Abstract
Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.
Highlights
The integro-differential equations stem from the mathematical modeling of many complex real-life problems
Liao [6, 7] found that the convergence of series solutions of nonlinear equations cannot be guaranteed by the early homotopy analysis method (HAM)
The homotopy analysis method is applicable for solving problems having strong nonlinearity [15], even if they do not have any small or large parameters, so it is more powerful than traditional perturbation methods
Summary
The integro-differential equations stem from the mathematical modeling of many complex real-life problems. Motsa et al [17,18,19] found that the spectral homotopy analysis method is more efficient than the homotopy analysis method as it does not depend on the rule of solution expression and the rule of ergodicity This method is more flexible than homotopy analysis method, since it allows for a wider range of linear and nonlinear operators, and one is not restricted to use the method of higher-order differential mapping for solving boundary value problems in bounded domains, unlike the homotopy analysis method. We apply spectral homotopy analysis method (SHAM) to solve higher-order nonlinear Fredholm type of integro-differential equations.
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