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Numerous environmental/engineering applications relying upon the theory of diffusion phenomena into chaotic environments have recently stimulated the interest toward the numerical solution of parabolic partial differential equations (PDEs). In the present paper, we outline a formulation of the mathematical problem underlying a quite general diffusion mechanism in the natural environments, and we shortly emphasize some remarks concerning the applicability of the (straightforward) finite difference method. An illustration example is also presented.

The simulation of viscous, compressible flows around complex geometries or similar applications often inherit the task of solving large, stiff systems of ODEs. Domain-based implicit-explicit (IMEX) type schemes offer the possibility to apply two different schemes to different parts of the computational domain. The goal hereby is to decrease the computational cost by increasing the admissible step sizes with no loss of stability and by reducing the system sizes of the linear solver within the implicit integrator. But which combination of methods reaches the largest gain in efficiency? Coupling of Runge-Kutta methods or different multistep methods has been investigated so far by other authors. Here, we inspect the adoption of the recently introduced exponential integrators called EPIRK and sEPIRK in the IMEX setting, since they are perfectly suited for large, stiff systems of ODEs.

The nonlinear partial differential equations have an important role in real life problems. To obtain the exact solutions of the nonlinear partial differential equations, a number of approximate methods are known in the literature. In this work, a time– space modulated nonlinearities of coupled Schrodinger equations are considered. We provide a large class of Jacobi-elliptic solutions via the auxiliary equation method with sixth order nonlinearity and the Chebyshev approximation.

Although every iterative scalar linear ordinary differential equation is of maximal symmetry algebra, the situation is different and far more complex for systems of linear ordinary differential equations, and an iterative system of linear equations need not be of maximal symmetry algebra. We illustrate these facts by examples and derive families of vector differential operators whose iterations are all linear systems of equations of maximal symmetry algebra. Some consequences of these results are also discussed.

Exponential propagation iterative methods of Runge-Kutta type (EPIRK) provide a flexible framework to derive efficient exponential integrators for different types of ODE systems. Different classes of EPIRK methods can be constructed depending on the properties of the equations to be solved. Both classically and stiffly accurate EPIRK schemes can be derived. Flexibility of the order conditions allows to optimize coefficients to construct more efficient schemes. Particularly well-performing fourth-order stiffly accurate methods have been derived and applied to a number of problems. A new efficient three-stage fourth order method is presented and tested here using numerical examples.

New class of solutions for nonlinear partial differential equations is introduced. We call them the concaten solutions. As an example we consider equations for the description of wave processes in the Fermi-Pasta-Ulam mass chain and construct the concatenon solutions for these equation. Stability of the concatenon-type solutions is investigated numerically. Interaction between the concatenon and solitons is discussed.

To investigate global uniformly asymptotically partial stability we propose a new approach based on construction of some functions other than the Lyapunov functions. Examples of the studies of stability with using the proposed approach are presented for solution of some system of ordinary nonlinear differential equations.

An analytical method for representation of solutions of nonlinear partial differential equations in the form of special series with recurrently computed coefficients is presented. The coefficients recurrent obtaining from linear differential equations is achieved by specificity of the considered equations. It turns out that due to the functional arbitrariness which possibly is contained in special series, one can prove global convergence of the constructed series to solution of considered nonlinear partial differential equations.

This paper proposes a fundamentally new method of investigation of a singularly perturbed Cauchy problem for a linear system of ordinary differential equations based on the spectral theory of equations with deviating argument.

The purpose of the talk is the presentation of some recent advances in the numerical solution of differential equations, with special emphasis to reaction-diffusion problems, Hamiltonian problems and ordinary differential equations with discontinuous right-hand side. As a special case, in this short paper we focus on the solution of reaction-diffusion problems by means of special purpose numerical methods particularly adapted to the problem: indeed, following a problem oriented approach, we propose a modified method of lines based on the employ of finite differences shaped on the qualitative behavior of the solutions. Constructive issues and a brief analysis are presented, together with some numerical experiments showing the effectiveness of the approach and a comparison with existing solvers.

This paper deals with the discussion of some approaches to analytical and numerical solutions of linear boundary value problems for Hamiltonian systems of ordinary differential equations. The discussed approaches are based on a general conception of reducing initial linear system to another, in a sense, simpler system.

There is a wide range of numerical methods for solving differential equations. Each method has advantages and disadvantages. Criteria to evaluate these methods include stability, highest degree, an extended stability region, simple structure, etc. Our numerical solution for ordinary differential equations, the second derivative hybrid method, is constructed from concrete methods of degree p ≤ 10. We illustrate our results with specific examples and compare the method constructed here with previously known methods.

The importance of ordinary differential equation and also systems of these equations in scientific world is a crystal-clear fact. Many problems in chemistry, physics, ecology, biology can be modeled by systems of ordinary differential equations. In solving these systems numerical methods are very important because most realistic systems of these equations do not have analytic solutions in applied sciences In this study, we apply Taylor collocation method and Adomian decomposition method to solve the systems of ordinary differential equations. In these both scheme, the solution takes the form of a convergent power series with easily computable components. So, we will be able to make a comparison between Adomian decomposition and Taylor collocation methods after getting these power series.

In this research, an modifiedsecond derivative multistep methods are constructed to solve ordinary differential equations (ODE). The idea of modification is based on central finite difference in order to get a new coefficient of the method. Numerical results are presented to illustrate the accuracy and efficiency of the modified method.

One of the most perspective ways to construct solutions of nonlinear partial differential equations is combination of numerical and analytical methods. In this paper analytical method of special series is used. Essence of this method is in expansion of solutions into a series by the powers of one or several special functions provided the coefficients of the series to be computed recurrently. In contrast to Fourier method, which also used for solving nonlinear partial differential equations, method of special series allows to obtain solutions with preliminary assigned accuracy because the developed approach leads to a sequence of finite systems of ordinary differential equations. The obtained sequence turns out to be linear even for nonlinear initial equation. This fact allows to obtain new results: in a number of cases it is possible to prove global convergence of constructed series including for unbounded domains, where numerical methods occur principal difficulties. This approach allows to obtain constr...

In this paper, a fourth-order explicit hybrid method with constant step-size is developed. The free parameters are chosen so that the error constant is small. Numerical comparisons are carried out to show the advantage of the new method for solving special second order ordinary differential equations.

The numerical solution of an initial value problem for nonlinear fractional differential equations is discussed. Using an integral equation reformulation of the initial value problem and graded grids, the attainable order of convergence of proposed algorithms is studied, theoretically and numerically.

Nowadays, the Differential Transformation Method (DTM) is widely used in ordinary differential equations, partial differential equations and integral equations. Basically, the DTM is a numerical method based on the Taylor series expansion. Via this method, the analytical solutions are constructed in the form of a polynomial. Instead of evaluating the derivatives symbolically, this method calculates the relative derivatives by an iteration procedure describe by the transformed equations. Previous studies show that the differential transformation method is a powerful method having high accuracy to solve lower order boundary value problem and higher order boundary value problems for linear and nonlinear differential equations. In the present study, we solve system of linear differential equations by using DTM. To illustrate the accuracy of the method we provide several numerical examples and compare the results with their exact solutions.

In this research, two modified predictor-corrector methods using arithmetic mean are constructed to solve ordinary differential equations (ODE). The third order predictor-corrector methods, namely, Adams-Bashforth and Adams-Moulton have been chosen and the modification has successfully produced two new methods. These two third order methods are used to solve ODE problems. Simulation output from both methods are compared with the original methods and third order Runge-Kutta method. Numerical results are presented to illustrate the accuracy and efficiency of the modified methods.

A two-stage explicit two-step Runge-Kutta-Nystrom (TSRKN) method is constructed for the numerical integration of special second-order IVPs. Algebraic order conditions of the method are obtained and third-order method is derived. A standard set of test problems are tested and comparisons on the numerical results are made with existing Runge-Kutta-Nystrom (RKN) and Runge-Kutta (RK) methods of the same order using constant step size. The numerical results show's its clear advantage in terms of function evaluation.

Two-derivative Runge-Kutta (TDRK) methods are a special case of multi-derivative Runge-Kutta methods first studied by Kastlunger and Wanner [1, 2]. These methods incorporate derivatives of order higher than the first in their formulation but we consider only the first and second derivatives. In this paper we first present our study of both explicit [3] and implicit TDRK methods on stiff ODE problems. We then extend the applications of these TDRK methods to various partial differential equations [4]. In particular, we show how a 2-stage implicit TDRK method of order 4 and stage order 4 can be adapted to solve diffusion equations more efficiently than the popular Crank-Nicolson method.

The stability and almost coercive stability of first order difference scheme for the approximate solution of the initial-boundary value problem for ultra-parabolic equations are studied.

1st International Conference on Analysis and Applied Mathematics (ICAAM) -- OCT 18-21, 2012 -- Gumushane, TURKEY WOS: 000309524400054 A numerical method is proposed for solving hyperbolic-Schrodinger partial differential equations with nonlocal boundary condition. The first and second orders of accuracy difference schemes are presented. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional hyperbolic-Schrodinger partial differential equation. The method is illustrated by numerical examples. Sci & Technol Res Council Turkey (TUBITAK), Gumushane Univ, Fatih Univ

We introduce a family of Linear Multistep Methods used as Boundary Value Methods for the numerical solution of initial value problems for second order ordinary differential equations of special type. The aim is to obtain P-stable methods with arbitrary order of accuracy. This result allows to overcome the order barrier established by Lambert and Watson which limited to p = 2 the maximum order of a P-stable Linear Multistep Method. In addition, an extension of the methods in the Exponential Fitting framework is also considered.

In this paper, we studied to obtain numerical solutions of partial differential equations with variable coefficient by Sumudu Transform Method (STM). We applied to three examples this method. Then, we ploted 2D and 3D graphics of these equations by means of programming language Mathematica. In order to solve these differential equations which is ordinary and partial, the integral transforms such as the Laplace, Hankel and Fourier were unusually used and therefore there are several works on literature and applications of them. But it has few been given by using STM. And thus, in recent years STM has been an important places in applied sciences such as especially engineering, physics, physical sciences. Moreover, it is important for solving ordinary and partial differential equations which is linear and nonlinear.

We employ the Adomian decomposition method to solve a high order linear fuzzy initial value problem involving ordinary differential equations. The Adomian decomposition method can be used for solving 4th order fuzzy differential equations directly without reduction to first order system. We illustrate the method in numerical experiment and compare the result with linear k-step methods.

The boundary value problems for 2k-th order partial differential equations are investigated. Well-posedness of two boundary value problems for partial differential equations in the cases even and odd k is established. Note that solvability results of the present paper are dependent on the evenness and oddness of number k.

With the numerical solution of ordinary differential equations(ODE), scientists engaged in the Middle Ages, beginning with the work of Clairaut. The domain of the numerical methods involved in many famous mathematicians - Euler, Runge, Kutta, Adams, Laplace, and others. They have constructed methods with different properties. In this paper we consider the construction of numerical methods with high accuracy and to this end is proposed to use multi-step multi-derivative and hybrid methods. As well as specific methods are constructed with a certain accuracy.

For a linear system of singular ordinary differential equations, necessary and sufficient conditions are presented for the existence of a unique Cm-smooth solution.

An approximate solution of the initial-boundary value problem for the delay parabolic partial differential equation is considered. Stable difference schemes of first and second orders of accuracy for this problem are investigated. Convergence estimates for the solution of these difference schemes in Holder norms are established. Theoretical statements are supported by numerical examples.

In this paper, a singly diagonally implicit Runge-Kutta-Nystrom (RKN) method is constructed for solving second-order ordinary differential equations (ODE) with oscillatory solutions. The produced method is 5-stage, algebraic order five, and phase-lag (or dispersion) order eight at a cost of five function evaluations per step. The method is more accurate compare to current existing similar type of methods for the numerical integration of second-order differential equations with periodic solutions by using constant step size.

In this paper, we investigate backward stochastic differential equations (BSDEs) driven by Brownian motion. We propose a class of Euler type numerical methods with one parameter based on random walk framework for one-dimensional Brownian motion. The convergence of such schemes is proved.

The minisymposium aims at presenting recent research in the area of numerical methods for time‐dependent partial differential equations. The topics of time integration, space discretizations, and convergence issues like that of convergence to a steady state are considered as well as new algorithms for detecting and treating singularities like shock waves.

In this paper, we propose the meshless approach for the numerical solution of time dependent partial differential algebraic equations (PDAEs) in terms of finite difference scheme generated from radial basis functions (RBF‐FD). Using the method, we can circumvent the influence from an index jump of PDAEs in some degree. Numerical example shows that the method has some advantages over some known methods, such as less computation and more accurate for PDAEs.

Consider the following problem related to an ordinary differential equation: ‘For given constans ua,ub,M,N, function f(x) in [a,b], find ∂u/∂n at x = a,b which satisfies u″+Mu′+Nu = f in (a,b), u(a) = ua,u(b) = ub.’ This kind of problem is called the “Dirichlet‐Neumann map problem”. This problem is usually solved in two steps. The solution is obtained in the first steps. In the second step, u′(a),u′(b) is computed by differentiating the solution. However, this two‐step procedure is inefficient because the solution in (a,b) obtained in the first step is not essentially required. In this paper, the author presents a new strategy for obtaining Neumann data directly via a boundary integral equation formulation. Using this strategy, an explicit analytical expression of the Dirichlet‐Neumann map of this problem can be directly obtained by solving 2×2 matrices. Furthermore, an extension of the strategy to partial differential equations in two‐dimensional space and numerical algorithms is also presented.

Discontinuous Galerkin (dG) finite‐element methods are well‐known to be suitable for solving convection‐dominated convection diffusion problems. High order Runge‐Kutta methods such as the RKDG and SSP methods have been developed and tested to work quite well (see e.g., Cockburn & Shu (2001), Gottlieb et al. (2001,2009)). An issue of concern remains the strong CFL restrictions on the discretization parameters. Restelli et al. (2006) proposed combining the dG methods with the semi‐Lagrangian methods in what they called semi‐Lagragian discontinuous Galerkin (SLDG). Proposed implementations of the SLDG methods however suffer from limited spatial accuracy as opposed to the RKDG methods. We hereby propose a method in the framework of Lie‐group exponential integrators (see Cellodoni et al. [1, 2]), that uses a modified version of the SLDG method as a building block for computing compositions of convection flows, and establish high temporal accuracy while maintaining the good properties of the dG formulations.

In this paper, we investigate second‐order generalized Laguerre function approximation. We introduce various orthogonal projections. We establish some approximation results, which play an important role in error analysis of fourth order differential equations on unbounded domain. As an example of their important applications, we propose the spectral scheme for some fourth order differential equations. their stability and convergence are proved. The numerical solution preserves the spectral accuracy.

In the present paper the two step difference scheme for the Cauchy problem for the stochastic hyperbolic equation is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of difference schemes for the numerical solution of two problems for hyperbolic equations are obtained. The theoretical statements for the solution of this difference schemes are supported by the results of numerical experiments.

In the present work, we consider the numerical solution of the multipoint nonlocal boundary value problem for elliptic‐parabolic differential equations. A finite difference method for solving the problem is studied. The first and second orders of accuracy difference schemes are presented. The stability, almost coercive stability and coercive stability for the solution of these difference schemes are established. The method is illustrated by numerical examples.

Second and third‐order Steffensen‐type methods have been developed in order to estimate the solution of nonlinear systems of equations without the evaluation of any jacobian matrix, being the efficiency indices of the resulting schemes much higher than their counterparts.

The time discretization of semilinear parabolic problems requires integrators that treat the stiff part in an implicit way. Traditionally, linearly implicit methods had been used for this purpose. More recently, exponential integrators proved to be competitive for this kind of problems. In this short note, we will put emphasis on methods that linearize the problem along the numerical trajectory. We will show that the error analysis can be carried out in the framework of logarithmic matrix norms which gives rise to convergence results that are independent of the spatial grid size.

This paper deals with two‐step methods of order two and order four with minimum truncation error for numerical integration of second order initial value problems. The methods depend upon a parameter p>0, and reduce to the Classical Numerov method for p = 0. As p becomes very large the truncation error tends to zero. The methods are unconditionally stable when applied to the test equation. To illustrate the order, accuracy and stability of the method, the test problem and non linear undamped duffing equations are solved. The results are compared with some other well known methods and the derived methods give better results than any other existing fourth order methods.