Numerical treatment of nonlinear mixed Volterra-Fredholm integro-differential equations of fractional order

Abstract

In recent years, considerable interest in fractional differential equations has been stimulated due to their numerous applications in the areas of physics and engineering. Many essential phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science are well described by differential equations of fractional order (Podlubny [8]). In certain circumstances, many exact solutions for linear fractional differential equation had been found. However, in general, there exists no method that yields an exact solution for nonlinear fractional differential equations. In this note, we consider the following nonlinear mixed Volterra-Fredholm integro-differential equation (VFIDEs) of fractional order. cD0+αu(t)φ(t)+λ∫0t ∫0Tk(x,s)F(u(s))dxds,(1) u(i)(0)=bi, i=0,…,n−1, n−1<α≤n,(2) where t ∈ Ω = [0; T], k : Ω × Ω → ℝ, ϕ : Ω → ℝ are known functions, F : C(Ω, ℝ) → ℝ is nonlinear function, and u(s) is unknown function to be determined, bi (i = 0,…, n − 1) and λ are constants, cD0+α is the Caputo fractional derivative of order α. At first, Eq. (1)-(2) is written into operator form and applied standard homotopy analysis method (Liao, 1992) to reduce it into a sequence of known integral equation problems. Solving the latter equations step by step using the truncated Taylor series, an approximate solution is obtained. Description of the method and discretization of the problems together with the control convergence parameter are thoroughly studied. In addition, one example is illustrated to show the accuracy and validity of the proposed approach. Numerical results reveal that the proposed method is highly accurate and quickly approaches the exact solution when the number of iteration increases.