Semi-analytical solution techniques

Differential Equations can only be solved exactly in limited number of cases. In general equations must be solved numerically or approximately. In this work we consider two approximate methods: Approximate-Iterative Method by V.K. Dzyadyk and Adomian's Decomposition Method. First we are interested in the construction of approximate polynomials solutions of ordinary differential equations for general non-linear right-hand sides. We use the Approximate Iterative Method of V.K. Dzyadyk that allows the construction of highly accurate polynomial solutions. In previous work analyticity of the right-hand side was assumed, but Dzyadyk's method can also be applied to non-analytic right-hand sides. Here we revisit some sample problems under much weaker conditions on the smoothness of the solution, assuming that it is only known that the solution belongs to Cr. Dzyadyk's method produces an a-priori error estimate for the solution of Cauchy problems. Using this estimate we obtain the degree of polynomial n and the number of iterations nu that are required to achieve the desired accuracy. The acquired polynomials are of low degree n with sufficiently high accuracy without significant computational effort. As a result such polynomials can be used in practical applications. Secondly, the Adomian Decomposition Method is being applied to nonlinear partial differential equations from hydrology. Two equations are considered. First we construct an approximate solution to the Boussinesq Equation, a particular example of the Porous Medium Equation and then will analyze the general case, the Porous Medium Equation, itself. For both equations, we first apply dimensional analysis to reduce the partial differential equation to an ordinary differential equation. These equations are solved with the Adomian Decomposition Method modified for boundary-value problems. At last the obtained solutions are compared with other results.