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

Abstract

In this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram-Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram-Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.

Numerical simulation of telegraph and Cattaneo fractional‐type models using adaptive reproducing kernel framework

“…For convenience, a brief review of the RKS method by illustrating a computational algorithm is provided at the beginning. [20][21][22][23][24][25] For the purpose of implementing the procedure, the desired domain Ω ητ is split out into mish points p × q , p, q∈N > 0, within a space-step-size Ω η = 1 p in the direction of η in [0,1] and a time-step-size Ω τ = 1 q in the direction of τ in [0,1]. Meanwhile, such mish points (η i , τ j ) of the independent space-time-domain are given simultaneously by (η i , τ j ) = (iΩ η )(jΩ τ ), i = 0,1,2,ÁÁÁ, p, j = 0,1,2,ÁÁÁ, q.…”

“…RKS method has many advantages, characteristics, and applications for treating physical, biological, and engineering problems, especially those containing nonlinear fractional PDEs associated with nonclassical initial and boundary conditions. [19][20][21][22][23][24][25] To select more, by using the RKS method, the numerical solution of time-fractional PDEs along with Robin's initialboundary conditions is discussed, which is employed to model many physical problems ranging from anomalous diffusion and subdiffusion processes to wave propagation in diverse media. 6 For greater privacy, consider the generalized time-fractional telegraph equation:…”

On the solutions of boundary value problems

“…Akgül et al [9] have worked on the representation for the reproducing kernel Hilbert space method for a nonlinear system. Allahviranloo et al [14] have investigated the reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions. For more details see [15][16][17][18][19][20][21][22][23].…”