Abstract
We are interested in the approximate analytical solutions of the wave-like nonlinear equations with variable coefficients. We use a wave operator, which provides a convenient way of controlling all initial and boundary conditions. The proposed choice of the auxiliary operator helps to find the approximate series solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method.
Highlights
We consider the equation utt − k (x, t) uxx = f (u, ux, uy, uxt) + g (x, t), (1)x > 0, t > 0 with initial conditions u (x, 0) = φ (x), (2)ut (x, 0) = ψ (x) and boundary condition u (0, t) = h (t), (3)where f, g, k, φ, ψ, and h are known functions
Ut (x, 0) = ψ (x), u (0, t) = h (t), has a unique solution and there exists an inverse of the operator L : utt − c2uxx
This operator can control all initial-boundary conditions in each step of HAM
Summary
Aslanov [21] used homotopy perturbation method to solve wave-like equations with initial-boundary conditions. Various methods for obtaining exact and approximate solutions to nonlinear partial differential equations have been proposed Among these methods are the homotopy perturbation and Adomian decomposition methods [25–29], the variational iteration method [30], homotopy analysis method [31], and others. We will further extend the applications of HAM to obtain an approximate series solution for the nonlinear wavelike equations with variable coefficients and with initialboundary conditions. Ut (x, 0) = ψ (x) , u (0, t) = h (t) , has a unique solution (see, e.g., [32]) and there exists an inverse of the operator L : utt − c2uxx This operator can control all initial-boundary conditions in each step of HAM.
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