Solving inverse non-linear fractional differential equations by generalized Chelyshkov wavelets

The purpose of this research is to employ a method involving Chelyshkov wavelets to construct a numerical solution to the inverse problem of determining the right-hand side function of a non-linear fractional differential equation by utilizing over-measured data. The novelty of this research is that this type of inverse problem is studied by Chelyshkov wavelets. Firstly, the problem is reduced into a system of algebraic equations with an unknown right-hand side by means of the orthonormal base of Chelyshkov wavelets. Secondly, by choosing suitable nodes, this system is transformed into a homogenous system of algebraic equations. The solution of the homogenous system allows us to determine the coefficients of the bases vectors for the solution of the non-linear fractional differential equation. In the final step, the right-hand side is obtained by substitut-ing the constructed solution into a non-linear fractional differential equation. The presented exam-ples illustrate that the numerical solution, obtained by this method, is remarkably close to the exact solution.(c) 2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

Automated summary

The purpose of this research is to construct a numerical solution to the inverse problem of determining the right-hand side function of a non-linear fractional differential equation using Chelyshkov wavelets and over-measured data. The novelty lies in the use of Chelyshkov wavelets for this type of inverse problem. The problem is reduced to a system of algebraic equations with an unknown right-hand side using the orthonormal base of Chelyshkov wavelets, and by choosing suitable nodes, the system is transformed into a homogenous system. The numerical solution obtained by this method is remarkably close to the exact solution, as demonstrated by the examples.