Solving a well-posed fractional initial value problem by a complex approach

Arran Fernandez,Necati Özdemir,Sümeyra Uçar

doi:10.1186/s13663-021-00696-2

Abstract

Nonlinear fractional differential equations have been intensely studied using fixed point theorems on various different function spaces. Here we combine fixed point theory with complex analysis, considering spaces of analytic functions and the behaviour of complex powers. It is necessary to study carefully the initial value properties of Riemann–Liouville fractional derivatives in order to set up an appropriate initial value problem, since some such problems considered in the literature are not well-posed due to their initial conditions. The problem that emerges turns out to be dimensionally consistent in an unexpected way, and therefore suitable for applications too.

Highlights

Fractional differential equations are a generalisation of the usual differential equations in which the order of differentiation is allowed to be “fractional”: that is to say, a natural number but any real or complex number
(2021) 2021:11 for example, the Fokas method [19,20,21] relies heavily on Cauchy’s theorem for deforming complex contour integrals, and the d-bar method [1, 22, 23] is based on using the complex d-bar derivative related to the Cauchy–Riemann equations. Some of these complex definitions and methods have recently been extended into fractional calculus, in some papers on complex methods for fractional differential equations [9, 12, 17] and the very recent fractionalisation of the complex d-bar derivative [18], historically the connections between complex analysis and fractional calculus have not been deeply explored
4 Conclusions and further work In this paper, we have refined the recent work of Şan [39, 40] on complex fixed point theorems for fractional initial value problems, replacing the initial conditions used there by new initial conditions which are more appropriate for the fractional problems

Summary

Introduction

Fractional differential equations are a generalisation of the usual differential equations in which the order of differentiation is allowed to be “fractional”: that is to say, a natural number but any real or complex number. 2 we derive and prove the main theorems of the paper: first figuring out an appropriate initial condition to associate with the differential equation (1), proving existence and uniqueness in detail, and rewriting the whole initial value problem in such a way that the solution function is in a more natural function space and the problem is more likely to be useful in applications. 3 we illustrate the results with several examples of different types, providing specific fractional initial value problems and conditions under which they have unique solutions.

Results

Conclusion