Collocation-homotopy method to initial-boundary value problems

In this paper, an algorithm based on the collocation and homotopy analysis methods, for solving initial-boundary value problems, is introduced. The application of this algorithm is based on the approximation and interpolation of the dependent variables by using suitable functions or polynomials according to their values in the collocation points corresponding to a suitable discretization of the space variable. Then the space derivatives are approximated using interpolation. Replacing them in the equation transforms the initial-boundary value problem into an initial value problem for ordinary differential equations. The obtained initial value problem is solved by homotopy analysis method. In the frame of the homotopy analysis method, the optimum value of convergence-parameter corresponding to each point is computed by a simple stochastic function minimizer, namely differential evolution method. Lagrange polynomials are usually adopted for the interpolation. In this framework, the Burgers model is considered as a prototype example.