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: −(tk1y′(t))′=t−ω1f1(t,y(t),z(t)), −(tk2z′(t))′=t−ω2f2(t,y(t),z(t)),where t > 0, subject to the following initial values, boundary values and four-point boundary values: y(0)=γ1, y′(0)=0, z(0)=γ2, z′(0)=0, y′(0)=0, y(1)=δ1, z′(0)=0, z(1)=δ2, y(0)=0, y(1)=n1z(v1), z(0)=0, z(1)=n2y(v2),where n1,n2,v1,v2∈(0,1) and k1≥0, k2≥0, ω1<1, ω2<1, γ1, γ2, δ1, δ2 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.

On some computational aspects of Hermite & Haar wavelets on a class of nonlinear singular BVPs

“…Computational work illustrate the validity and accuracy of the procedure. It will be interesting to see what happens to the proposed method in this paper when we try to solve coupled Lane-Emden equations, nonlinear SBVPs, nonlinear PDEs, fractional PDEs etc [1,2,3,4,11,19,31,39,40,41]. We can also see how these newly developed wavelets will behave when we couple them with SVM kernels [44].…”

Biorthogonal flatlet multiwavelet collocation method for solving the singular nonlinear system with initial and boundary conditions

“…MEðf ðxÞÞ MEðgðxÞÞ BFMCM(m 5 2, J 5 2) 0 0 BFMCM(m 5 4, J 5 1) 0 0 HWCM(J 5 3) (Verma et al, 2020) 2.2204E-16 1.1102E-16 HWCM(J 5 4) (Verma et al, 2020) 0 0 Source(s): Authors' own creation BFMCM for solving singular nonlinear system 8 > > < > > :…”

“…The absolute errors between the approximate and exact solution are shown in Tables 1 and 2. We also compare the numerical results of the maximum absolute errors obtained by the present method and the results obtained by the HWCM (Verma et al, 2020) in Table 3. Figure 1 represents the behavior of the solution with and without noise for f (x) and g(x).…”

“…Recently, a method for solving the above equations has been presented by Verma et al (2020) and Barnwal and Pathak (2019). But the presented process is very costly and time consuming.…”

A note on variation iteration method with an application on Lane–Emden equations

“…Several techniques (analytical/numerical) developed to deal with this type of equations, like monotone iterative technique coupled with upper/lower solutions (Singh and Verma, 2017;Singh et al, 2015;Verma, 2011Verma, , 2012Pandey and Verma, 2011), shooting method (Russell and Shampine, 1975), finite-difference method (Chawla and Katti, 1985;Pandey, 1997;Pandey and Singh, 2004), B-splines and cubic splines method (Chawla and Subramanian, 1988;Kanth and Bhattacharya, 2006), Haar wavelet (Verma and Tiwari, 2019;Swati et al, 2020;Verma et al, 2020b), etc.…”