Abstract
The present paper is concerned with the implementation of the optimal homotopy asymptotic method to find the approximate solutions of two-dimensional fractional order Volterra integro-differential equations. The technique’s applicability and validity are tested through some numerical examples. The fractional order derivatives are calculated using Caputo’s sense. Results obtained by the proposed technique are compared with the Legendre wavelet method. The proposed method provides us with efficient and more accurate solutions than the other existing methods in the literature. Error analysis and convergence of the proposed method are also provided in the paper.
Highlights
Riemann and Liouville defined the concept of fractional order integro-differential equations
Many numerical methods have received considerable importance toward resolving these difficulties, such as the wavelet methods,5–7 fractional differential transform method,8,9 Adomian decomposition method,10,11 variational iteration method,12,13 shifted Chebyshev polynomial method,14 homotopy perturbation method,15 spline collocation method,16 solitary wave solutions of the modified Benjamin–Bona–Mahony equation,17 solution of the fractional modified Fornberg–Whitham equation arising in water waves,18 solitary wave solution of fractional Kudryashov–Sinelshchikov equation,19 accurate solution of the time-fractional Kaup–Kupershmidt equation arising in capillary gravity waves,20 time-fractional modified Fornberg–Whitham equation for analyzing the behavior of water waves,21 and fractional Burgers–Fisher and generalized Fisher’s equations
The numerical result shows that the optimal homotopy asymptotic method (OHAM) provides more accuracy than the other methods, and the approximate solution becomes very close to the exact solution
Summary
Riemann and Liouville defined the concept of fractional order integro-differential equations. N − 1 < α < n, n ∈ N, where Dαz represents the Caputo fractional operator for a fractional order of α with respect to z, h(z, t) and k(z, t, r, s) are the known analytical functions, and u(z, t) is an unknown solution. The Caputo partial fractional derivative operator Dαz of order α with respect to z is defined as follows:. M ≥1 and at ρ = 1, the series is observed to be convergent, so the approximate solution containing auxiliary constants is given as u(z, t, ρ, cl) = u0(z, t) + ∑ um(z, t, cl)ρm, l = 1, 2, . Differentiating the functional J(cm) with respect to the different values of cm, we get the following system of equations containing cl, l = 1, 2, .
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