Partial Differential Equations in Applied Mathematics (Dec 2024)
Innovative approach to unbounded boundary value problems: IRPSM-Padé algorithm
Abstract
In this effort, a new method called the improved residual power series method with Padé approximants (IRPSM-Padé) has been introduced to solve the boundary value problems (BVPs) on an unbounded domain. It was known from the previous studies that the IRPSM is only acceptable for solving BVPs in a finite domain for small values of the independent variable. To overcome this difficulty, the IRPSM has been modified with the Padé approximants. Combining the results obtained by IRPSM and Padé approximants delivers an inspiring tool to handle BVPs on an unbounded domain. The applications of IRPSM-Padé have been introduced with the help of well-known boundary-layer Blasius problems over a stretching sheet arising in incompressible fluids. MATHEMATICA and Maple software are used for the computation of this analysis. In the first example, the results achieved by IRPSM-Padé have been associated with the results achieved by ADM-Padé and DTM-Padé solutions. A good agreement has been illustrated among the results obtained by IRPSM-Padé, ADM-Padé and DTM-Padé solutions. In the second example, the results achieved by IRPSM and IRPSM-Padé have been associated with the exact solution. A good agreement has been showed between the results achieved by IRPSM-Padé and exact solutions. Furthermore, it is also verified that the IRPSM technique is not suitable to solve BVPs on an unbounded domain. The IRPSM-Padé requires less computational work without linearization, discretization, or perturbation. This confirms that the IRPSM-Padé is a promising tool for solving infinite BVPs in applied fields.
