This is a preview and has not been published.

Newton-Kantorovich Method for Solving One of the Non-Linear Sturm-Liouville Problems




Non-linear Sturm-Liouville problems, Newton-Kantorovich method, Finite Difference Method, Adomian’s decomposition method


Due to its importance in physics and applied mathematics, the non-linear Sturm-Liouville problems
witnessed massive attention since 1960. A powerful Mathematical technique called the Newton-Kantorovich
method is applied in this work to one of the non-linear Sturm-Liouville problems. To the best of the authors’
knowledge, this technique of Newton-Kantorovich has never been applied before to solve the non-linear
Sturm-Liouville problems under consideration. Accordingly, the purpose of this work is to show that this
important specific kind of non-linear Sturm-Liouville differential equations problems can be solved by
applying the well-known Newton-Kantorovich method. Also, to show the efficiency of applying this method
to solve these problems, a comparison is made in this paper between the Newton-Kantorovich method and the
Adomian decomposition method applied to the same non-linear Sturm-Liouville problems under consideration
in this work. As a result of this comparison, the results of the Newton-Kantorovich method agreed with the
results obtained by applying Adomian’s decomposition method.


Download data is not yet available.


Aal-Rkhais HA, Kamil AH, Al Oweidi KF. The Approximation of Weighted Hölder Functions by Fourier-Jacobi Polynomials to the Singular Sturm-Liouville Operator. Baghdad Sci J. 2022(Aug.16); 19(6):1387.

Kurseeva VY, Moskaleva M, Valovik DV. Asymptotical analysis of a nonlinear Sturm-Liouville problem: Linearisable and non-linearisable solutions. Asymptot Anal. 2019; 119(10): 1-21.

He J, Yang L. Existence of positive solutions for systems of nonlinear Sturm-Liouville differential equations with weight functions. . Electron J Differ Equ. 2019; 2019(111): 1-24.

Al-Khaled K, Hazaimeh A. Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems. Symmetry. 2020 Jul 16; 12(7): 1179.

Regmi S, Argyros IK, George S, Argyros CI. Extended Newton–Kantorovich Theorem for Solving Nonlinear Equations. Foundations. 2022; 2: 504-511.

Boichuk AA, Chuiko SM. On approximate solutions of nonlinear boundary-value problems by the Newton-Kantorovich method. J Math Sci. 2021; 258(5): 594–617.

Hussan EA, Abbas MK. The Solution of Diffusion And Exothermic Zero Equation by Using Newton – Kantorovich Method. AL-Qadisiyha J Sci. 2014; 19(2): 173-185.

Hosseini K, Ilie M, Mirzazadeh M, Yusuf A, Sulaiman AT, Baleanu D, et al. An effective computational method to deal with a time-fractional nonlinear water wave equation in the Caputo sense. Math Comput Simul. 2021; 187: 248-260.

Hosseini K, Ilie M, Mirzazadeh M, Baleanu D. An analytic study on the approximate solution of a nonlinear time-fractional Cauchy reaction–diffusion equation with the Mittag–Leffler law. Math Methods Appl Sci. 2021; 44(8): 6247-6258.

Hosseini K, Sadri K, Mirzazadeh M, Ahmadian A, Yu-Ming Chu. Reliable methods to look for analytical and numerical solutions of a nonlinear differential equation arising in heat transfer with the conformable derivative. Math Methods Appl. Sci. 2021; Early View. doi.10.1002/mma.7582.

Hosseini K, Ilie M, Mirzazadeh M, Baleanu D, Park C, Salahshour S. The Caputo–Fabrizio time-fractional Sharma–Tasso–Olver–Burgers equation and its valid approximations. Commun. Theor. Phys. 2022; 74(7): 075003.

Somali S, Gokmen G. Adomian Decomposition Method for Nonlinear Sturm-Liouville Problems. Surv Math Appl. 2007; 2: 11-20.

AL-Jizani KH, Al-Delfi JKK. An Analytic Solution for Riccati Matrix Delay Differential Equation using Coupled Homotopy-Adomian Approach. Baghdad Sci J. 2022; 4: 800-804.