Approximation of functions in Hölder’s class and solution of nonlinear Lane–Emden differential equation by orthonormal Euler wavelets

In this article, a method has been developed for the solution of a non-linear Lane-Emden differential equation based on orthonormal Euler wavelet series. By dilatation and translation of orthogonal Euler polynomials, the orthonormal Euler wavelets are constructed. The convergence analysis of the orthonormal Euler wavelet series is studied in the H¨older’s class. The orthonormal Euler wavelet approximations of solution functions of the non-linear Lane-Emden differential equation in H¨older’s class are determined by partial sums of their orthonormal Euler wavelet series. In concisely, two approximations E^{(۱)}_{۲^{k−۱},M}(f) and E^{(۲)}_{۲^{k−۱},M}(f)of solution functions of classes H^α_۲ [۰, ۱) and H^ϕ_۲[۰, ۱) by (۲^k, M)^{th} partial sums of their orthonormal Euler wavelet expansions have been estimated. There are several applications of non-linear differential equations, which include the non-linear Lane-Emden differential equations. The solution of the non-linear Lane-Emden differential equation obtained by the orthonormal Euler wavelets method is compared to its solution obtained by the ODE-۴۵ method. It has been shown that the solutions produced by the orthonormal Euler wavelets are more accurate than those produced by the ODE-۴۵ method. This is a result of the wavelet analysis research article.

Keywords:

Orthonormal Euler wavelet , H^α_۲[۰ , H^ϕ_۲[۰ , ۱) class , Approximation of function and nonlinear Lane–Emden differential equations