PPubMed2 Feb 2014Cited 7×

Nature Inspired Computational Technique for the Numerical Solution of Nonlinear Singular Boundary Value Problems Arising in Physiology

Suheel Abdullah Malik, Ijaz Mansoor Qureshi, Muhammad Amir, Ihsanul Haq

Abstract

'Ihsanul Haq'] We present a hybrid heuristic computing method for the numerical solution of nonlinear singular boundary value problems arising in physiology. The approximate solution is deduced as a linear combination of some log sigmoid basis functions. A fitness function representing the sum of the mean square error of the given nonlinear ordinary differential equation (ODE) and its boundary conditions is formulated. The optimization of the unknown adjustable parameters contained in the fitness function is performed by the hybrid heuristic computation algorithm based on genetic algorithm (GA), interior point algorithm (IPA), and active set algorithm (ASA). The efficiency and the viability of the proposed method are confirmed by solving three examples from physiology. The obtained approximate solutions are found in excellent agreement with the exact solutions as well as some conventional numerical solutions.

§ The Valyu brief

The paper, in five minutes

Read from the full text via Valyu Contents

A hybrid GA‑IPA‑ASA heuristic method solves nonlinear singular BVPs in physiology with higher accuracy than existing techniques.

What’s new

i.Proposed a hybrid heuristic computing approach combining genetic algorithm, interior point algorithm, and active set algorithm.
ii.Introduced log‑sigmoid basis functions to represent the approximate solution of nonlinear singular BVPs.
iii.Demonstrated superior accuracy on three physiological singular BVP examples compared with modified decomposition, B‑spline, and other evolutionary methods.
iv.Provided detailed algorithm parameter settings and extensive error analysis.

How they did it

The solution is expressed as a linear combination of log‑sigmoid basis functions; a fitness function equal to the sum of mean‑square errors of the ODE and its boundary conditions is constructed. Unknown parameters are optimized using hybrid schemes (GA‑IPA, GA‑ASA) that integrate genetic algorithms with interior‑point and active‑set optimizers.

What they found

Maximum absolute error for GA‑IPA on Example 1: 3.99 × 10⁻⁶ (smaller than Approach I 1.05 × 10⁻⁵ and Approach II 2.00 × 10⁻⁶).
Maximum absolute error for GA‑ASA on Example 1: 2.13 × 10⁻⁶ (comparable to B‑spline method 4.97 × 10⁻⁷).
Maximum absolute error for GA on Example 2 (m=0.25): 1.11 × 10⁻⁴, outperforming finite‑difference method (1.17 × 10⁻⁴).
Maximum absolute error for GA‑IPA on Example 3 (m=0.75): 1.14 × 10⁻⁵, better than finite‑difference method 1.25 × 10⁻⁵.

Caveats

✕Study limited to three singular BVP examples from physiology.
✕Only log‑sigmoid basis functions were employed; other basis sets were not tested.
✕Hybrid schemes evaluated only for GA‑IPA and GA‑ASA; other evolutionary algorithms were not explored.
✕No theoretical convergence proof or complexity analysis is provided.

§ The Valyu brief

Reading the full paper and taking notes. This takes a few seconds…

§ Ask this paper

Ask a question about this paper

Valyu reads the full text and answers from what the paper actually says.

§ Related work in other archives

Nature Inspired Computational Technique for the Numerical Solution of Nonlinear Singular Boundary Value Problems Arising in Physiology

The paper, in five minutes

Ask a question about this paper

Similar papers in all four archives

Similar papers in all four archives

The paper, in five minutes