DISCRETE MULTI-GALERKIN METHODS FOR DERIVATIVE-DEPENDENT HAMMERSTEIN TYPE WEAKLY SINGULAR NONLINEAR FREDHOLM INTEGRAL EQUATIONS WITH GREEN'S KERNEL

Krishna Murari Malav; Kapil Kant; Joydip Dhar; Gnaneshwar Nelakanti

doi:10.11948/20240324

2025 Volume 15 Issue 3

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Krishna Murari Malav, Kapil Kant, Joydip Dhar, Gnaneshwar Nelakanti. DISCRETE MULTI-GALERKIN METHODS FOR DERIVATIVE-DEPENDENT HAMMERSTEIN TYPE WEAKLY SINGULAR NONLINEAR FREDHOLM INTEGRAL EQUATIONS WITH GREEN'S KERNEL[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1616-1640. doi: 10.11948/20240324

Citation:

Krishna Murari Malav, Kapil Kant, Joydip Dhar, Gnaneshwar Nelakanti. DISCRETE MULTI-GALERKIN METHODS FOR DERIVATIVE-DEPENDENT HAMMERSTEIN TYPE WEAKLY SINGULAR NONLINEAR FREDHOLM INTEGRAL EQUATIONS WITH GREEN'S KERNEL[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1616-1640. doi: 10.11948/20240324

DISCRETE MULTI-GALERKIN METHODS FOR DERIVATIVE-DEPENDENT HAMMERSTEIN TYPE WEAKLY SINGULAR NONLINEAR FREDHOLM INTEGRAL EQUATIONS WITH GREEN'S KERNEL

1.
Department of Engineering Sciences, ABV-Indian Institute of Information Technology and Management, Gwalior 474015, India
2.
Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India

Author Bio: Email: km8949@gmail.com(K. M. Malav); Email: jdhar@iiitm.ac.in(J. Dhar); Email: gnanesh@maths.iitkgp.ernet.in(G. Nelakanti)
Corresponding author: Email: sahukapil8@gmail.com(K. Kant)

Abstract

Abstract

In this article, we study discrete multi-Galerkin and iterated discrete multi-Galerkin methods for solving the derivative-dependent nonlinear Hammerstein type Fredholm integral equations, where the nonlinear function within the integration is dependent on the derivative, and the kernel function is of Green's type. We achieve the error bounds by substituting all integrals in the multi-Galerkin method with numerical quadrature and obtain the superconvergence results for derivative-dependent Fredholm-Hammerstein integral equations using piecewise polynomials as basis functions. By applying the numerical quadrature rule, we prove that the iterated discrete multi-Galerkin method provides superior convergence rates over the discrete multi-Galerkin method with $\mathbb{O}(h^{\min(d+1,~m+2{m_{1}},~m+2{m_{2}})})$, where $h$ represents the norm of the partitions. Numerical results are presented to validate the theoretical findings, with figures illustrating a comparison of the error analysis between the proposed methods and those discussed in [22].
- Hammerstein type integral equation /
- piecewise polynomials /
- two-point boundary value problem /
- discrete multi-Galerkin methods /
- Green's kernel /
- derivative-dependent /
- superconvergence results
MSC: 45B05, 45G10, 65R20

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References

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