Finite Difference and Iteration Methods for Fractional Hyperbolic Partial Differential Equations with the Neumann Condition

Abstract

The numerical and analytic solutions of the mixed problem for multidimensional fractional hyperbolic partial differential equations with the Neumann condition are presented. The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation with the Neumann condition is presented. Stability estimates for the solution of this difference scheme and for the first-and second-order difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations. He's variational iteration method is applied. The comparison of these methods is presented.

Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

“…Section 7.6 is devoted to fractional hyperbolic differential and difference equations. It is based on results of [94][95][96][97]. Finally, Section 7.7 is devoted to singular perturbation hyperbolic problems.…”

“…In [94], a procedure of modified Gauss elimination method was used for obtaining the solution of difference scheme (313) in the case of one-dimensional fractional hyperbolic partial differential equations. The theoretical statements for the solution of this difference scheme were supported by the results of the numerical experiment.…”

DGJ Method for Linear and Nonlinear Third order Fractional Differential Equation

“…There are some methods for approximate solutions of fractional differential equations due to space and time variables [4,6,7,5]. These methods are the radial basis function, Chebyshew Tau method, thin plate splines method, variational iteration method, finite difference schemes method and DGJ method [9,10,11,12,23]. DGJ method was used for evolution and nonlinear functional equation and fractional order nonlinear systems [13,14,15].…”

Positive solutions for a fractional boundary value problem

“…Note that numerous works [6,7,8,9,10,11,12,26] were dedicated to the research questions of local and non local boundary value problems for partial di¤erential equations with boundary operators of high (integer and fractional) order. In [6], the initial boundary value problem for partial di¤erential equations of higher order with the caputo fractional derivative was studied in the case when the order of the fractional derivative belongs to the interval (0,1).…”