Haar wavelets collocation method for a system of nonlinear singular differential equations

Abstract

Purpose The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane–Emden equations: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mrow><m:mo>−</m:mo><m:mo stretchy="true">(</m:mo><m:msup><m:mi>t</m:mi><m:mrow><m:msub><m:mi>k</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:msup><m:mi>y</m:mi><m:mo>′</m:mo><m:mo stretchy="true">(</m:mo><m:mi>t</m:mi><m:mo stretchy="true">)</m:mo><m:mo stretchy="true">)</m:mo><m:mo>′</m:mo><m:mo>=</m:mo><m:msup><m:mi>t</m:mi><m:mrow><m:mo>−</m:mo><m:msub><m:mi>ω</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:msup><m:msub><m:mi>f</m:mi><m:mn>1</m:mn></m:msub><m:mo stretchy="true">(</m:mo><m:mi>t</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo stretchy="true">(</m:mo><m:mi>t</m:mi><m:mo stretchy="true">)</m:mo><m:mo>,</m:mo><m:mi>z</m:mi><m:mo stretchy="true">(</m:mo><m:mi>t</m:mi><m:mo stretchy="true">)</m:mo><m:mo stretchy="true">)</m:mo><m:mo>,</m:mo></m:mrow></m:math> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mrow><m:mo>−</m:mo><m:mo stretchy="true">(</m:mo><m:msup><m:mi>t</m:mi><m:mrow><m:msub><m:mi>k</m:mi><m:mn>2</m:mn></m:msub></m:mrow></m:msup><m:mi>z</m:mi><m:mo>′</m:mo><m:mo stretchy="true">(</m:mo><m:mi>t</m:mi><m:mo stretchy="true">)</m:mo><m:mo stretchy="true">)</m:mo><m:mo>′</m:mo><m:mo>=</m:mo><m:msup><m:mi>t</m:mi><m:mrow><m:mo>−</m:mo><m:msub><m:mi>ω</m:mi><m:mn>2</m:mn></m:msub></m:mrow></m:msup><m:msub><m:mi>f</m:mi><m:mn>2</m:mn></m:msub><m:mo stretchy="true">(</m:mo><m:mi>t</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo stretchy="true">(</m:mo><m:mi>t</m:mi><m:mo stretchy="true">)</m:mo><m:mo>,</m:mo><m:mi>z</m:mi><m:mo stretchy="true">(</m:mo><m:mi>t</m:mi><m:mo stretchy="true">)</m:mo><m:mo stretchy="true">)</m:mo><m:mo>,</m:mo></m:mrow></m:math> where t &gt; 0, subject to the following initial values, boundary values and four-point boundary values: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mrow><m:mi>y</m:mi><m:mo stretchy="true">(</m:mo><m:mn>0</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:msub><m:mi>γ</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:mo> </m:mo><m:mi>y</m:mi><m:mo>′</m:mo><m:mo stretchy="true">(</m:mo><m:mn>0</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mo> </m:mo><m:mi>z</m:mi><m:mo stretchy="true">(</m:mo><m:mn>0</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:msub><m:mi>γ</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:mo> </m:mo><m:mi>z</m:mi><m:mo>′</m:mo><m:mo stretchy="true">(</m:mo><m:mn>0</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>,</m:mo></m:mrow></m:math> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mrow><m:mi>y</m:mi><m:mo>′</m:mo><m:mo stretchy="true">(</m:mo><m:mn>0</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mo> </m:mo><m:mi>y</m:mi><m:mo stretchy="true">(</m:mo><m:mn>1</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:msub><m:mi>δ</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:mo> </m:mo><m:mi>z</m:mi><m:mo>′</m:mo><m:mo stretchy="true">(</m:mo><m:mn>0</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mo> </m:mo><m:mi>z</m:mi><m:mo stretchy="true">(</m:mo><m:mn>1</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:msub><m:mi>δ</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo></m:mrow></m:math> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mrow><m:mi>y</m:mi><m:mo stretchy="true">(</m:mo><m:mn>0</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mo> </m:mo><m:mi>y</m:mi><m:mo stretchy="true">(</m:mo><m:mn>1</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:msub><m:mi>n</m:mi><m:mn>1</m:mn></m:msub><m:mi>z</m:mi><m:mo stretchy="true">(</m:mo><m:msub><m:mi>v</m:mi><m:mn>1</m:mn></m:msub><m:mo stretchy="true">)</m:mo><m:mo>,</m:mo><m:mo> </m:mo><m:mi>z</m:mi><m:mo stretchy="true">(</m:mo><m:mn>0</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mo> </m:mo><m:mi>z</m:mi><m:mo stretchy="true">(</m:mo><m:mn>1</m:mn><m:mo stretchy="true">)</m:mo><m:mo>=</m:mo><m:msub><m:mi>n</m:mi><m:mn>2</m:mn></m:msub><m:mi>y</m:mi><m:mo stretchy="true">(</m:mo><m:msub><m:mi>v</m:mi><m:mn>2</m:mn></m:msub><m:mo stretchy="true">)</m:mo><m:mo>,</m:mo></m:mrow></m:math> where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:msub><m:mi>n</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>n</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>v</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>v</m:mi><m:mn>2</m:mn></m:msub><m:mo>∈</m:mo><m:mo stretchy="true">(</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mn>1</m:mn><m:mo stretchy="true">)</m:mo></m:mrow></m:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:msub><m:mi>k</m:mi><m:mn>1</m:mn></m:msub><m:mo>≥</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mtext> </m:mtext><m:msub><m:mi>k</m:mi><m:mn>2</m:mn></m:msub><m:mo>≥</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mtext> </m:mtext><m:msub><m:mi>ω</m:mi><m:mn>1</m:mn></m:msub><m:mo>&lt;</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mtext> </m:mtext><m:msub><m:mi>ω</m:mi><m:mn>2</m:mn></m:msub><m:mo>&lt;</m:mo><m:mn>1</m:mn></m:mrow></m:math> , γ 1 , γ 2 , δ 1 , δ 2 are real constants. Design/methodology/approach To deal with singularity, Haar wavelets are used, and to deal with the nonlinear system of equations that arise during computation, the Newton-Raphson method is used. The convergence of these methods is also established and the results are compared with existing techniques. Findings The authors propose three methods based on uniform Haar wavelets approximation coupled with the Newton-Raphson method. The authors obtain quadratic convergence for the Haar wavelets collocation method. Test problems are solved to validate various computational aspects of the Haar wavelets approach. The authors observe that with only a few spatial divisions the authors can obtain highly accurate solutions for both initial value problems and boundary value problems. Originality/value The results presented in this paper do not exist in the literature. The system of nonlinear singular differential equations is not easy to handle as they are singular, as well as nonlinear. To the best of the knowledge, these are the first results for a system of nonlinear singular differential equations, by using the Haar wavelets collocation approach coupled with the Newton-Raphson method. The results developed in this paper can be used to solve problems arising in different branches of science and engineering.

Topics & Concepts