ABSTRACT
Today, the studies of singular initial value problems in the second order ordinary differential equations (ODEs) of Lane-Emden type have wide applications in mathematical physics and astrophysics [49, 57, 162, 207, 222, 205, 209, 99, 11]. The well-known Lane-Emden equation has been used to model several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres, the theory of thermionic currents, and the modeling of clusters of galaxies [49, 57, 162]. A substantial amount of work has been done on these types of problems for various structures. The singular behavior
equation equivalent to the Lane-Emden equations of first or second kind. The newly established Volterra integro-differential equation will be solved by using the orthogonal wavelets. Many researchers started using various wavelets for analyzing problems of high computational complexity. It is proven that wavelets are powerful tools to explore a new direction in solving differential equations and integral equations. In this chapter, we have applied Legendre multi-wavelets and Chebyshev wavelets to solve Volterra type integro-differential equations. In Section 7.2, we establish a Volterra integro-differential equation equivalent to the Lane-Emden equation of first and second kind and the newly established Volterra integro-differential equation will be solved by using the Legendre multi-wavelet method (LMWM). The Legendre multi-wavelet method has been applied to solve the integral equations and integro-differential equations of different forms [125, 137, 217, 38, 229]. In Section 7.3, another type of Lane-Emden equation has been considered and solved by Chebyshev wavelet method. The Chebyshev wavelet method has been applied to solve the integral equations and integro-differential equations of different forms [20, 5, 3, 230]. Abd-Elameed et al. have solved singular differential equations and boundary value problems by using Chebyshev wavelets in [3, 4]. The Chebyshev wavelet method for boundary value problems has been discussed in [7]. Biazar et al. have solved a system of integro-differential equations by Chebyshev wavelet method [40]. Also Lane-Emden type differential equations have been solved by Chebyshev polynomials [58]. Properties and function approximation of Legendre multi-wavelets and Chebyshev wavelets have been discussed in Chapter 2. These wavelet methods convert the Volterra integro-differential equation to a system of algebraic equations in the aid of the Gauss-Legendre rule and that algebraic equations system again can be solved numerically by Newton’s method. Illustrative examples have been discussed to demonstrate the validity and applicability of the present methods.
