On Separate Fractional Sum-Difference Equations with n-Point Fractional Sum-Difference Boundary Conditions via Arbitrary Different Fractional Orders

Saowaluck Chasreechai,Thanin Sitthiwirattham

doi:10.3390/math7050471

Saowaluck Chasreechai, Thanin Sitthiwirattham

Open Access

https://doi.org/10.3390/math7050471

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Abstract

In this article, we study the existence and uniqueness results for a separate nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and the Schauder’s fixed point theorem. Our problem contains two nonlinear functions involving fractional difference and fractional sum. Moreover, our problem contains different orders in n + 1 fractional differences and m + 1 fractional sums. Finally, we present an illustrative example.

Highlights

Fractional calculus has recently been an attractive field to researchers because it is a powerful tool for explaining many engineering and scientific disciplines as the mathematical modeling of systems and processes which appear in nature, for example, ecology, biology, chemistry, physics, mechanics, networks, flow in porous media, electrical, control systems, viscoelasticity, mathematical biology, fitting of experimental data, and so forth
In 2017, Sumelka and Voyiadjis [4] proposed a concept of short memory connected with the definition of damage parameter evolution in terms of fractional calculus for hyperelastic materials
We study the existence and unique results of the solution for a separate nonlinear Caputo fractional sum-difference equation with fractional sum-difference boundary conditions

Summary

Introduction

Fractional calculus has recently been an attractive field to researchers because it is a powerful tool for explaining many engineering and scientific disciplines as the mathematical modeling of systems and processes which appear in nature, for example, ecology, biology, chemistry, physics, mechanics, networks, flow in porous media, electrical, control systems, viscoelasticity, mathematical biology, fitting of experimental data, and so forth. Mathematics 2019, 7, 471 where t ∈ N2−μ1 −μ2 −μ3 ,b+2−μ1 −μ2 −μ3 , 0 < μ1 , μ2 , μ3 < 1, 1 < μ2 + μ3 < 2, 1 < μ1 + μ2 + μ3 < 2, f : N0 × R → [0, +∞) is a continuous function, and ∆μ is the Riemann-Liouville fractional difference operator of order μ. Sitthiwirattham [19,20] investigated three-point fractional sum boundary value problems for sequential fractional difference equations of the forms β α [ φ ( ∆ x )]( t ) = f ( t + α + β − 1, x ( t + α + β − 1)),.

Preliminaries

Main Results

An Example

Conclusions