Picard-Reproducing Kernel Hilbert Space Method for Solving Generalized Singular Nonlinear Lane-Emden Type Equations

Abstract

Abstract. An iterative method is discussed with respect to its effectiveness and capability of solving singular nonlinear Lane-Emden type equations using reproducing kernel Hilbert space method combined with the Picard iteration. Some new error estimates for application of the method are established. We prove the convergence of the combined method. The numerical examples demonstrates a good agreement between numerical results and analytical predictions.

Numerical Study of Astrophysics Equations by Meshless Collocation Method Based on Compactly Supported Radial Basis Function

“…Other researchers try to solve the Lane-Emden type equations with several methods, For example, Yıldırım andÖziş (Yıldırım andÖziş, 2007(Yıldırım andÖziş, , 2009) by using HPM and VIM methods, Benko et al (Benko et al, 2008) by using Nyström method, Iqbal and javad (Iqbal and Javad, 2011) by using Optimal HAM, Boubaker and Van Gorder (Boubaker and Van-Gorder, 2012) by using Boubaker polynomials expansion scheme, Daşcıoglu and Yaslan (Akyüz-Daşcıoglu and Ç erdik Yaslan, 2011) by using Chebyshev collocation method, Yüzbaşı (Yüzbaşı, 2011;Yüzbaşı and Sezer, 2013) by using Bessel matrix and improved Bessel collocation method, Boyd (Boyd, 2011) by using Chebyshev spectral method, Bharwy and Alofi (Bharwy and Alofi, 2012) by using Jacobi-Gauss collocation method, Pandey et al by using Legendre and Brenstein operation matrix, Rismani and monfared (Rismani and Monfared, 2012) by using Modified Legendre spectral method, Nazari-Golshan et al (Nazari-Golshan et al, 2013) by using Homotopy perturbation with Fourier transform, Doha et al (Doha et al, 2013) by using second kind Chebyshev operation matrix algorithm, Carunto and bota (Caruntu and Bota, 2013) by using Squared reminder minimization method, Mall and 5 Chakaraverty (Mall and Chakraverty, 2014) by using Chebyshev Neural Network based model, Gürbüz and sezer (Gürbüz and Sezer, 2014) by using Laguerre polynomial, Kazemi-Nasab et al (Kazemi-Nasab et al, 2015) by using Chebyshev wavelet finite difference method, Hosseini and Abbasbandy (Hosseini and Abbasbandy, 2015) by using combination of spectral method and ADM method and Azarnavid et al (Azarnavid et al, 2015) by using Picard-Reproducing Kernel Hilbert Space Method 4. ICSRBF method 4.1.…”

Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

“…Reproducing Kernel Hilbert Space (RKHS) was introduced by Minggen et al [8,9], and it was developed in different areas, including approximation theory, statistics, machine learning theory, group representation theory, and various areas of complex analysis. Reproducing Kernel Hilbert Space Method (RKHSM) is a kernel based approximation method which was applied for solving nonlinear boundary value problems [7][8][9][10][11][12], generalized singular nonlinear Lane-Emden type equations [13], integrodifferential equations [14][15][16], integrodifferential fractional equations [17], Bratus Problem [18], and so forth.…”

“…In previous works like [13][14][15], the Gram-Schmidt orthogonalization process has been considered to implement RKHSM. Since this process is unstable numerically and it may take a lot of time to run the algorithm, here, we put away this process and act with another way.…”

“…Since this process is unstable numerically and it may take a lot of time to run the algorithm, here, we put away this process and act with another way. Our approach combines the methods mentioned in [13][14][15][16][17]. More specifically, on the contrary to [13][14][15], without use of the orthogonalization process, the RKSHM is applied successfully to solve the nonlinear problem (1).…”

Variable Shape Parameter Strategy in Local Radial Basis Functions Collocation Method for Solving the 2D Nonlinear Coupled Burgers’ Equations

“…By following this utilization of kernels, we can solve the PDEs. An overview of kernel methods prior to the year 2006 is presented in [6], while their recent variations are in [7][8][9][10][11][12] and the related references.…”