied to the corresponding fractional sturm liouville problem and error estimates of the eigenvalue approximations are given extensive numerical results fully confirm our theoretical study

finite size scaling of directed percolation above the upper critical dimension critical phenomena such as second order phase transitions are characterized by singularities causing a discontinuous behavior of various quantities at the transition point e g the speciflc heat the susceptibility and the correlation length these singularities are described by power laws deflning the well known critical exponents studying the phase transition of a given system the aim of investigation is to identify the set of critical exponents which characterizes together with certain universal scaling functions the so called universality class since most systems are not analytically tractable their critical behavior is often investigated via numerical methods for example monte carlo simulations or transfer matrix calculations in these cases the obtained data are limited to flnite system sizes therefore flnite size scaling fss is widely used to extrapolate to the behavior of the inflnite systems in particular fss is an e cient method

a fast finite difference method for three dimensional time dependent space fractional diffusion equations and its efficient implementation fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the non local property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved with gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive in this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time dependent space fractional diffusion equations in three space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices the fast method has a computational work count of o nlogn per iteration and a memory requirement of o n while retaining the same accuracy as the underlying finite difference method solved with gaussian elimination numerical experiments of a three dimensional space fractional diffusion equation show the utility of the fast method

one phase inverse stefan problem solved by adomian decomposition method abstract in this paper the solution of one phase inverse stefan problem is presented the problem consists of the reconstruction of the function which describes the distribution of temperature on the boundary when the position of the moving interface is well known the proposed solution is based on the adomian decomposition method and the least square method

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

a petrov galerkin spectral element method for fractional elliptic problems abstract we develop a new c 0 continuous petrov galerkin spectral element method for one dimensional fractional elliptic problems of the form 0 d x u x u x f x 1 2 subject to homogeneous boundary conditions we employ the standard modal spectral element basis and the jacobi poly fractonomials as the test functions zayernouri and karniadakis 2013 we formulate a new procedure for assembling the global linear system from elemental local mass and stiffness matrices the petrov galerkin formulation requires performing elemental local construction of mass and stiffness matrices in the standard domain only once moreover we efficiently obtain the non local history stiffness matrices in which the non locality is presented analytically for uniform grids we also investigate two distinct choices of basis test functions i local basis test functions and ii local basis with global test functions we show that the former choice leads to a better conditioned system and accuracy while the latter results in ill conditioned linear systems and therefore that is not an efficient and a proper choice of test function we consider smooth and singular solutions where the singularity can occur at boundary points as well as in the interior domain we also construct two non uniform grids over the whole computational domain in order to capture singular solutions finally we perform a systematic numerical study of non local effects via full and partial history fading in order to further enhance the efficiency of the scheme

block preconditioning strategies for time space fractional diffusion equations abstract we present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time space fractional diffusion equations in contrast to the traditional time marching procedure the discretization via finite difference is considered in a fully coupled time space framework the resulting fully coupled discretized linear system is a summation of two kronecker products the four preconditioning methods are based on block diagonal banded block triangular and kronecker product splittings of the coefficient matrix all preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator numerical experiments show the efficiency of the four block preconditioners and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one

spectral direction splitting methods for two dimensional space fractional diffusion equations a numerical method for a kind of time dependent two dimensional two sided space fractional diffusion equations is developed in this paper the proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization the direction splitting approach renders the underlying two dimensional equation into a set of one dimensional space fractional diffusion equations at each time step then these one dimensional equations are solved by using the spectral method based on weak formulations a time error estimate is derived for the semi discrete solution and the unconditional stability of the fully discretized scheme is proved some numerical examples are presented to validate the proposed method

the adjoint method for an inverse design problem in the directional solidification of binary alloys this paper presents a systematic procedure based on the adjoint method for solving a class of inverse directional alloy solidification design problems in which a desired growth velocityvfis achieved under stable growth conditions to the best of our knowledge this is the first time that a continuum adjoint formulation is proposed for the solution of an inverse problem with simultaneous heat and mass transfer thermo solutal convection and phase change in this paper the interfacial stability is considered to imply a sharp solid liquid freezing interface this condition is enforced using the constitutional undercooling criterion in the form of an inequality constraint between the thermal and solute concentration gradients gandgc respectively at the freezing front the main unknowns of the design problem are the heating and or cooling boundary conditions on the mold walls the inverse design problem is formulated as a functional optimization problem the cost functional is defined by the square of thel2norm of the deviation of the freezing interface temperature from the temperature corresponding to thermodynamic equilibrium a continuum adjoint system is derived to calculate the adjoint temperature concentration and velocity fields such that the gradient of the cost functional can be expressed analytically the cost functional minimization process is realized by the conjugate gradient method via the finite element method solutions of the continuum direct sensitivity and adjoint problems the developed formulation is demonstrated with an example of designing the directional solidification of a binary aqueous solution in a rectangular mold such that a stable vertical interface advances from left to right with a desired growth velocity

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

fourth order compact scheme for space fractional advection diffusion reaction equations with variable coefficients abstract in this work a new fourth order compact approximation is derived for riemann liouville space fractional derivatives modified wave numbers are obtained for various approximations of fractional derivatives considering these modified wave numbers fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations a novel fourth order compact scheme is developed for space fractional advection diffusion reaction equations with variable coefficients in one and two dimensions in the proposed compact scheme the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations this splitting of second derivative approximations allows us to obtain a similar system of linear equations as in zhao and tian 2017 which presents only second order accurate finite difference method for advection diffusion reaction equations in one dimension the stability of the proposed compact scheme is demonstrated numerically furthermore the proposed compact scheme is employed to space fractional black scholes equation for pricing european options as an application of fractional derivatives in mathematical finance numerical illustrations are presented to validate the theoretical claims

induction heating processes optimization a general optimal control approach a general automatic optimization procedure coupled to a finite element induction heating process simulation has been developed the mathematical model and the numerical methods are presented along with results validating the model the first part of this paper presents the direct induction heating mathematical model the related main numerical choices and especially the ultra weak coupling procedure the general optimization problem is then presented with the full detailed transposition of the ultra weak coupling procedure to the adjoint problem numerical results provided at the end prove the efficiency and robustness of the adjoint model in optimizing induction heating processes

stability and convergence analysis of the quadratic spline collocation method for time dependent fractional diffusion equations abstract a quadratic spline collocation method combined with the crank nicolson time discretization is proposed for time dependent two sided fractional diffusion equations by carefully analyzing the mathematical properties of the coefficient matrix the new scheme is proved to be unconditionally stable in the sense of discrete l2 norm for 2 where is the order of the space fractional derivative of the fractional diffusion equation and 1 2576 see lemma 3 1 furthermore the fractional order spline interpolation error over the collocation points is studied and subsequently we show that the spline collocation solution of the fractional diffusion equation converges to the exact one with order o h 3 2 under the discrete l2 norm where and h are the temporal and spatial step sizes respectively finally numerical experiments are given to verify the theoretical results

an analysis of implicit conservative difference solver for fractional klein gordon zakharov system abstract in this paper we propose an efficient linearly implicit conservative difference solver for the fractional klein gordon zakharov system first of all we present a detailed derivation of the energy conservation property of the system in the discrete setting then by using the mathematical induction it is proved that the proposed scheme is uniquely solvable subsequently by virtue of the discrete energy method and a cut off function technique it is shown that the proposed solver possesses the convergence rates of o t 2 h 2 in the sense of l and l2 norms respectively and is unconditionally stable finally numerical results testify the effectiveness of the proposed scheme and exhibit the correctness of theoretical results

wellposedness of variable coefficient conservative fractional elliptic differential equations fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second order differential equations there are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second order differential equations these are i fractional differential operators are nonlocal operators and ii the adjoint of a fractional differential operator is not the negative of itself the wellposedness of a galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously many subsequent works were carried out to extend the analysis to other numerical methods a constant diffusivity coefficient has been assume

galerkin finite element approximation of symmetric space fractional partial differential equations abstract in this paper symmetric space fractional partial differential equations ssfpde with the riesz fractional operator are considered the ssfpde is obtained from the standard advection dispersion equation by replacing the first order and second order space derivatives with the riesz fractional derivatives of order 2 0 1 and 2 1 2 respectively we prove that the variational solution of the ssfpde exists and is unique using the galerkin finite element method and a backward difference technique a fully discrete approximating system is obtained which has a unique solution according to the lax milgram theorem the stability and convergence of the fully discrete schemes are derived finally some numerical experiments are given to confirm our theoretical analysis

inhomogeneous dirichlet boundary value problems of space fractional diffusion equations and their finite element approximations we prove the wellposedness of the galerkin weak formulation and petrov galerkin weak formulation for inhomogeneous dirichlet boundary value problems of constant or variable coefficient conservative caputo space fractional diffusion equations we also show that the weak solutions to their riemann liouville analogues do not exist in general in addition we develop an indirect finite element method for the dirichlet boundary value problems of caputo fractional differential equations which reduces the computational work for the numerical solution of variable coefficient fractional diffusion equations from o n 3 to o n and the memory requirement from o n 2 to o n on any quasiuniform space partition we further prove a nearly sharp error estimate for the method which is expressed in terms of the smoothness of the prescribed data of the problem only we carry out numerical experiments to investigate the performance of the method in comparison with the galerkin finite element method

universal finite size scaling behavior and universal dynamical scaling behavior of absorbing phase transitions with a conserved field we analyze numerically three different models exhibiting an absorbing phase transition we focus on the finite size scaling as well as the dynamical scaling behavior an accurate determination of several critical exponents allows one to validate certain hyperscaling relations using these hyperscaling relations it is possible to express the avalanche exponents of a self organized critical system in terms of the ordinary exponents of a continuous absorbing phase transition

optimal control of thermally convected fluid flows we examine the optimal control of stationary thermally convected fluid flows from the theoretical and numerical point of view we use thermal convection as control mechanism that is control is effected through the temperature on part of the boundary control problems are formulated as constrained minimization problems existence of optimal control is given and a first order necessary condition of optimality from which optimal solutions can be obtained is established we develop numerical methods to solve the necessary condition of optimality and present computational results for control of cavity and channel type flows showing the feasibility of the proposed approach

a crank nicolson adi spectral method for a two dimensional riesz space fractional nonlinear reaction diffusion equation in this paper a new alternating direction implicit galerkin legendre spectral method for the two dimensional riesz space fractional nonlinear reaction diffusion equation is developed the temporal component is discretized by the crank nicolson method the detailed implementation of the method is presented the stability and convergence analysis is strictly proven which shows that the derived method is stable and convergent of order 2 in time an optimal error estimate in space is also obtained by introducing a new orthogonal projector the present method is extended to solve the fractional fitzhugh nagumo model numerical results are provided to verify the theoretical analysis

the numerical solution of the inverse stefan problem in two space variables a method is given for solving the inverse stefan problem for the heat equation in two space variables the approach is based on using a complete family of solutions to the heat equation thus reducing the dimensionality of the problem by one and minimizing the maximal defect in the initial boundary data subject to regularizing constraints thus making the problem well posed in the sense that the solution now depends continuously on the data two numerical examples are given which make use of a linear semi infinite programming method recently developed by hettich and zencke

a functional optimization formulation and implementation of an inverse natural convection problem abstract a formulation and a fem implementation for the solution of an ill posed inverse design natural convection problem is proposed in particular we consider an incompressible viscous liquid material occupying a given domain where convection is driven by buoyancy thermal boundary conditions are only prescribed in the part h 0 of the boundary in addition the temperature distribution is also prescribed in the part i of the boundary h 1 where the heat flux is known i e i is a boundary with over specified thermal boundary conditions the inverse convection problem is posed as an optimization problem in l 2 i 0 t max for the calculation of the boundary heat flux q o x t with x t h 0 0 t max the optimization scheme minimizes the discrepancy m x t x t q o l 2 1 0 t max between the temperature x t q o calculated from the solution of a direct problem for each flux q 0 and the measured or desired temperature m x t at the boundary i the gradient of the cost functional is obtained from the adjoint fields the adjoint problem is defined from the sensitivity operators obtained by linearization of the equations governing the direct problem the conjugate gradient method is employed to perform the optimization process the fem is used for the calculation of the direct adjoint and sensitivity thermal and fluid flow fields finally the method is demonstrated with the solution of some two dimensional inverse problems with known solutions the numerical results are discussed and future work with potential applications is addressed

domain decomposition methods for space fractional partial differential equations abstract a two level additive schwarz preconditioner is proposed for solving the algebraic systems resulting from the finite element approximations of space fractional partial differential equations sfpdes it is shown that the condition number of the preconditioned system is bounded by c 1 h where h is the maximum diameter of subdomains and is the overlap size among the subdomains numerical results are given to support our theoretical findings

optimal control of coefficients in parabolic free boundary problems modeling laser ablation abstract inverse stefan type free boundary problem for the second order parabolic equation arising in modeling of laser ablation of biomedical tissues is analyzed where information on the coefficients heat flux on the fixed boundary and density of heat sources are missing and must be found along with the temperature and free boundary new pde constrained optimal control framework in hilbert besov spaces introduced in abdulla 2013 is employed where the missing data and the free boundary are components of the control vector and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary discretization by finite differences is pursued and convergence of the discrete optimal control problems to the original problem is proved numerical algorithm based on the frechet differentiability and the gradient method in hilbert besov spaces is implemented the results of the numerical experiments for the identification of the free boundary and diffusion coefficient are presented

control of the freezing interface motion in two dimensional solidification processes using the adjoint method the aim of this work is to calculate the optimum history of boundary cooling conditions that in two dimensional conduction driven solidification processes results in a desired history of the freezing interface location motion the freezing front velocity and heat flux on the solid side of the front define the obtained solidification microstructure that can be selected such that desired macroscopic mechanical properties and soundness of the final cast product are achieved the so called two dimensional inverse stefan design problem is formulated as an infinite dimensional minimization problem the adjoint method is developed in conjunction with the conjugate gradient method for the solution of this minimization problem

the accuracy and stability of cn wsgd schemes for space fractional diffusion equation abstract in this paper we discuss the accuracy and the stability of the cn wsgd when it is applied to the initial boundary value problem for a space fractional diffusion equation in the cn wsgd scheme the weighted and shifted grunwald difference wsgd scheme is used to discretize the space fractional derivative and the crank nicolson cn scheme is used to discretize the temporal derivative there is a free parameter in a cn wsgd scheme that affects the stability and the accuracy of the scheme in general a cn wsgd scheme has temporally and spatially second order accuracy and there exists a scheme that has spatially third order accuracy however the third order accuracy scheme is not always stable in this paper we present and prove some theoretical results old and new on the stability of several cn wsgd schemes and consider choosing a value for the parameter such that the corresponding cn wsgd scheme is unconditioned stable and has optimal upper bound for the accuracy numerical results are presented to verify the theoretical results

using the swarm intelligence algorithms in solution of the two dimensional inverse stefan problem in the paper a procedure for solving the two dimensional inverse stefan problem is presented in considered problem the heat transfer coefficient is identified with the aid of known measurements of temperature in selected points of the region as the additional information direct stefan problem is solved by using the alternating phase truncation method goal of the paper is to compare two swarm intelligence algorithms the ant colony optimization algorithm and the artificial bee colony algorithm applied for minimizing a functional expressing the error of approximate solution

a unified spectral method for fpdes with two sided derivatives part ii stability and error analysis abstract we present the stability and error analysis of the unified petrov galerkin spectral method developed in 1 for linear fractional partial differential equations with two sided derivatives and constant coefficients in any 1 d dimensional space time hypercube d 1 2 3 subject to homogeneous dirichlet initial boundary conditions specifically we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method finally we perform several numerical simulations to compare the theoretical and computational rates of convergence

finite difference finite element method for two dimensional time space fractional bloch torrey equations with variable coefficients on irregular convex domains abstract in magnetic resonance imaging of the human brain the diffusion process of tissue water is considered in the complex tissue environment of cells membranes and connective tissue models based on fractional order bloch torrey equations are known to provide insights into tissue structures and the microenvironment in this paper we consider new two dimensional multi term time and space fractional bloch torrey equations with variable coefficients on irregular convex domains which involve the caputo time fractional derivative and the riemann liouville space fractional derivative an unstructured mesh galerkin finite element method is used to discretize the spatial fractional derivative while for each time fractional derivative we use the l 1 scheme on a temporal graded mesh the stability and convergence of the fully discrete scheme are proved numerical examples are given to verify the efficiency of our method

backward fractional advection dispersion model for contaminant source prediction the forward fractional advection dispersion equation fade provides a useful model for non fickian transport in heterogeneous porous media the space fade captures the long leading tail skewness and fast spreading typically seen in concentration profiles from field data this paper develops the corresponding backward fade model to identify source location and release time the backward method is developed from the theory of inverse problems and then explained from a stochastic point of view the resultant backward fade differs significantly from the traditional backward advection dispersion equation ade because the fractional derivative is not self adjoint and the probability density function for backward locations is highly skewed finally the method is validated using tracer data from a well known field experiment where the peak of the backward fade curve predicts source release time while the median or a range of percentiles can be used to determine the most likely source location for the observed plume the backward ade cannot reliably identify the source in this application since the forward ade does not provide an adequate fit to the concentration data

on the solution of an ill posed design solidification problem using minimization techniques in finite and infinite dimensional function spaces this paper provides a comparative study of two alternative methodologies for the solution of an inverse design solidification problem it is the one dimensional solidification problem of calculating the boundary heat flux history that achieves a desired freezing front velocity and desired heat fluxes at the freezing front the front velocity h t and flux history qms t on the solid side of the front control the obtained cast structure as such the potential applications of the proposed methods to the control of casting processes are enormous the first technique utilizes a finite dimensional approximation of the unknown boundary heat flux function q0 t the second technique uses the adjoint method to calculate in l2 the derivative of the cost functional tm t h t t q0 that expresses the square error between the calculated t h t t q0 and the given freezing front temperature tm both steepest descent sdm and conjugate gradient methods cgm are examined a front tracking fem technique is used for the discretization of the state space a detailed numerical analysis of the space and time discretization of the parameter and state spaces of the effect of the end condition of the adjoint problem and of other parameters in the solution are examined

a fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation abstract we develop a fast solver for the spectral element method sem applied to the two sided fractional diffusion equation on uniform geometric and graded meshes by approximating the singular kernel with a degenerate kernel we construct a hierarchical matrix h matrix to represent the stiffness matrix of the sem and provide error estimates which we verify numerically we can solve efficiently the h matrix approximation problem using a hierarchical lu h lu decomposition method which reduces the computational cost to o r 2 n d log 2 n o r 3 n d log n where r is the rank of the h lu decomposition n d is the total number of degrees of freedom and n is the number of elements however we lose the high accuracy of the sem to prevent this we solve the corresponding preconditioned system by using the h matrix approximation problem as a preconditioner hence recovering the high order accuracy of the sem numerical results show that the condition number of the preconditioned system is independent of the polynomial degree p and grows with the number of elements but at modest values of the rank r here r is the rank of submatrices of the h matrix approximation it is below order 10 in our experiments which represents a reduction of more than 11 orders of magnitude from the unpreconditioned system this reduction is higher in the two sided fractional derivative compared to one sided fractional derivative the corresponding cost is o r 2 n d log 2 n o r 3 n d log n o n d 2 moreover by using a structured mesh uniform or geometric mesh we can further reduce the computational cost to o r 2 n d log 2 n o r 3 n d log n o p 2 n log n for the preconditioned system furthermore we compare the present method with the adaptive cross approximation aca which is kernel independent and relies on the coefficient matrix the comparison shows that the accuracy using the present algorithm is better than the one using aca for the h matrix approximation problem and furthermore we have observed that the aca method may fail to converge in some cases