Numerical Results for Linear Sequential Caputo Fractional Boundary Value Problems

Bhuvaneswari Sambandham,Vinodh K Chellamuthu,Aghalaya S Vatsala

doi:10.3390/math7100910

Bhuvaneswari Sambandham, Vinodh K Chellamuthu + Show 1 more

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https://doi.org/10.3390/math7100910

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Abstract

The generalized monotone iterative technique for sequential 2 q order Caputo fractional boundary value problems, which is sequential of order q, with mixed boundary conditions have been developed in our earlier paper. We used Green’s function representation form to obtain the linear iterates as well as the existence of the solution of the nonlinear problem. In this work, the numerical simulations for a linear nonhomogeneous sequential Caputo fractional boundary value problem for a few specific nonhomogeneous terms with mixed boundary conditions have been developed. This in turn will be used as a tool to develop the accurate numerical code for the linear nonhomogeneous sequential Caputo fractional boundary value problem for any nonhomogeneous terms with mixed boundary conditions. This numerical result will be essential to solving a nonlinear sequential boundary value problem, which arises from applications of the generalized monotone method.

Highlights

It is known that in [1] the half order fractional diffusion model has been established to be useful and economical
We make a similar assumption throughout this paper. This guarantees the uniqueness of the solution of all the linear sequential Caputo fractional boundary value problems that we discuss in this paper
In order to compute the solution of sequential Caputo fractional nonlinear boundary value problems of order 2q by the generalized monotone method, it is necessary to compute the solution of the corresponding linear equation with different nonhomogeneous terms

Summary

Introduction

It is known that in [1] the half order fractional diffusion model has been established to be useful and economical. It should be noted that in the generalized monotone method for a given nonlinear sequential Caputo boundary value problem, all the linear iterates will have the same Green’s function. In our first three examples we have q considered the sequential boundary operator −sc Da+ u( x ) The generalization of these examples for any nonhomogeneous term will be very useful in solving the nonlinear sequential boundary value problem by the generalized monotone method. We plan to develop numerical code for the q linear sequential Caputo differential equation of the form −sc Da+ u( x ) − (λ) u = f ( x ), with Caputo mixed boundary conditions. We have included the code for some examples in the Appendix A

Preliminary Results

Main Results

Conclusions