TY  - JOUR
AU  - Pippal, Sarita
PY  - 2026
DA  - 2026/09/17
TI  - New Iteration Method (NIM) for Solving Fourth-Order Two-Point Nonlinear Boundary Value Problems
JO  - Journal of Numerical Simulations in Physics and Mathematics
T2  - Journal of Numerical Simulations in Physics and Mathematics
JF  - Journal of Numerical Simulations in Physics and Mathematics
VL  - 2
IS  - 2
SP  - 90
EP  - 103
DO  - 10.62762/JNSPM.2026.798641
UR  - https://www.icck.org/article/abs/JNSPM.2026.798641
KW  - new iteration method
KW  - nonlinear differential equations
KW  - two-point boundary value problems
KW  - fourth-order boundary value problems
KW  - iterative series solution
AB  - In this study, the New Iteration Method (NIM) is employed to obtain approximate analytical solutions of nonlinear fourth-order ordinary differential equations of the general form \[ \mathcal{L}(u) + \mathcal{N}(u) = g(x), \qquad x \in [a,b], \] subject to the two-point boundary conditions \[ \mathcal{B}_1(u)=0, \qquad \mathcal{B}_2(u)=0, \] where $\mathcal{L}$ denotes a linear differential operator, $\mathcal{N}$ represents a nonlinear operator, and $g(x)$ is a given continuous function. The NIM framework constructs a rapidly convergent iterative sequence $\{u_n\}_{n=0}^{\infty}$ whose limit approximates the exact solution. The convergence behavior and numerical performance of the method are investigated through several test problems, including problems with polynomial, trigonometric, and exponential nonlinearities. The obtained solutions are compared with exact solutions and benchmark results available in the literature. The comparative analysis demonstrates that NIM yields highly accurate approximations, showing excellent agreement with exact or reference solutions. These results confirm that the proposed iterative scheme is mathematically consistent, computationally efficient, and robust for solving higher-order nonlinear boundary value problems.
SN  - 3068-9082
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Pippal2026New,
  author = {Sarita Pippal},
  title = {New Iteration Method (NIM) for Solving Fourth-Order Two-Point Nonlinear Boundary Value Problems},
  journal = {Journal of Numerical Simulations in Physics and Mathematics},
  year = {2026},
  volume = {2},
  number = {2},
  pages = {90-103},
  doi = {10.62762/JNSPM.2026.798641},
  url = {https://www.icck.org/article/abs/JNSPM.2026.798641},
  abstract = {In this study, the New Iteration Method (NIM) is employed to obtain approximate analytical solutions of nonlinear fourth-order ordinary differential equations of the general form \[ \mathcal{L}(u) + \mathcal{N}(u) = g(x), \qquad x \in [a,b], \] subject to the two-point boundary conditions \[ \mathcal{B}\_1(u)=0, \qquad \mathcal{B}\_2(u)=0, \] where \$\mathcal{L}\$ denotes a linear differential operator, \$\mathcal{N}\$ represents a nonlinear operator, and \$g(x)\$ is a given continuous function. The NIM framework constructs a rapidly convergent iterative sequence \$\{u\_n\}\_{n=0}^{\infty}\$ whose limit approximates the exact solution. The convergence behavior and numerical performance of the method are investigated through several test problems, including problems with polynomial, trigonometric, and exponential nonlinearities. The obtained solutions are compared with exact solutions and benchmark results available in the literature. The comparative analysis demonstrates that NIM yields highly accurate approximations, showing excellent agreement with exact or reference solutions. These results confirm that the proposed iterative scheme is mathematically consistent, computationally efficient, and robust for solving higher-order nonlinear boundary value problems.},
  keywords = {new iteration method, nonlinear differential equations, two-point boundary value problems, fourth-order boundary value problems, iterative series solution},
  issn = {3068-9082},
  publisher = {Institute of Central Computation and Knowledge}
}