Existence and Uniqueness Analysis for a Class of Singular Non-Linear Two-Point Boundary Value Problems by an Optimal Iterative ‎Sequence

Document Type : Research Paper

Author

Department of Mathematics, Imam Khomeini International University, Qazvin, Iran.

Abstract

The convergence of this
iterative sequence is then controlled by an embedded parameter. The fastest convergence occurs for an optimal embedded parameter which maximizes a special function. This optimization problem brings a sequence with high rate of the convergence to the
unique solution in the finite region where $\frac{\partial f}{\partial y}$ has to be positive.
Some illustrative examples are given to confirm the validity and reliability of this constructive theory.

Keywords


 [1] R. P. Agarwal, D. ORegan, Nonlinear superlinear singular and nonsingular second order boundary value problems, Journal of Differential Equations 143 (1998) 60-95.
[2] R. P. Agarwal, D. ORegan, An upper and lower solution approach for singular boundary value problems with sign changing nonlinearities, Mathematical Methods in the Applied Sciences 25 (2002) 491-506.
[3] J. V. Baxley, Some singular nonlinear boundary value problems,
SIAM Journal on Mathematical Analysis 22 (1991) 463-479.
[4] J. V. Baxley, G. S. Gersdorff, Singular reaction-diffusion boundary value problems,
Journal of Differential Equations 115 (1995) 441-457.
[5] A. Dinmohammadi, A. Razani, E. Shivanian, Analytical solution to the nonlinear singular boundary value problem arising in biology,
Boundary Value Problems 1 (2017) 63-74.
[6] A. Dinmohammadi, E. Shivanian, A. Razani, Existence and uniqueness of solutions for a class of singular nonlinear twopoint boundary value problems with signchanging nonlinear terms,
Numerical Functional Analysis and Optimization 38 (2017) 344-359.
[7] A. Ebaid, A new analytical and numerical treatment for singular two-point boundary value problems via the adomian decomposition method,
Journal of computational and applied mathematics 235 (2011) 1914-1924.
[8] A. Fink, J. A. Gatica, G. E. Hernandez, P. Waltman, Approximation of solutions of singular second order boundary value problems,
SIAM Journal on Mathematical Analysis 22 (1991) 440-462.
[9] W. F. Ford, J. A. Pennline, Singular nonlinear two-point boundary value problems: Existence and uniqueness,
Nonlinear Analysis: Theory, Methods & Applications 71 (2009) 1059-1072.
[10] J. Gatica, V. Oliker, P. Waltman, Singular nonlinear boundary value problems for second-order ordinary differential equations,
Journal of Differential Equations 79 (1989) 62-78.
[11] R. Jia, J. Shao, Existence and uniqueness of periodic solutions of second-order nonlinear differential equations,
Journal of Inequalities and Applications 1 (2013) 1-13.
[12] R. Kannan, D. O’regan, Singular and nonsingular boundary value problems with sign changing,
J. Inequal. Appl. 5 (2000) 621-637.
[13] H. B. Keller, Numerical methods for twopoint boundary-value problems,
Waltham, 1968.
[14] H. Lu, Z. Bai, Positive radial solutions of a singular elliptic equation with sign changing nonlinearities,
Applied Mathematics Letters 19 (2006) 555-567.
[15] J. A. Pennline, Improving convergence rate in the method of successive approximations,
Mathematics of Computation 37 (1981) 127-134.
[16] J. A. Pennline, Constructive existence and uniqueness for some nonlinear two-point boundary value problems,
Journal of Mathematical Analysis and Applications 96 (1983) 584-598.
[17] J. A. Pennline, Constructive existence and uniqueness for two-point boundary-value problems with a linear gradient term,
Applied Mathematics and Computation 15 (1984) 233-260.
[18] A. Tieno, On a class of singular boundary value problems which contains the boundary conditions,
Journal of Differential Equations 113 (1994) 1-16.
[19] H. Wang, Y. Li, Existence and uniqueness of solutions to two point boundary value problems for ordinary differential equations,
Zeitschrift fur angewandte Mathematik und Physik ZAMP 47 (1996) 373-384.
[20] J. Wang, On positive solutions of singular nonlinear two-point boundary value prob
lems, Journal of Differential Equations 107 (1994) 163-174.
[21] J. Zhou, The existence and uniqueness of the solution for nonlinear elliptic equations in hilbert spaces,
Journal of Inequalities and Applications 1 (2015) 1-23.