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Abstract
In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.
Highlights
Fractional calculus is a useful extension of the classical calculus by allowing derivatives and integrals of arbitrary orders
Fractional calculus has become a popular topic for researchers in mathematics, physics, and engineering because the fractional differential equations govern the behavior of many physical systems with more precision [2]
The proposed technique is based upon expanding the unknown functions by Chebyshev wavelets (CWs) for transforming the main system to a system of algebraic equations using the mentioned operational matrix (OM) and applying the Galerkin technique
Summary
Fractional calculus is a useful extension of the classical calculus by allowing derivatives and integrals of arbitrary orders. We remind that the main advantage of using fractional differential (integral) equations for modeling applied problems is their nonlocal property [3], i.e., in a fractional dynamical system, the state depends on all the previous situations so far [3] Another interesting extension to fractional order calculus is considering the fractional order to be a known time-dependent function α(t) [4]. The proposed technique is based upon expanding the unknown functions by CWs for transforming the main system to a system of algebraic equations using the mentioned OM and applying the Galerkin technique. In this way, a new method is introduced to compute nonlinear terms in such systems. Proof Equations (2.6) and (2.7) together with Definition 2.6 complete the proof
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