A deficient spline function approximation to systems of first order differential equations

The stability of the vector-valued spline function approximations S(x) of degree m deficiency 3, i.e., S∈C^m?3, to systems of first order differential equations are investigated. The method will be shown to be A-stable for m=4, unstable and hence divergent for m?6. The method is stable form=5.     

A deficient spline function approximation to fourth-order differential equations

An approximate spline solution is developed for the initial value problem of a fourth-order ordinary differential equation. The approximation is based on deficient spline polynomials of degree M = 8 and deficiency 4. The existence and uniqueness of the solution, which satisfies a Lipschitz condition, are proved. The consistency, stability, and consequently convergence of the solution are established. Furthermore, the method is proved to be of order 9, and the errors are limited by the relation S⁽ⁱ⁾(x) − y⁽ⁱ⁾(x) = O(h^{9 − i}), I = 0(1)8.     

Deficient spline function approximation to second- order differential equations

An approximation with spline functions of degree m and deficiency 3 is developed for solving second-order initial-value problems. The stability of the approximate solution is investigated and it is demonstrated that the method is divergent for m 7. Convergence is shown for m = 5 and m = 6. Moreover, the method is of order (m + 1) and error bounds are of the form: [boxV ]S⁽ⁱ⁾(x)−yⁱx[boxV ]∞=0h^m+-i), i=01m A computational example is presented to illustrate the efficiency of the method.     

Numerical solution of stiff and convection-diffusion equations using adaptive spline function approximation

The singular perturbation mathematical model plays an important role in modelling fluid processes which arise in applied mechanics. We have either, the stiff system of initial value problems or convection-diffusion problems. When conventional numerical methods are used to obtain the solution, the stepsize must be limited to small values. Any attempt to use a larger step-size results in the production of nonphysical oscillations in the solution.In this paper we have constructed an adaptive spline function to solve initial and boundary value problems of ordinary and partial differential equations. The numerical methods based on the spline relations when applied to the test models produce oscillation free solutions. The numerical results are presented and discussed.     

On the approximation of stochastic differential equations

The stability properties of stochastic differential equations with respetct to the perturbation of the coefficients and of the driving processes are investigated in the topology of uniform convergence in probability     

On Wong-Zakai approximation of stochastic differential equations

An approximation theorem of stochastic differential equations driven by semimartingales is proved, based on approximation of semimartingales by a sequence of processes with piecewise monotonic sample functions.     

Liouville-Green asymptotic approximation for a class of matrix differential equations and semi-discretized partial differential equations

A Liouville-Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y^″=[f(t)A+G(t)]Y on [a,+∞), where A and G(t) are matrices and f(t) is scalar. This includes the case of an “asymptotically constant” (not necessarily diagonalizable) coefficient A (when f(t)≡1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown. Several examples, some concerning semi-discretized wave and convection-diffusion equations, are given.     

A method for constructing solutions to linear systems of partial differential equations

We obtain some conditions of solvability in Sobolev spaces for the systems of linear partial differential equations and deduce the corresponding formulas for solutions to these systems. The solutions are given as the sum of the series whose terms are the iterations of some pseudodifferential operators constructed explicitly.     

On the approximation of stochastic partial differential equations i

The stability of abstract stochastic partial differential equations with respect to the simultaneous perturbation of the driving processes and of the differential operators is investigated. The results obtained here will be applied to concrete stochastic partial differential equations in the continuation of this paper     

A result of systems of nonlinear complex algebraic differential equations

In this article, we mainly investigate the behavior of systems of complex differential equations when we add some condition to the quality of the solutions, and obtain an interesting result, which extends Gackstatter and Laine's result concerning complex differential equations to the systems of algebraic differential equations.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

A new Chebyshev polynomial approximation for solving delay differential equations

The purpose of this study is to give a Chebyshev polynomial approximation for the solution of mth-order linear delay differential equations with variable coefficients under the mixed conditions. For this purpose, a new Chebyshev collocation method is introduced. This method is based on taking the truncated Chebyshev expansion of the function in the delay differential equations. Hence, the resulting matrix equation can be solved, and the unknown Chebyshev coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of this investigation are discussed.     

A link of stochastic differential equations to nonlinear parabolic equations

Using Girsanov transformation,we derive a new link from stochastic differential equations of Markovian type to nonlinear parabolic equations of Burgers-KPZ type,in such a manner that the obtained BurgersKPZ equation characterizes the path-independence property of the density process of Girsanov transformation for the stochastic differential equation.Our assertion also holds for SDEs on a connected differential manifold.     

On nonlinear systems consisting of different types of differential equations

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity. Supported by the Hungarian NFSR under grant OTKA T 049819.     

Application to differential transformation method for solving systems of differential equations

In this paper, we present an analytical solution for different systems of differential equations by using the differential transformation method. The convergence of this method has been discussed with some examples which are presented to show the ability of the method for linear and non-linear systems of differential equations. We begin by showing how the differential transformation method applies to a non-linear system of differential equations and give two examples to illustrate the sufficiency of the method for linear and non-linear stiff systems of differential equations. The results obtained are in good agreement with the exact solution and Runge–Kutta method. These results show that the technique introduced here is accurate and easy to apply.     

Method of weighted residuals as applied to nonlinear differential equations

A new method for solving a class of nonlinear boundary-value problems is presented. In this method, the nonlinear equation is linearized by guessing an initial solution and using it to evaluate the nonlinear terms. Next, a method of weighted residuals is applied to transform the linearized form of the boundary value problem to an initial value problem. The second (improved) solution is obtained by integrating the initial value problem by a fourth order Runge-Kutta scheme. The entire process is repeated until a desired convergence criterion is achieved.     

Galerkin finite element approximation of symmetric space-fractional partial differential equations

In this paper, symmetric space-fractional partial differential equations (SSFPDE) with the Riesz fractional operator are considered. The SSFPDE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order 2β ∈ (0, 1) and 2α ∈ (1, 2], respectively. We prove that the variational solution of the SSFPDE exists and is unique. Using the Galerkin finite element method and a backward difference technique, a fully discrete approximating system is obtained, which has a unique solution according to the Lax-Milgram theorem. The stability and convergence of the fully discrete schemes are derived. Finally, some numerical experiments are given to confirm our theoretical analysis.     

A numerical approximation of parabolic stochastic partial differential equations driven by a Poisson random measure

The paper deals with SPDEs driven by Poisson random measures in Banach spaces and its numerical approximation. We investigate the accuracy of space and time approximation. As the space approximation we consider spectral methods and as time approximation the implicit Euler scheme and the explicit Euler scheme. AMS subject classification (2000) 60H15, 35R30     

On a class of stochastic differential equations arising from the stochastic approximation theory

The paper dealt with generalized stochastic approximation procedures of Robbins-Monro type. We consider these procedures as strong solutions of some stochastic differential equations with respect to semimartingales and investigate their almost sure convergence and mean square convergence