Two computational approaches for solving a fractional obstacle system in Hilbert space

Shatha Hasan,Asad Freihet,Mohammed Al-Smadi,Shaher Momani

doi:10.1186/s13662-019-1996-5

Abstract

The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0<alphaleq1, concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.

Highlights

The concept of variational inequality problems (VIPs) has become an influential mathematical methodology for qualitatively analyzing free boundary, equilibrium, optimization, complementarity, obstacle, unilateral, and environmental network issues in numerous disciplines including economics, finance, management, mechanics, elasticity and engineering [1,2,3]
In 1966, Hartman and Stampacchia introduced the theory of VIP for studying a class of partial differential equations with applications that were derived mainly from mechanics
The obstacle model is essential in the development of the VIPs theory that arises in a variety of pure and differential applied sciences

Summary

Introduction

The concept of variational inequality problems (VIPs) has become an influential mathematical methodology for qualitatively analyzing free boundary, equilibrium, optimization, complementarity, obstacle, unilateral, and environmental network issues in numerous disciplines including economics, finance, management, mechanics, elasticity and engineering [1,2,3]. Where 0 < α ≤ 1, D2aα is the Caputo-fractional derivative, μ1, μ2 ∈ R, the parameter r is real finite constant, g(x) is an analytical continuous function on [c, d], f (x) is a continuous on [a, b], the function u(x) is unknown smooth to be obtained such that u(i)(x), i = 0, 1, is continuous function at internal points c and d of [a, b] In this aspect, the reproducing-kernel method (RKM) is applied for finding smooth approximations to the solution of the modified obstacle system of fractional order (1.4) and (1.5), and its derivative. To apply the fractional RPS technique, there are three cases to obtain the approximate solution, un(x), for the obstacle BVPs (1.4) and (1.5) depending on the corresponding intervals

BDαa g

Exact solution

Conclusion