an approximation method to circular arcs abstract modern cad cam systems are not able to dispose the circles represented by the implicit equations or the parametric equations with trigonometric function parametric rational polynomials are used to approximate the circular arcs a new approximation method by means of quartic bezier curves is presented to effectively approximate the circular arcs it is shown that our method provides the approximation of order eight and yields a smaller error than previous methods by equally spaced subdivision of circular arcs the curvature continuous spline approximation algorithm of the circular arcs is given

the valuation of foreign currency options under stochastic interest rates abstract we consider the valuation of options written on a foreign currency when interest rates are stochastic and the matrix of the diffusion representing the global economy is strongly coercive we solve the associated variational inequality for the value function numerically by the finite element method in the european case a comparison is made to the exact solution the corresponding result for the american option is also presented

strongly polynomial algorithms for the high multiplicity scheduling problem a high multiplicity scheduling problem consists of many jobs which can be partitioned into relatively few groups where all the jobs within each group are identical polynomial and even strongly polynomial algorithms for the standard scheduling problem in which all jobs are assumed to be distinct become exponential for the corresponding high multiplicity problem in this paper we study various high multiplicity problems of scheduling unit time jobs on a single machine we provide strongly polynomial algorithms for constructing optimal schedules with respect to several measures of efficiency completion time lateness tardiness the number of tardy jobs and their weighted counterparts the algorithms require a number of operations that are polynomial in the number of groups rather than in the total number of jobs as a by product we identify a new family of nxn transportation problems which are solvable in o n log n time by a simple greedy algorithm

far field boundary conditions for black scholes equations in this paper we consider the partial differential equations approach for valuing european style options in order to solve the equations numerically by finite difference or finite element methods it is necessary to introduce an artificial boundary in order to make the computational domain bounded we derive pointwise bounds for the error caused by various boundary conditions imposed on the artificial boundary the error estimates can be used to determine a priori a suitable location for the artificial boundary in terms of a given error tolerance

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

an analytic study of fisher s equation by using adomian decomposition method in this paper we develop a framework to obtain exact solutions to fisher s equation and to a nonlinear diffusion equation of the fisher type by employing adomian decomposition method the proposed scheme is supported by examining nonlinear diffusion equations of the fisher type

two boundedness and monotonicity preserving methods for a generalized fisher kpp equation two finite difference methods are proposed for a generalized fisher kpp equation these methods are capable of preserving the skew symmetry of the equation these methods preserve the positivity boundedness and monotonicity of solutions these methods are solvable these methods are convergent a semi explicit finite difference method and an implicit finite difference method are proposed for a generalized fisher kpp equation with two space variables it is proved that these two methods could preserve the skew symmetry of this equation moreover under a condition on the step sizes the semi explicit method is capable of preserving the positivity the boundedness and the spatial and temporal monotonicity of initial approximations and the implicit method is able to preserve these properties with no restriction on the step sizes the stability and convergence of these two methods are also analyzed respectively finally some numerical simulations are provided to verify the validity of our analytical results

a front dynamics approach to curvature dependent flow a front dynamics approach is developed to study the evolution of planar curves whose normal speed depends on curvature the formulation is similar to whitham s shock dynamics theory for the propagation of shock wave in gases but assumes a different propagation rule equations that describe the motion of the front are obtained and these are evolution equations for the normal direction and local arc length of the front the solution of these equations leads to the front positions using an appropriate integration along rays a similarity solution of the equations is found for the evolution of an initial corner free boundary problems for the motion of a junction connecting front segments are discussed a numerical method is presented to calculate the evolution of any number of front segments the segments can be closed or open connected to wall boundaries or not or connected to other segments at 3 segment junctions several sample problems are considered to illustrate the method an extension of the metho

nonplanar traveling fronts in reaction diffusion equations with combustion and degenerate fisher kpp nonlinearities abstract this paper is concerned with nonplanar traveling fronts in reaction diffusion equations with combustion nonlinearity and degenerate fisher kpp nonlinearity our study contains two parts in the first part we establish the existence of traveling fronts of pyramidal shape in r 3 and in the second part we establish the existence and stability of v shaped traveling fronts in r 2

a new method for evaluating options based on multiquadric rbf fd method abstract in this paper a new local meshless approach based on radial basis functions rbfs is presented to price the options under the black scholes model the global rbf approximations derived from the conventional global collocation method usually lead to ill conditioned matrices employing the idea of local approximants of the finite difference fd method and combining it with the radial basis function rbf method can result in a local meshless approach such as rbf fd it removes the difficulty of ill conditionness of the original method the new proposed approach is unconditionally stable as it is shown by von neumann stability analysis it is fast and produces high accurate results as shown in numerical experiments moreover we took into account the variation of shape parameter and analyzed numerically the behavior of the rbf fd method

high order computational methods for option valuation under multifactor models abstract many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second order discretization schemes a higher order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high order convergence rates for the solution of multi factor option problems these new schemes employ galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two asset option problems and time integration of the resulting semi discrete systems requires the computation of a single matrix exponential the computations indicate that this combination of high order finite elements and exponential time integration leads to efficient algorithms for multi factor problems highly accurate european prices are obtained with relatively coarse meshes and high order convergence rates are also observed for options with the american early exercise feature various numerical examples are provided for illustrating the accuracy of the option prices for heston s and bates stochastic volatility models and for two asset problems under merton s jump diffusion model

iterative process for g2 multi degree reduction of bezier curves 2 continuity abstract in this paper the issue of multi degree reduction of bezier curves with c 1 and g 2 continu ity at the end points of the curve is considered an iterative method which is the first of this type is derived it is shown that this algorithm converges and can be applied itera tively to get the required accuracy some examples and figures are given to demonstrate the efficiency of this method

multi asset black scholes model as a variable second class constrained dynamical system abstract in this paper we study the multi asset black scholes model from a structural point of view for this we interpret the multi asset black scholes equation as a multidimensional schrodinger one particle equation the analysis of the classical hamiltonian and lagrangian mechanics associated with this quantum model implies that in this system the canonical momentums cannot always be written in terms of the velocities this feature is a typical characteristic of the constrained system that appears in the high energy physics to study this model in the proper form one must apply dirac s method for constrained systems the results of the dirac s analysis indicate that in the correlation parameters space of the multi assets model there exists a surface called the kummer surface k where the determinant of the correlation matrix is null on which the constraint number can vary we study in detail the cases with n 2 and n 3 assets for these cases we calculate the propagator of the multi asset black scholes equation and show that inside the kummer k surface the propagator is well defined but outside k the propagator diverges and the option price is not well defined on k the propagator is obtained as a constrained path integral and their form depends on which region of the kummer surface the correlation parameters lie thus the multi asset black scholes model is an example of a variable constrained dynamical system and it is a new and beautiful property that had not been previously observed

an exact solution of a quasilinear fisher equation in cylindrical coordinates lie symmetry method is applied to analyse fisher equation in cylindrical coordinates symmetry algebra is found and symmetry invariance is used to reduce the equation to a first order ode the first order ode is further analysed to obtain exact solution of fisher equation in explicit form

on traveling wave solutions of fisher s equation in two spatial dimensions it is shown for the quadratic fisher equation in two spatial dimensions that along with a plane wave there exist several other traveling waves with nontrivial front geometry some of the solutions are found in explicit form others are constructed approximately the dispersion relationship and velocity curvature dependence generated by these solutions are studied

on the use of boundary conditions for variational formulations arising in financial mathematics the general intractability of derivative security valuation models to present techniques both analytic and numerical arguably remains one of the preeminant problem of mathematical finance it is the focus of this paper to examine the applicability of a promising recent development namely radial basis functions rbf to the problem of option valuation a black scholes framework is considered for american and european options written on a one and two risky assets the performance of rbf and finite differencing algorithms are examined with respect to artificial boundary conditions computational domain domain decomposition and mesh scaling

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the effects of obstacle size on periodic travelling waves in oscillatory reaction diffusion equations many natural populations undergo multi year cycles and field studies have shown that these can be organized into periodic travelling waves ptws mathematical studies have shown that large scale landscape obstacles represent a natural mechanism for wave generation here we investigate how the amplitude and wavelength of the selected waves depend on the obstacle size we firstly consider a large circular obstacle in an infinite domain for a reaction diffusion system of l u type we use perturbation theory to derive a leading order approximation to the wave generated by the obstacle this shows the dependence of the wave properties on both parameter values and obstacle size we find that the limiting values of the amplitude and wavelength are approached algebraically with distance from the obstacle edge rather than exponentially in the case of a flat boundary we use our results to predict the properties of waves generated by a large circular obstacle for an oscillatory predator prey system via a reduction of the predator prey model to normal form close to hopf bifurcation our predictions compare well with numerical simulations we also discuss the implications of these results for wave stability and briefly investigate the effects of obstacles with elliptical geometries

polynomiality for bin packing with a constant number of item types we consider the bin packing problem with d different item sizes si and item multiplicities ai where all numbers are given in binary encoding this problem formulation is also known as the one dime

a note on iterative process for g2 multi degree reduction of bezier curves abstract in the paper a rababah s mann iterative process for g 2 multi degree reduction of bezier curves applied mathematics and computation 217 2011 8126 8133 rababah and mann proposed an iterative method for multi degree reduction of bezier curves with c 1 and g 2 continuity at the endpoints in this paper we provide a theoretical proof for the existence of the unique solution in the first step of the iterative process while the proof in their paper applies only in some special cases also we give a complete convergence proof for the iterative method we solve the problem by using convex quadratic optimization

an explicit positivity preserving finite difference scheme for the classical fisher kolmogorov petrovsky piscounov equation abstract in this work we introduce a family of explicit finite difference schemes to approximate positive solutions of the classical fisher kolmogorov petrovsky piscounov equation from population dynamics a neumann stability analysis reveals that our methods are indeed stable under certain choices of the model and numerical parameters and numerical comparisons with analytical solutions are carried out in some detail

pricing real estate index options by compactly supported radial polynomial basis point interpolation abstract this paper presents a novel method to price the real estate index options which are modeled based on the framework proposed by fabozzi et al 2012 the cs pbf method combines compactly supported radial basis functions csrbf and polynomial basis functions pbf to yield the interpolation functions which can guarantee interpolation shape functions with kronecker property and overcome possible singularity associated with the pbf method compared with the csrbf method and the finite difference fd method the cs pbf method is more accurate and efficient for the real estate index option meanwhile a local mesh refinement technique is employed for dealing with the non smooth options payoffs which is very effective and stable to improve the computational accuracy for the cs pbf method finally the cs pbf method is extended to price american option of the real estate index

radial basis function partition of unity methods for pricing vanilla basket options meshfree methods based on radial basis function rbf approximation are becoming widely used for solving pde problems they are flexible with respect to the problem geometry and highly accurate a disadvantage of these methods is that the linear system to be solved becomes dense for globally supported rbfs a remedy is to introduce localisation techniques such as partition of unity rbf partition of unity methods rbf pum allow for a significant sparsification of the linear system and lower the computational effort in this work we apply a global rbf method as well as rbf pum to problems in option pricing we consider one and two dimensional vanilla options in order to price american options we employ a penalty approach a penalty term suitable for american multi asset call options has been designed rbf pum is shown to be competitive compared with a finite difference method and a global rbf method it is as accurate as the global rbf method but significantly faster the results for rbf pum look promising for extension to higher dimensional problems

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

best polynomial degree reduction on q lattices with applications to q orthogonal polynomials we study the best polynomial degree reduction with respect to the q l2 norm we study a finite analogue with respect to finite q lattices we present applications to q orthogonal polynomials we show that a weighted least squares approximation of q bezier coefficients provides the best polynomial degree reduction in the q l2 norm we also provide a finite analogue of this result with respect to finite q lattices and we present applications of these results to q orthogonal polynomials

finite difference approximations of multidimensional unsteady convection diffusion reaction equations in this paper the numerical approximation of unsteady convection diffusion reaction equations with finite difference method on a special grid is studied in the convection or reaction dominated regime we extend the method 19 which was designed for multidimensional steady convection diffusion reaction equations to unsteady problems we investigate two possible different ways of combining the discretization in time and in space where the sequence of the discretizations is interchanged discretization in time is performed by using crank nicolson and backward euler finite difference schemes while for the space discretization we consider the method 19 numerical tests are presented to show good performance of the method

bifurcation and spatiotemporal patterns in a homogeneous diffusive predator prey system abstract a diffusive predator prey system with holling type ii predator functional response subject to neumann boundary conditions is considered hopf and steady state bifurcation analysis are carried out in details in particular we show the existence of multiple spatially non homogeneous periodic orbits while the system parameters are all spatially homogeneous our results and global bifurcation theory also suggest the existence of loops of spatially non homogeneous periodic orbits and steady state solutions these results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion abstract we consider a simplified chemotaxis model of tumor angiogenesis described by a keller segel system on the two dimensional infinite cylindrical domain x y r s where s is the circle of perimeter 0 the domain models a virtual channel where newly generated blood vessels toward the vascular endothelial growth factor will be located the system is known to allow planar traveling wave solutions of an invading type in this paper we establish the nonlinear stability of these traveling invading waves when chemical diffusion is present if is sufficiently small the same result for the corresponding system in one dimension was obtained by li li wang 2014 17 our result solves the problem remained open in chae choi kang lee 2018 3 at which only linear stability of the planar traveling waves was obtained under certain artificial assumption

a pseudospectral method of solution of fisher s equation in this paper we develop an accurate and efficient pseudospectral solution of fisher s equation a prototypical reaction diffusion equation the solutions of fisher s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient there is an ongoing effort to better adapt pseudospectral methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations the collocation method employed is based on chebyshev gauss lobatto quadrature points we compare results for a single domain as well as for a subdivision of the main domain into subintervals instabilities that occur in the numerical solution for a single domain analogous to those found by others are attributed to round off errors arising from numerical features of the discrete second derivative matrix operator however accurate stable solutions of fisher s equation are obtained with a multidomain pseudospectral method a detailed comparison of the present approach with the use of the sine interpolation is also carried out

on the bin packing problem with a fixed number of object weights abstract we consider a bin packing problem where the number of different object weights is fixed to c we analyze a simple approximate approach and show that it leads to an asymptotically exact polynomial algorithm with absolute error 0 if c 2 at most 1 if 1 6 a consequence of our analysis is a new upper bound on the gap between the optimal value of the problem at hand and the round up of the optimal value of the linear relaxation of its gilmore gomory formulation

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

finite difference methods for pricing american put option with rationality parameter numerical analysis and computing abstract in this paper finite difference methods for pricing american option with rationality parameter are proposed the irrational exercise policy arising in american options is characterized in terms of a rationality parameter the model is formulated in terms of a new nonlinear black scholes equation that requires specific numerical methods although the solution converges to the solution of the classical american option price when the parameter tends to infinity for finite values of the parameter the classical boundary conditions cannot apply and we propose specific ones a logarithmic transformation is used to improve properties of the numerical solution that is constructed by explicit finite difference method numerical analysis provides stability conditions for the methods and its positivity properties of intensity function are studied from the point of view of numerical solution concerning the numerical methods for the original problem we propose the method for time discretization thus including explicit fully implicit and crank nicolson schemes as particular cases the nonlinear term is treated by a newton method the convergence rate is illustrated by numerical examples

haar wavelet method for solving fisher s equation in this paper we develop an accurate and efficient haar wavelet solution of fisher s equation a prototypical reaction diffusion equation the solutions of fisher s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient there is an ongoing effort to better adapt haar wavelet methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations moreover the use of haar wavelets is found to be accurate simple fast flexible convenient small computation costs and computationally attractive

an rbf fd sparse scheme to simulate high dimensional black scholes partial differential equations abstract the objective of this research is to investigate the numerical solution of high dimensional black scholes partial differential equations pdes to do this the weights of the radial basis function finite difference rbf fd scheme using inverse multi quadric function are used with an emphasis on the hotzone the proposed rbf fd technique is capable to price multi asset options efficiently

radial basis functions method for valuing options from the view point of probability this study presents a theoretical framework to show the convergence of the rbfs method for valuing options it will be proved to be equivalent to a multinomial tree approach in which the underlying variable can move from its initial value to an infinity of possible values of the next time step specially the probability of a move in a short period time follows the normal distribution when using the gaussian basis kernel it is a precise simulation of the behavior of the underlying variable which provides a more reasonable explanation of high accuracy of the rbfs method this helps open a new area of research in developing the expected numerical method for derivative securities in which the underlying asset follows other stochastic process by using corresponding radial basis kernel the paper also illustrates the approach by using it to value stock options and its greek letters

singularity structure symmetries and integrability of generalized fisher type nonlinear diffusion equation in this letter the integrability aspects of a generalized fisher type equation with modified diffusion in 1 1 and 2 1 dimensions are studied by carrying out a singularity structure and symmetry analysis it is shown that the painleve property exists only for a special choice of the parameter m 2 a backlund transformation is shown to give rise to the linearizing transformation to the linear heat equation for this case m 2 a lie symmetry analysis also picks out the same case m 2 as the only system among this class having a nontrivial infinite dimensional lie algebra of symmetries and that the similarity variables and similarity reductions lead in a natural way to the linearizing transformation and physically important classes of solutions including known ones in the literature thereby giving a group theoretical understanding of the system for nonintegrable cases in 2 1 dimensions associated lie symmetries and similarity reductions are indicated

penalty methods for the numerical solution of american multi asset option problems we derive and analyze a penalty method for solving american multi asset option problems a small non linear penalty term is added to the black scholes equation this approach gives a fixed solution domain removing the free and moving boundary imposed by the early exercise feature of the contract explicit implicit and semi implicit finite difference schemes are derived and in the case of independent assets we prove that the approximate option prices satisfy some basic properties of the american option problem several numerical experiments are carried out in order to investigate the performance of the schemes we give examples indicating that our results are sharp finally the experiments indicate that in the case of correlated underlying assets the same properties are valid as in the independent case

explicit g2 constrained degree reduction of bezier curves by quadratic optimization in this paper we revisit g 2 constrained degree reduction of bezier curves which has been solved in our previous work by using iterative methods we propose an explicit and effective method for g 1 constrained degree reduction and c 1g 2 constrained degree reduction our main idea is to express the distance function defined in the l 2 norm as a strictly convex quadratic function of two variables which becomes a quadratic optimization problem we can explicitly obtain the unique solution by solving two linear equations such that the distance function is minimized the existence of the unique solution is also proved

on the approximation of optimal stopping problems with application to financial mathematics the determination of the value function associated with a given reward and stochastic process represents an important class of stochastic control problem in particular the expectations of such processes may be represented as solutions of variational inequalities of evolutionary type typically characterized by their high number of degrees of freedom unbounded domains and lack of natural boundary conditions in this paper we introduce two methodologies for computing the value function of optimal stopping associated with general stochastic processes our results are implemented utilizing finite elements and are validated using problems taken from financial mathematics

stability of planar traveling waves in a keller segel equation on an infinite strip domain abstract we consider a simplified model of tumor angiogenesis described by a keller segel equation on the two dimensional domain x y r s where s is the circle of perimeter it is known that the system allows planar traveling wave solutions of an invading type in case that is sufficiently small we establish the nonlinear stability of traveling wave solutions in the absence of chemical diffusion if the initial perturbation is sufficiently small in some weighted sobolev space when chemical diffusion is present it can be shown that the system is linearly stable lastly we prove that any solution with our front condition eventually becomes planar under certain regularity conditions

pricing european and american options by radial basis point interpolation a rbpi method for pricing european and american options is proposed rbpi has not yet been used for option pricing the overall efficiency is improved by coupling rbpi with other approaches three different algorithms for american options are tested and compared numerical results demonstrate the effectiveness of the techniques employed we propose the use of the meshfree radial basis point interpolation rbpi to solve the black scholes model for european and american options the rbpi meshfree method offers several advantages over the more conventional radial basis function approximation nevertheless it has never been applied to option pricing at least to the very best of our knowledge in this paper the rbpi is combined with several numerical techniques namely an exponential change of variables which allows us to approximate the option prices on their whole spatial domain a mesh refinement algorithm which turns out to be very suitable for dealing with the non smooth options payoff and an implicit euler richardson extrapolated scheme which provides a satisfactory level of time accuracy finally in order to solve the free boundary problem that arises in the case of american options three different approaches are used and compared the projected successive overrelaxation method psor the bermudan approximation and the penalty approach numerical experiments are presented which demonstrate the computational efficiency of the rbpi and the effectiveness of the various techniques employed

an analytical study for fisher type equations by using homotopy perturbation method in this paper homotopy perturbation method is applied to fisher type equations the solutions introduced in this study are in recursive sequence forms which can be used to obtain the closed form of the solutions if they are required the method is tested on various examples which are revealing the effectiveness and the simplicity of the method

a numerical analysis of variational valuation techniques for derivative securities in this paper we consider the partial differential equations approach for valuing european and american style options on multiple assets we use a method of lines finite element implementation available in the software package femlab in order to solve the variational inequality that characterizes the american style option as well as the partial differential equation that defines the european style option for two and three state variables a detailed study of the approximation error is provided including a theoretical estimate an asymptotic analysis the space time distribution and the dependence on the size of the truncation domain

numerical simulations of time dependent partial differential equations when a time dependent partial differential equation pde is discretized in space with a spectral approximation the result is a coupled system of ordinary differential equations odes in time this is the notion of the method of lines mol and the resulting set of odes is stiff the stiffness may be even exacerbated sometimes the linear terms are the primarily responsible for the stiffness with a rapid exponential decay of some modes as in a dissipative pde or a rapid oscillation of some modes as in a dispersive pde therefore for a time dependent pde which combines low order nonlinear terms with higher order linear terms it is desirable to use a higher order approximation both in space and in time along our research we have focused on a particular case of spectral methods the so called pseudo spectral methods to solve numerically time dependent pdes using different techniques an integrating factor in de la hoz and vadillo 2010 an exponential time differencing method in de la hoz and vadillo 2008 and differentiation matrices in the theoretical frame of matrix differential equations in de la hoz and vadillo 2012 2013a b this paper which is a unified review of those contributions aims at providing a better understanding of those methods by illustrating their variety and more importantly their power furthermore we also give emphasis to choosing adequate schemes to advance in time

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

conditional full stability of positivity preserving finite difference scheme for diffusion advection reaction models this work has been partially supported by the ministerio de economia y competitividad spanish grant mtm2017 89664 p

fisher equation for anisotropic diffusion simulating south american human dispersals the fisher equation is commonly used to model population dynamics this equation allows describing reaction diffusion processes considering both population growth and diffusion mechanism some results have been reported about modeling human dispersion always assuming isotropic diffusion nevertheless it is well known that dispersion depends not only on the characteristics of the habitats where individuals are but also on the properties of the places where they intend to move then isotropic approaches cannot adequately reproduce the evolution of the wave of advance of populations solutions to a fisher equation are difficult to obtain for complex geometries moreover when anisotropy has to be considered and so few studies have been conducted in this direction with this scope in mind we present in this paper a solution for a fisher equation introducing anisotropy we apply a finite difference method using the crank nicholson approximation and analyze the results as a function of the characteristic parameters finally this methodology is applied to model south american human dispersal

stabilized explicit runge kutta methods for multi asset american options american derivatives have become very popular instruments in financial markets however they are more complicated to price than european options since at each time level we have to determine not only the option value but also whether or not it should be exercised several procedures have been proposed to dissolve these difficulties but they usually involve the solution of nonlinear partial differential equations pdes in the case of multi dimensional problems solving these equations is a very challenging task in this paper we propose stabilized explicit runge kutta serk methods to solve this kind of problems they can easily be applied to many different classes of problems with large dimensions and they have low memory demand since these methods are explicit they do not require algebra routines to solve large nonlinear systems associated to odes as for example lapack and blas packages or multigrid or iterative methods applied together with newton type algorithms and are especially well suited for the method of lines mol discretizations of parabolic pdes

curved fronts in the belousov zhabotinskii reaction diffusion systems in r 2 in this paper we consider a diffusion system with the belousov zhabotinskii bz for short chemical reaction following brazhnik and tyson 4 and perez munuzuri et al 45 who predicted v shaped fronts theoretically and discovered v shaped fronts by experiments respectively we give a rigorous mathematical proof of their results we establish the existence of v shaped traveling fronts in r 2 by constructing a proper supersolution and a subsolution furthermore we establish the stability of the v shaped front in r 2 c 2018 elsevier inc all rights reserved

a penalty method for american options with jump diffusion processes the fair price for an american option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem we develop an implicit discretization method for pricing such american options the jump diffusion correlation integral term is computed using an iterative method coupled with an fft while the american constraint is imposed by using a penalty method we derive sufficient conditions for global convergence of the discrete penalized equations at each timestep finally we present numerical tests which illustrate such convergence