solution of boundary value problems for integro differential equations by using differential transform method in this study we extended the differential transform method dtm to solve the integro differential equations new theorems for the transformation of integrals are introduced and prooved the commonly encountered linear and nonlinear integro differential equations that appear in literature are solved as an illustration for the efficiency of the method some numerical results are also given to demonstrate the superiority of the method to other common techniques the results we obtained by using this technique is much more accurate compared to the existing ones

a roadmap for discretely energy stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity abstract we present a framework for devising discretely energy stable schemes for general dissipative systems based on a generalized auxiliary variable the auxiliary variable a scalar number can be defined in terms of the energy functional by a general class of functions not limited to the square root function adopted in previous approaches the current method has another remarkable property the computed values for the generalized auxiliary variable are guaranteed to be positive on the discrete level regardless of the time step sizes or the external forces this property of guaranteed positivity is not available in previous approaches a unified procedure for treating the dissipative governing equations and the generalized auxiliary variable on the discrete level has been presented the discrete energy stability of the proposed numerical scheme and the positivity of the computed auxiliary variable have been proved for general dissipative systems the current method termed gpav generalized positive auxiliary variable requires only the solution of linear algebraic equations within a time step with appropriate choice of the operator in the algorithm the resultant linear algebraic systems upon discretization involve only constant and time independent coefficient matrices which only need to be computed once and can be pre computed several specific dissipative systems are studied in relative detail using the gpav framework ample numerical experiments are presented to demonstrate the performance of the method and the robustness of the scheme at large time step sizes

analytical solution of mhd slip flow past a constant wedge within a porous medium using dtm pade abstract the objective of present study is to investigate the two dimensional magnetohydrodynamic mhd flow of a viscous fluid over a constant wedge immersed in a porous medium with velocity slip condition the flow is induced by suction injection and also by the mainstream flow that is assumed to vary in a power law manner with co ordinate distance along the boundary similarity transformations are used to convert the governing nonlinear boundary layer equations into a third order falkner skan equation this equation is solved analytically by using a novel analytical method called dtm pade technique which is a combination of the differential transformation method and the pade approximation this method is applied to give solutions of equation with boundary condition at infinity graphical results are presented to investigate the effects of the velocity slip parameter hartmann number permeability suction injection parameter and nonlinear pressure gradient on the flow field further the results of the present analysis have been compared with the corresponding results available in literature our results have been found in excellent agreement

multiple scalar auxiliary variable msav approach and its application to the phase field vesicle membrane model we consider in this paper gradient flows with disparate terms in the free energy that cannot be efficiently handled with the scalar auxiliary variable sav approach and we develop the multiple sc

on the solutions of the difference equation xn 1 xn 1 1 axnxn 1 we investigate the solutions of the difference equation x n 1 x n 1 1 ax nx n 1 n 0 1 2 where a 0 x 1 and x 0 are real numbers such that ax 1x 0 1

the scalar auxiliary variable sav approach for gradient flows abstract we propose a new approach which we term as scalar auxiliary variable sav approach to construct efficient and accurate time discretization schemes for a large class of gradient flows the sav approach is built upon the recently introduced ieq approach it enjoys all advantages of the ieq approach but overcomes most of its shortcomings in particular the sav approach leads to numerical schemes that are unconditionally energy stable and extremely efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step the scheme is not restricted to specific forms of the nonlinear part of the free energy so it applies to a large class of gradient flows numerical results are presented to show that the accuracy and effectiveness of the sav approach over the existing methods

two classes of linearly implicit local energy preserving approach for general multi symplectic hamiltonian pdes abstract two classes of efficient and robust schemes are proposed for the general multi symplectic hamiltonian systems using the invariant energy quadratization ieq approach the schemes are linear second order accurate local energy preserving and preserve the global energy they are not restricted to specific forms of the nonlinear part of the state function and only require solving linear equations at each time step we applied the new schemes to various multi symplectic hamiltonian pdes to demonstrate their effectiveness computational efficiency and accuracy

on the existence of positive solutions of fourth order difference equations in this paper by means of krasnosel skii s fixed point theorem the existence of positive solutions of fourth order difference equations is considered

on the application of multiquadric bases in conjunction with the ltdrm method to solve nonlinear diffusion equations as transient diffusion equations with nonlinear source terms have broad engineering applications many numerical solution approaches have been proposed in the past one of the numerically efficient and accurate methods was the laplace transform dual reciprocity method ltdrm as a result of a series of studies to further improve the accuracy and efficiency of the ltdrm we shall present in this paper the ltdrm in conjunction with the multiquadric bases which were shown to possess a better convergence rate than other interpolation functions such as linear interpolation functions or augmented thin plate splines atps three test examples are included in this paper to support the previous theoretical work and to provide evidence that the multiquadric bases are indeed superior when they are used in conjunction with the ltdrm

retrieving topological information for phase field models the phase field approach has become a popular tool in modeling interface motion microstructure evolution and more recently the shape transformation of vesicle membranes under elastic bending energy while it is advantageous to employ phase field models in numerical simu lations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes detecting topological events may also become important for many applications such as those in the simulation of blood cells motivated by such considerations a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events it can also be used as a control method to avoid unphys ical changes of topology due to the numerical methods should it become necessary for particular practical applications through numerical experiments we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes

linear and unconditionally energy stable schemes for the binary fluid surfactant phase field model abstract in this paper we consider the numerical solution of a binary fluid surfactant phase field model in which the free energy contains a nonlinear coupling entropy a ginzburg landau double well potential and a logarithmic flory huggins potential the resulting system consists of two nonlinearly coupled cahn hilliard type equations we develop a first and a second order time stepping schemes for this system using the invariant energy quadratization approach in particular the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi explicitly both schemes are linear and lead to symmetric positive definite systems in space at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable in the discrete sense various 2d and 3d numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

the approximate solution of high order linear difference equations with variable coefficients in terms of taylor polynomials this paper presents a numerical method for the approximate solution of mth order linear difference equations with variable coefficients under the mixed conditions in terms of taylor polynomials about any point in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model abstract in this paper we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles in which the free energy of the system consists of an elastic bending energy a surface area constraint and a volume constraint one major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi discrete level we develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic pde system based on the invariant energy quadratization approach in particular the resulted semi discretizations lead to linear systems in space with symmetric positive definite operators at each time step thus can be efficiently solved in addition the proposed schemes are rigorously proved to be unconditionally energy stable various numerical experiments in 2d and 3d are presented to demonstrate the stability and accuracy of the proposed schemes

a phase field model and its numerical approximation for two phase incompressible flows with different densities and viscosities modeling and numerical approximation of two phase incompressible flows with different densities and viscosities are considered a physically consistent phase field model that admits an energy law is proposed and several energy stable efficient and accurate time discretization schemes for the coupled nonlinear phase field model are constructed and analyzed ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios

a phase field model for vesicle substrate adhesion in this paper a phase field model is developed for vesicle adhesion involving complex substrate and vesicle geometries the model takes into account an adhesion potential that depends on the distance of vesicle to the substrate a variational problem is solved in a 3d computational domain by minimizing the contribution of bending elastic energy and the adhesion energy under the constraints of total surface area and volume described via a phase function an adaptive finite element method is used to efficiently compute the numerical solutions of the model the computational results are validated through comparison of several axisymmetric shapes with the sharp interface ode solution moreover we compute shapes for non axisymmetric situations to support the observation that concave substrates favor adhesion

axisymmetric indentation of a thin incompressible elastic layer axisymmetric indentation of an incompressible elastic layer by a frictionless rigid sphere is considered estimates of the contact radius and the force required to produce a given indentation are given when the thickness of the layer is small compared to the contact radius the elastic layer is either bonded to or slips along a rigid substrate the estimates are obtained by asymptotically matching a lubrication type expansion valid in the contact region to an edge layer expansion studied using the wiener hopf technique a substantially larger force is required to equally indent a bonded layer compared to a slipping layer in the former case the force is inversely proportional to the cube of the layer thickness rather than to the layer thickness itself if the layer is free to slip

solutions of the system of differential equations by differential transform method in this study three dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time moreover as an application of two and three dimensional differential transform exact solutions of linear and non linear systems of partial differential equations have been investigated the results of the present method are compared very well with those obtained by decomposition method differential transform method can easily be applied to linear or non linear problems and reduces the size of computational work with this method exact solutions may be obtained without any need of cumbersome work and it is an useful tool for analytical and numerical solutions

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

decoupled energy stable schemes for phase field vesicle membrane model we consider the numerical approximations of the classical phase field vesicle membrane models proposed a decade ago in du et al 2004 6 we first reformulate the model derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish the energy dissipation law then we develop a stabilized decoupled time discretization scheme for the coupled nonlinear system the scheme is unconditionally energy stable and leads to linear and decoupled elliptic equations to be solved at each time step stability analysis and ample numerical simulations are presented thereafter

a multi step differential transform method and application to non chaotic or chaotic systems the differential transform method dtm is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form in this paper we propose a reliable new algorithm of dtm namely multi step dtm which will increase the interval of convergence for the series solution the multi step dtm is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations this new algorithm is applied to lotka volterra chen and lorenz systems then a comparative study between the new algorithm multi step dtm classical dtm and the classical runge kutta method is presented the results demonstrate reliability and efficiency of the algorithm developed

laguerre polynomial approach for solving linear delay difference equations this paper presents a numerical method for the approximate solution of mth order linear delay difference equations with variable coefficients under the mixed conditions in terms of laguerre polynomials the aim of this article is to present an efficient numerical procedure for solving mth order linear delay difference equations with variable coefficients our method depends mainly on a laguerre series expansion approach this method transforms linear delay difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system maple

a new method for theoretical analysis of static indentation test a new mathematical method was developed to study the indentation problem of an infinite elastic layer overlaid on a rigid foundation rigid flat ended cylindrical or spherical indenters are pressed onto the upper surface of the elastic layer causing a small deformation mode shear stresses between the indenter and the layer are assumed negligible and the layer is assumed to be either bonded or unbonded to the rigid foundation the problem is equivalent to a mixed boundary value problem of the theory of elasticity instead of using the fredholm integral equation reported in the literature the new approach obtained closed form solutions through an infinite series convergence can be achieved using less than 10 terms of the series

fast provably unconditionally energy stable and second order accurate algorithms for the anisotropic cahn hilliard model abstract in this paper we consider numerical approximations for solving the anisotropic cahn hilliard model we combine the scalar auxiliary variable sav approach with the stabilization technique to arrive at a novel stabilized sav approach where three linear stabilization terms which are shown to be crucial to remove the oscillations caused by the anisotropic coefficient are added to enhance the stability while keeping the required accuracy the schemes are very easy to implement and fast in the sense that all nonlinear terms are treated in a semi explicit way and one only needs to solve three decoupled linear equations with constant coefficients at each time step we further prove the unconditional energy stabilities rigorously and present numerous 2d and 3d numerical simulations to demonstrate the stability and accuracy

strong form meshless implementation of taylor series method a new meshless approach is investigated by using taylor series expansion and technique of differential transform method which is called meshless implementation of taylor series method mitsm in particular strong form meshless implementation of taylor series method smitsm is studied in this paper then the basis functions are used to solve a 1d second order ordinary differential equation and 2d laplace equation by using the smitsm comparisons are made with the analytical solutions and results of symmetric smoothed particle hydrodynamics ssph method we also compared the effectiveness of the smitsm and ssph method by considering various particle distributions nonhomogeneous terms and number of terms in the basis functions it is observed that the mitsm has the conventional convergence properties and at the expense of cpu time yields smaller l 2 error norms than the ssph method especially in the existence of nonsmooth nonhomogeneous problems

highly efficient invariant conserving explicit runge kutta schemes for nonlinear hamiltonian differential equations abstract a unified framework of invariant conserving explicit runge kutta schemes for the nonlinear hamiltonian odes and pdes are proposed by utilizing the invariant energy quadratization technique first the nonlinear hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant for the nonlinear ode the reformulation is then discretized using a class of relaxation runge kutta schemes for the nonlinear pde the reformulation is first discretized in the space direction by adopting the fourier pseudo spectral discretization which preserves the semi discrete quadratic conservation laws then the semi discrete system is integrated in the time direction using the explicit relaxation runge kutta method the obtained method can conserve different quadratic conservation laws to machine accuracy which improves the numerical stability during long time computations besides the proposed method keeps the same convergence rate of the standard runge kutta scheme when computed with time step size rescaling and gives a rate of convergence reduced by at most one when computed without step size rescaling numerical experiments for several odes and pdes are provided to illustrate the advantages of the proposed algorithms over long time and verify the theoretical analysis

solution of differential difference equations by using differential transform method abstract in this work we successfully extended differential transform method dtm by presenting and proving new theorems to the solution of differential difference equations ddes theorems are presented in the most general form to cover a wide range of ddes being linear or nonlinear and constant or variable coefficient in order to show the power and the robustness of the method and to illustrate the pertinent features of related theorems examples are presented

legendre spectral collocation method for fredholm integro differential difference equation with variable coefficients and mixed conditions in this article the legendre spectral collocation method has been applied to solve fredholm integro differential difference equations with variable coefficients the proposed method is based on the gauss legendre points with the basis functions of lagrange polynomials usually this type of integral equations are very difficult to solve analytically as well as numerically the presented method applied to the integral equation reduces to solve the system of algebraic equations also the numerical results obtained by legendre spectral collocation method have been compared with the results obtained by existing methods illustrative examples have been discussed to demonstrate the validity and applicability of the presented method

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

a novel linear second order unconditionally energy stable scheme for a hydrodynamic q tensor model of liquid crystals abstract the hydrodynamic q tensor model has been used for studying flows of liquid crystals and liquid crystal polymers it can be derived from a variational approach together with the generalized onsager principle in which the total energy decreases in time in this paper we develop a novel linear second order semi discrete scheme in time to solve the governing system of equations in the model the scheme is developed following the novel energy quadratization strategy so that it is linear and unconditionally energy stable at the semi discrete level this scheme is further discretized in space using a second order finite difference method and implemented on a gpu for high performance computing the convergence rate in time is established using a mesh refinement test several numerical examples are presented to demonstrate the usefulness of the model and the effectiveness of the numerical scheme in simulating defect dynamics in flows of liquid crystals

simulating the deformation of vesicle membranes under elastic bending energy in three dimensions in this paper we study the three dimensional deformation of a vesicle membrane under the elastic bending energy with prescribed bulk volume and surface area both static and dynamic deformations are considered a newly developed energetic variational formulation is employed to give an effective eulerian description efficient time and spatial discretizations are considered and implemented numerical experiments illustrate some fascinating phenomena that are of interests in real applications

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

comparing numerical methods for solving fourth order boundary value problems in this study we present a numerical comparison between differential transform method and the adomian decomposition method for solving fourth order boundary value problems three examples are given the numerical results reveal that differential transform method is very efficient and accurate

mixed function method for obtaining exact solution of nonlinear differential difference equations and coupled nddes in this paper we construct a new mixed function method for the first time by using this new method we study the two nonlinear differential difference equations named the generalized hybrid lattice and two component volterra lattice equations some new exact solutions of mixed function type such as discrete solitary wave solutions discrete kink and anti kink wave solutions and discrete breather solutions with kink and anti kink character are obtained and their dynamic properties are also discussed by using software mathematica we show their profiles

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

numerical study of the solution of the burgers and coupled burgers equations by a differential transformation method in this paper the differential transformation method dtm is employed to obtain the numerical analytical solutions of the burgers and coupled burgers equations we begin by showing how the differential transformation method applies to the linear and nonlinear parts of any pde and give some examples to illustrate the sufficiency of the method for solving such nonlinear partial differential equations we also compare it against three famous methods namely the homotopy perturbation method the homotopy analysis method and the variational iteration method these results show that the technique introduced here is accurate and easy to apply

decoupled non iterative and unconditionally energy stable large time stepping method for the three phase cahn hilliard phase field model abstract in this paper we consider numerical approximations for a three phase phase field model where three fourth order cahn hilliard equations are nonlinearly coupled together through a lagrange multiplier term and a sixth order polynomial bulk potential by combining the recently developed sav approach with the linear stabilization technique we arrive at a novel stabilized sav scheme at each time step the scheme requires solving only four linear biharmonic equations with constant coefficients making it the first to the best of the author s knowledge totally decoupled second order accurate linear and unconditionally energy stable scheme for the model we further prove the unconditional energy stability rigorously and demonstrate the stability and the accuracy of the scheme numerically through the comparisons with the non stabilized sav scheme for simulating numerous benchmark numerical examples in 2d and 3d

a survey stability and boundedness of volterra difference equations abstract volterra difference equations arise in the modeling of many real phenomena in this survey stability and boundedness problems of some volterra nonlinear difference equations are investigated stability conditions and boundedness are formulated in terms of the characteristics of equations

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

efficient and stable exponential time differencing runge kutta methods for phase field elastic bending energy models the willmore flow formulated by phase field dynamics based on the elastic bending energy model has been widely used to describe the shape transformation of biological lipid vesicles in this paper we develop and investigate some efficient and stable numerical methods for simulating the unconstrained phase field willmore dynamics and the phase field willmore dynamics with fixed volume and surface area constraints the proposed methods can be high order accurate and are completely explicit in nature by combining exponential time differencing runge kutta approximations for time integration with spectral discretizations for spatial operators on regular meshes we also incorporate novel linear operator splitting techniques into the numerical schemes to improve the discrete energy stability in order to avoid extra numerical instability brought by use of large penalty parameters in solving the constrained phase field willmore dynamics problem a modified augmented lagrange multiplier approach is proposed and adopted various numerical experiments are performed to demonstrate accuracy and stability of the proposed methods

numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach summary we present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model which is derived from the variation of a free energy functional consisting of a temperature dependent bulk potential and a conformational entropy with a gradient dependent anisotropic coefficient we introduce a novel invariant energy quadratization approach to transform the free energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations based on the reformulated equivalent governing system we develop a first and a second order semi discretized scheme in time for the system in which all nonlinear terms are treated semi explicitly the resulting semi discretized equations consist of a linear elliptic equation system at each time step where the coefficient matrix operator is positive definite and thus the semi discrete system can be solved efficiently we further prove that the proposed schemes are unconditionally energy stable convergence test together with 2d and 3d numerical simulations for dendritic crystal growth are presented after the semi discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes copyright 2016 john wiley sons ltd

a new lagrange multiplier approach for gradient flows abstract we propose a new lagrange multiplier approach to design unconditional energy stable schemes for gradient flows the new approach leads to unconditionally energy stable schemes that are as accurate and efficient as the recently proposed sav approach shen xu and yang 2018 but enjoys two additional advantages i schemes based on the new approach dissipate the original energy as opposed to a modified energy in the recently proposed sav approach shen xu and yang 2018 and ii they do not require the nonlinear part of the free energy to be bounded from below as is required in the sav approach the price we pay for these advantages is that a nonlinear algebraic equation has to be solved to determine the lagrange multiplier we present ample numerical results to validate the new approach and as a particular example of challenging applications we consider a block copolymer bcp coupled cahn hilliard model and carry out new and nontrivial simulations which are consistent with experiment results

numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method abstract the molecular beam epitaxial model is derived from the variation of a free energy that consists of either a fourth order ginzburg landau double well potential or a nonlinear logarithmic potential in terms of the gradient of a height function one challenge in solving the mbe model numerically is how to develop proper temporal discretization for the nonlinear terms in order to preserve energy stability at the time discrete level in this paper we resolve this issue by developing a first and second order time stepping scheme based on the invariant energy quadratization ieq method the novelty is that all nonlinear terms are treated semi explicitly and the resulted semi discrete equations form a linear system at each time step moreover the linear operator is symmetric positive definite and thus can be solved efficiently we then prove that all proposed schemes are unconditionally energy stable the semi discrete schemes are further discretized in space using finite difference methods and implemented on gpus for high performance computing various 2d and 3d numerical examples are presented to demonstrate stability and accuracy of the proposed schemes

an energy stable linear diffusive crank nicolson scheme for the cahn hilliard gradient flow abstract we propose and analyze a linearly stabilized semi implicit diffusive crank nicolson scheme for the cahn hilliard gradient flow in this scheme the nonlinear bulk force is treated explicitly with two second order stabilization terms this treatment leads to linear elliptic system with constant coefficients and provable discrete energy dissipation rigorous error analysis is carried out for the fully discrete scheme when the time step size and the space step size are small enough second order accuracy in time is obtained with a prefactor controlled by some lower degree polynomial of 1 e here e is the thickness of the interface numerical results together with an adaptive time stepping are presented to verify the accuracy and efficiency of the proposed scheme

solution of difference equations by using differential transform method abstract this work presents semi numerical analytical solutions to both linear and nonlinear difference equations with variable coefficients by applying differential transform method dtm and extending it with the related theorems and their proofs to make clear and illustrate the features and capabilities of the presented method examples are carried out and numerical results with comparison to exact solutions and other methods are given

a phase field model for rate independent crack propagation robust algorithmic implementation based on operator splits the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field following our recent work c miehe f welschinger m hofacker thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations international journal for numerical methods in engineering doi 10 1002 nme 2861 on phase field type fracture we propose in this paper a new variational framework for rate independent diffusive fracture that bases on the introduction of a local history field it contains a maximum reference energy obtained in the deformation history which may be considered as a measure for the maximum tensile strain obtained in history it is shown that this local variable drives the evolution of the crack phase field the introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology in particular it allows for the construction of a new algorithmic treatment of diffusive fracture here we propose an extremely robust operator split scheme that successively updates in a typical time step the history field the crack phase field and finally the displacement field a regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added the proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids we demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples

the global dynamics of discrete semilinear parabolic equations a class of scalar semilinear parabolic equations possessing absorbing sets a lyapunov functional and a global attractor are considered the gradient structure of the problem implies that provided all steady states are isolated solutions approach a steady state as t to infty the dynamical properties of various finite difference and finite element schemes for the equations are analysed the existence of absorbing sets bounded independently of the mesh size is proved for the numerical methods discrete lyapunov functions are constructed to show that under appropriate conditions on the mesh parameters numerical orbits approach steady state solutions as discrete time increases however it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented furthermore it is also shown that the explicit euler scheme introduces spurious solutions with period 2 in the timestep as a result the absorbing set is destroyed and there is initial data leading to blow up of the scheme however small the mesh parameters are taken to obtain stabilization to a steady state for this scheme it is necessary to restrict the timestep in terms of the initial data and the space step implicit schemes are constructed for which absorbing sets and lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step both one step and multistep bdf methods are studied

an application of the ltdrm to transient diffusion problems with nonlinear material properties and nonlinear boundary conditions in this paper we shall present an application of the laplace transform dual reciprocity method ltdrm to the solution of transient diffusion problems with nonlinear material properties and nonlinear boundary conditions two methods with and without utilizing the kirchhoff transformation will be discussed the nonlinear terms in both governing equation and boundary conditions are linearized first followed by the removal of the time variable from the linearized differential system with the laplace transformation and then the problem is solved by the dual reciprocity method as a result a solution at any specific time can be obtained without step by step calculations in time domain and computations of domain integrals to show the efficiency and accuracy of the present method several problems are analyzed and their solutions are compared with those obtained either analytically or numerically

energy decaying extrapolated rk sav methods for the allen cahn and cahn hilliard equations we construct and analyze a class of extrapolated and linearized runge kutta rk methods which can be of arbitrarily high order for the time discretization of the allen cahn and cahn hilliard

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

analytical parametric analysis of the contact problem of human buttocks and negative poisson s ratio foam cushions abstract analytical investigations on the contact problems between two homogeneous and isotropic soft bodies were performed to simulate the contact of human buttocks and seat cushions the cushion materials poisson s ratio were allowed to be negative the human buttocks were modeled as an ideal sphere with radius 15 cm and assumed to have a low young s modulus and a poisson s ratio close to 0 5 these parameters were held constant during our analysis peak contact pressure was reduced by adjusting the contour curvature of cushions according to hertz theory as expected moreover analysis by both the hertz model and a finite thickness 3d elasticity model showed that using negative poisson s ratio cushions could further reduce the pressure negative poisson s ratio cushions may be beneficial in the prevention of pressure sores or ulcers in the sick and in reduction of pressure induced discomfort in seated people

a beale kato majda blow up criterion for a hydrodynamic system modeling vesicle and fluid interactions abstract in this paper we establish an analog of the beale kato majda type criterion for singularities of smooth solutions of a hydrodynamic system modeling vesicle and fluid interactions the result shows that the maximum norm of the vorticity alone controls the breakdown of smooth solutions

nonexistence of unbounded nonoscillatory solutions of partial difference equations abstract we develop criteria for the nonexistence of eventually positive negative and nondecreasing nonincreasing solutions of the partial difference equation m n y m n p m n y m k n l q m n y m k n l and m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

efficient and accurate numerical schemes for a hydro dynamically coupled phase field diblock copolymer model in this paper we consider numerical approximations of a hydro dynamically coupled phase field diblock copolymer model in which the free energy contains a kinetic potential a gradient entropy a ginzburglandau double well potential and a long range nonlocal type potential we develop a set of second order time marching schemes for this system using the invariant energy quadratization approach for the double well potential the projection method for the navierstokes equation and a subtle implicit explicit treatment for the stress and convective term the resulting schemes are linear and lead to symmetric positive definite systems at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

linear second order and unconditionally energy stable schemes for the viscous cahn hilliard equation with hyperbolic relaxation using the invariant energy quadratization method abstract in this paper we consider numerical approximations for the viscous cahn hilliard equation with hyperbolic relaxation this type of equations processes energy dissipative structure the main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms allowing large time marching step while preserving the energy stability i e the energy dissipative structure at the time discrete level we resolve this issue by developing two second order time marching schemes using the recently developed invariant energy quadratization approach where all nonlinear terms are discretized semi explicitly in each time step one only needs to solve a symmetric positive definite spd linear system all the proposed schemes are rigorously proven to be unconditionally energy stable and the second order convergence in time has been verified by time step refinement tests numerically various 2d and 3d numerical simulations are presented to demonstrate the stability accuracy and efficiency of the proposed schemes

linearly first and second order unconditionally energy stable schemes for the phase field crystal model in this paper we develop a series of linear unconditionally energy stable numerical schemes for solving the classical phase field crystal model the temporal discretizations are based on the first order euler method the second order backward differentiation formulas bdf2 and the second order crank nicolson method respectively the schemes lead to linear elliptic equations to be solved at each time step and the induced linear systems are symmetric positive definite we prove that all three schemes are unconditionally energy stable rigorously various classical numerical experiments in 2d and 3d are performed to validate the accuracy and efficiency of the proposed schemes

on linear schemes for a cahn hilliard diffuse interface model numerical schemes to approximate the cahn hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems in this work we propose two type of linear schemes based on different ways to approximate the double well potential term the first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy stable linear schemes for a modified energy we present first and second order in time linear schemes to approximate the ch problem detailing their advantages over other linear schemes that have been previously introduced in the literature furthermore we compare all the schemes through several computational experiments

solutions of integral and integro differential equation systems by using differential transform method in this study differential transform method dtm is applied to both integro differential and integral equation systems the method is further expanded with a formulation to treat fredholm integrals if the system considered has a solution in terms of the series expansion of known functions this powerful method catches the exact solution so as to show this capability and robustness some systems of integral and integro differential equations are solved as numerical examples

error analysis of full discrete invariant energy quadratization schemes for the cahn hilliard type equation abstract in this paper we present the error analysis for a fully discrete scheme of the cahn hilliard type equation along with numerical verifications the numerical schedule is developed by first transforming the cahn hilliard type equation into an equivalent form using the invariant energy quadratization ieq technique then the equivalent form is discretized by using the linear implicit crank nicolson method for the time variable and the fourier pseudo spectral method for the spatial variables the resulted full discrete scheme is linear and unconditionally energy stable which makes it easy to implement by constructing an appropriate interpolation equation the uniform boundedness of the numerical solution is obtained theoretically then we prove that the numerical solutions converge with the order o t 2 h m where t is the temporal step and h is the spatial step with m the regularity of the exact solution several numerical examples are presented to confirm the theoretical results and demonstrate the effectiveness of the presented linear scheme the numerical strategies and analytical tools in this paper could be readily applied to study other phase field models or gradient flow problems

an efficient computational approach to solving singular initial value problems for lane emden type equations in the paper we present the new computational approach for solving singular initial problems for the lane emden type equations with nonlinear terms based on the differential transformation method dtm and the modified general formula for the adomian polynomials with differential transformation components to illustrate the capability and efficiency of the method three examples are given the results obtained in this work are also compared with numerical results of heydari et al 2012 wazwaz 2005 singh et al 2009 and the standard dtm as well these results show that the proposed technique without linearization or small perturbation is very effective and convenient for solving of linear and nonlinear differential equations the general differential transformation formula is proved the general formula is presented for any analytic nonlinear terms no discretization linearization or perturbation is required nonlinear differential equations are solved using the proposed formula

linear first and second order unconditionally energy stable numerical schemes for the phase field model of homopolymer blends abstract in this paper we develop a series of efficient numerical schemes to solve the phase field model for homopolymer blends the governing system is derived from the energetic variational approach of a total free energy that consists of a nonlinear logarithmic flory huggins potential and a gradient entropy with a concentration dependent de gennes type coefficient the main challenging issue to solve this kind of models numerically is about the time marching problem i e how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level we solve this issue in this paper by developing the first and second order temporal approximation schemes based on the invariant energy quadratization method where all nonlinear terms are treated semi explicitly consequently the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step the unconditional energy stabilities are further proved various numerical simulations of 2d and 3d are presented to demonstrate the stability and the accuracy of the proposed schemes

convergence and error analysis for the scalar auxiliary variable sav schemes to gradient flows we carry out convergence and error analysis of the scalar auxiliary variable sav methods for l 2 and h 1 gradient flows with a typical form of free energy we first derive h 2 bounds under certain assumptions suitable for both the gradient flows and the sav schemes which allow us to establish the convergence of the sav schemes under mild conditions we then derive error estimates with further regularity assumptions we also discuss several other gradient flows which cannot be cast in the general framework used in this paper but for which convergence and error analysis can still be established using a similar procedure

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

numerical comparison of methods for solving a special fourth order boundary value problem in this paper we implement relatively new analytical techniques the differential transform method adomian decomposition and homotopy perturbation method for solving a fourth order boundary value problem in these schemes the solution takes the form of a convergent series with easily computable components this paper will present a numerical comparison between the three methods for solving a fourth order boundary value problem the numerical results demonstrates that the new methods are quite accurate and efficient for this kind of problems

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

the stress analysis of thin contact layers a singular perturbation method for integral equations in this paper a contact problem is considered with a circular indenter pressed normally against a semi infinite elastic composite that consists of a contact layer with uniform thickness welded tog

a phase field approach in the numerical study of the elastic bending energy for vesicle membranes in this paper we compute the equilibrium configurations of a vesicle membrane under elastic bending energy with prescribed volume and surface area a variational phase field method is developed for such a problem discrete finite difference approximations and numerical simulations are carried out in the axial symmetrical cases different energetic bifurcation phenomena are discussed

a mathematical approximation for the solution of a static indentation test abstract the classical contact problem of the indentation of a thin compressible linear elastic layer bonded to a rigid substrate is considered closed form mathematical approximations of the deformation are presented for the cases of plane indentation by a rectangular block and three dimensional indentation by a plane ended axisymmetric cyclinder the approximations are analyzed in the context of a static indentation test by comparison of applied load values to those obtained using a classical integral transform solution in the case of plane indentation the mathematical and classical predictions agree to within 2 relative error for aspect ratios between 0 1 and 1 0 and apparent poisson ratio between 0 0 and 0 3 comparisons for the axisymmetric case indicate a similar pattern the main advantage of the new approach is that it yields closed form approximations of the static indentation solution which can also capture the essential singular behavior