System of Lane-Emden equations as IVPs BVPs and Four Point BVPs & Computation with Haar Wavelets

TL;DR

This paper introduces a Haar wavelet collocation method to solve various forms of Lane-Emden equations, demonstrating high accuracy and second-order convergence for initial, boundary, and four-point boundary value problems.

Contribution

The work develops a Haar wavelet-based numerical approach for solving Lane-Emden equations with different boundary conditions, including four-point BVPs, and establishes its convergence and high accuracy.

Findings

Achieves exact solutions at resolution J=4 for IVPs and BVPs.

Attains highly accurate solutions with errors around 10^{-16} for four-point BVPs.

Demonstrates second-order convergence of the method.

Abstract

In this work we present Haar wavelet collocation method and solve the following class of system of Lane-Emden equation defined as \begin{eqnarray*} -(t^{k_1} y'(t))'=t^{-\omega_1} f_1(t,y(t),z(t)),\\ -(t^{k_2} z'(t))'=t^{-\omega_2} f_2(t,y(t),z(t)), \end{eqnarray*} where $t>0$, subject to initial values, boundary values and four point boundary values: \begin{eqnarray*} \mbox{Initial Condition:}&&y(0)=\gamma_1,~y'(0)=0,~z(0)=\gamma_2,~z'(0)=0,\\ \mbox{Boundary Condition:}&&y'(0)=0,~y(1)=\delta_1,~z'(0)=0,~z(1)=\delta_2,\\ \mbox{Four~point~Boundary~Condition:}&&y(0)=0,~y(1)=n_1z(v_1),~z(0)=0,~z(1)=n_2y(v_2), \end{eqnarray*} where $n_1$, $n_2$, $v_1$, $v_2$ $\in (0,1)$ and $k_1\geq 0$, $k_2\geq0$, $\omega_1<1$, $\omega_2<1$ are real constants. Results are compared with exact solutions in the case of IVP and BVP. In case of four point BVP we compare the result with other methods. Convergence of these methods is also established and found to be of second order. We observe that as resolution is increased to $J=4$ we get the exact values for IVPs and BVPs. For four point BVPs also at $J=4$, we get highly accurate solutions, e.g., the $L^\infty$ error is of order $10^{-16}$ or $10^{-17}$.