Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space

Abstract

This article is concerned with a method for solving nonlocal initial-boundary value problems for parabolic and hyperbolic integro-differential equations in reproducing kernel Hilbert space. Convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method and some error estimates for the numerical approximation in reproducing kernel Hilbert space are presented. Finally, two numerical examples are considered to illustrate the computation efficiency and accuracy of the proposed method.

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

“…Theorem Suppose y n ( x , t ) is the approximate solution of the problem (1) in space and y ( x , t ) is the exact solution then , where ( x , t ) ∈ D and ∥ y ( x , t ) − y n ( x , t )∥ ∞ = max ( x , t ) ∈ D = | y ( x , t ) − y n ( x , t )| and ∥ ∂ x ∂ t y ( x , t ) − ∂ x ∂ t y n ( x , t )∥ ∞ = max ( x , t ) ∈ D = | ∂ x ∂ t y ( x , t ) − ∂ x ∂ t y n ( x , t )| and C 1 , C 2 , C 3 , C 4 are positive constants , , and . n = n 1 × n 2 where is number of collocation points in region D . Proof For more details refer to [28–32]. In each [ x i , x i + 1 ] × [ t j , t j + 1 ] ⊂ D we have, …”

A kernel-based method for Volterra delay integro-differential equations

“…The theory of reproducing kernels was first proposed by Zaremba [1]. This theory has played an important role in a number of successful applications in numerical analysis and has successfully been used for constructing approximate solutions to several linear and nonlinear problems such as singular nonlinear second-order periodic boundary value problems [9], nonlinear system of second order boundary value problems [10], one-dimensional variable-coefficient Burgers equation [4], the coefficient inverse problem [6], nonlinear agestructured population equation [3], the generalized regularized long wave equation [14], nonlinear delay differential equations of fractional order [11], variational problems depending on indefinite integrals [8], nonlocal initial-boundary value problems for parabolic and hyperbolic integro-differential equations [7] and the generalized Black-Scholes equation [15]. Cui and Lin in [5] provide an excellent overview of the existing reproducing kernel methods for solving various model problems such as integral and integro-differential equations.…”

Numerical solution for a class of parabolic integro-differential equations subject to integral boundary conditions

“…In [15], Merad and Martín-Vaquero presented a computational study for two-dimensional hyperbolic integrodifferential equations with purely integral conditions, in which, they demonstrated the existence and uniqueness of the solution and proposed a numerical approach based on Galerkin method. Authors in [11], utilized reproducing kernels approach to solve parabolic and hyperbolic integrodifferential equations subject to integral and weighted integral conditions. More recently, Bencheikh et al [1] implemented numerical method, based on operational matrices of orthonormal Bernstein polynomials, to approximate the solution of an integrodifferential parabolic equation with purely nonlocal integral conditions.…”