Abstract
In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.
Highlights
Fractional differential equations are used to describe a wide range of phenomena in natural science, and because of its numerous applications in physical, chemical, and biological sciences, fractional calculus has captivated the scientific community
Different types of fractional derivatives have appeared in the literature that strengthen and generalize the classical fractional operators defined by the aforementioned authors [5, 6]
Researchers have recently succeeded in extending several standard wavelet methods to numerical solutions for fractional differential equations
Summary
Fractional differential equations are used to describe a wide range of phenomena in natural science, and because of its numerous applications in physical, chemical, and biological sciences, fractional calculus has captivated the scientific community. Researchers have recently succeeded in extending several standard wavelet methods to numerical solutions for fractional differential equations. Numerical integration and numerical solutions of fractional ordinary and fractional partial differential equations are some of the other applications of wavelet methods in applied mathematics. The Riemann-Liouville operators of fractional order are generalized by introducing the fractional-order differentiation and integration of a function by another function [28, 29]. In this paper, taking motivation by the work cited above, we developed a new numerical method for solving linear and nonlinear boundary value problems in φ-FDEs. The rest of the paper is organized as follows: We start Section 2 with an overview of the fractional calculus followed by a discussion of the classical Haar-wavelet and an approximation of the functions by the Haar-wavelet.
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