## General Mathematics   ## Homotopy Analysis Method for Solving a Class of Nonlinear Mixed Volterra-Fredholm Integro-Differential Equations of Fractional Order

Authors: Zaid Laadjal

In this paper, we describe the solution approaches based on Homotopy Analysis Method for the follwing Nonlinear Mixed Volterra-Fredholm integro-differential equation of fractional order $$^{C}D^{\alpha }u(t)=\varphi (t)+\lambda \int_{0}^{t}\int_{0}^{T}k(x,s)F\left( u(s\right) )dxds,$$ $$u^{(i)}(0)=c_{i},i=0,...,n-1,$$ where $t\in \Omega =\left[ 0;T\right] ,\ k:\Omega \times \Omega \longrightarrow \mathbb{R},$ $\varphi :\Omega \longrightarrow \mathbb{R},$ are known functions,\ $F:C\left(\Omega, \mathbb{R}\right) \longrightarrow \mathbb{R}$ is nonlinear function, $c_{i} (i=0,...,n-1),$ and $\lambda$ are constants, $^{C}D^{\alpha }$ is the Caputo derivative of order $\alpha$ with $n-1<\alpha \leq n.$ In addition some examples are used to illustrate the accuracy and validity of this approach.

Comments: 10 Pages.

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### Submission history

[v1] 2018-10-24 14:59:24
[v2] 2018-11-17 09:34:46

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