Numerical solution of a fractional coupled system with the Caputo-Fabrizio fractional derivative

Abstract

Within this work, we discuss the existence of solutions for a coupled system of linear fractional differential equations involving Caputo-Fabrizio fractional orders. We prove the existence and uniqueness of the solution by using the Picard-Lindelöf method and fixed point theory. Also, to compute an approximate solution of problem, we utilize the Adomian decomposition method (ADM), as this method provides the solution in the form of a series such that the infinite series converge to the exact solution. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.

Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative

“…By utilizing the Laplace transform method, we successfully identified the exact solution for equation (17) as , = − + − 1 Lu4ˆ . We can now utilize the following formula to calculate the maximum norm of error estimates for numerical analysis using the modified Gauss elimination technique for equation ( 10 Table 2 provides the approximate numerical results for the proposed model (19), which was obtained using the theta difference approach for different values of each 9, H, : and , when 0 < < 1, 0 < < ˆ.…”

“…Therefore, in the proposed model ( 1), the Caputo fractional derivative [16], is used because it is a natural extension of the classical derivative with a memory effect, and it has the desirable property of having a well-defined derivative for constant functions. The fractional derivative has been used to model several real-world events [17,18].…”

Numerical Approach Based on the Haar Wavelet Collocation Method for Solving a Coupled System with the Caputo–Fabrizio Fractional Derivative

“…For instance, the Caputo-Fabrizio fractional integral and derivative [1] avoid the singularity problem; this property makes it popular in the scientific community. The main problem facing researchers in solving Caputo-Fabrizio fractional differential equations and systems [6] is the difficulty in finding an analytical solution, which leads them to use numerical methods. In fact, it is known that in many numerical methods, the solutions contain discontinuities at some points, which negatively affects the accuracy and convergence rate.…”

“…Indeed, the Haar wavelet has the advantages of simplicity, orthogonality, and compact support. As a support for the Haar wavelet method, in the present paper, we apply and investigate it to solve the following system [6] for 0 ≤ t ≤ 1:…”

“…In fact, Ikram Mansouri et al [6] considered the questions of the existence of a unique solution for this system, where the Adomian Decomposition Method (ADM) is applied to provide an approximate solution to it. However, the Adomian Decomposition Method has a polynomial decay of the convergence rate, and it is expensive to compute its terms and requires a large number of terms to obtain the exact solution.…”