Approximate series solution of nonlinear singular boundary value problems arising in physiology.

Area of Science:

• Numerical Analysis
• Applied Mathematics
• Computational Science

Background:

• Nonlinear singular boundary value problems (BVPs) pose significant challenges in various scientific and engineering fields.
• Traditional methods often struggle with convergence and complexity for such problems.

Purpose of the Study:

• To introduce an efficient and modified Adomian decomposition method (ADM) for solving nonlinear singular BVPs.
• To develop a recursive scheme that incorporates all boundary conditions upfront, avoiding undetermined coefficients.

Main Methods:

• A modified Adomian decomposition method (ADM) is employed.
• All boundary conditions are utilized to derive an integral equation before establishing the recursive scheme.
• The method generates solution components without requiring the solution of intermediate nonlinear algebraic or transcendental equations.

Main Results:

• An efficient recursive scheme for solving nonlinear singular BVPs is successfully developed.
• The approximate solution is obtained as a series with easily calculable components.
• The uniqueness, convergence, and error analysis of the proposed method are rigorously established.

Conclusions:

• The proposed modified ADM provides an efficient, accurate, and reliable approach for nonlinear singular BVPs.
• It overcomes limitations of the classical ADM by eliminating the need to solve for undetermined coefficients.
• Numerical examples confirm the method's effectiveness.