Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method

Abstract

In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective.

A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems

“…However, in the traditional reproducing kernel method needs a dense sequence on the interval. So, this new approach varies from the traditional method in [27,[36][37][38][39]42].…”

“…Recently, the RKM has been applied for a different type of problem. For example, fractional order nonlocal boundary value problems [36], Riccati differential equations [37], forced Duffing equations with a nonlocal boundary conditions [38], Burgers' equation with a fractional order Caputo derivative [39], time-fractional Kawahara equation [40], fractional order Boussinesq equation [41], nonlinear fractional Volterra integro-differential equations [42].…”

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

“…4 Rashidinia et al 5 developed the Sinc-Galerkin method for the numerical solution of SBVP. There are different researches such as other studies 6,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] in the literature for the numerical solutions of differential equations.…”

New Challenges Arising in Engineering Problems with Fractional and Integer Order

“…In [4], obtaining approximate solutions for a fractional order Burgers' equation was been presented in a reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces were identified according to the inner products and norms.…”