# Nonlinear Fredholm Functional-Integral Equation of First Kind with Degenerate Kernel and Integral Maxima

## DOI:

https://doi.org/10.11113/mjfas.v19n1.2788## Keywords:

Fredholm functional-integral equation, nonlinear equation, degenerate kernel, integral maxima, boundary conditions, regularization, small parameter## Abstract

In this note, the problems of solving by the aid of regularization method for a nonlinear Fredholm functional-integral equation of the first kind with degenerate kernel and integral maxima are considered. In using the regularization method, we denote the nonlinear function as a new unknown function. This method we combine with the method of the degenerate kernel. Fredholm functional-integral equation of the first kind is ill-posed (non-correct) problem. We have used boundary conditions to ensure the uniqueness of the solution. We transform the implicit functional equation with integral maxima to another type of nonlinear functional-integral equation of the second kind by some integral transforms. Using the method of successive approximations and the method of compressing mapping, the theorem on single valued solvability of the problem is proved. It is obtained the necessary and sufficient conditions of existence and uniqueness of the solution of the problem with integral maxima. Two simple examples are analyzed with an exact and approximate solution.

## References

. Assanova, A. T., Bakirova, E. A. and Kadirbayeva, Zh. M. (2019). Numerical implementation of solving a boundary value problem for a system of loaded differential equations with parameter. News of the NAS RK. Series Phys.-Math., 3(325), 77-84.

. Assanova, A. T., Bakirova, E. A. and Kadirbayeva, Z. M. (2020). Numerical solution to a control problem for integro-differential equations, Comput. Math. and Math. Phys., 60(2), 203-221.

. Assanova, A. T., Imanchiyev, A. E., Kadirbayeva, Zh. M. (2018). Numerical solution of systems of loaded ordinary differential equations with multipoint conditions, Comput. Math. Math. Phys., 58(4), 508-516.

. Bakirova, E. A., Tleulesova, A. B. and Kadirbayeva, Zh. M. (2017). On one algorithm for finding a solution to a two-point boundary value problem for loaded differential equations with impulse effect. Bulletin of the Karaganda University. Mathematics, 87(3), 43-50.

. Boyd, J. P. (1989). Chebyshev and Fourier Spectral Methods. Springer-Verlag, Berlin.

. Brunner, H. (2004). Collocation methods for volterra integral and related functional differential equations. Cambridge University Press, Cambridge.

. Funaro, D. (1992). Polynomial approximations of differential equations. Springer-Verlag.

. Guo, B. (1998). Spectral methods and their applications. World Scientific, Singapore.

. Kadirbayeva, Zh. M. (2017). On the method for solving linear boundary-value problem for the system of loaded differential equations with multipoint integral condition. Mathematical Journ., 17(4), 50-61.

. Pimenov, V. G. and Hendy, A. S. (2016). Fractional analog of crank-nicholson method for the two sided space fractional partial equation with functional delay. Ural Math. J., 2(1), 48-57.

. Pimenov, V. G. (2018). Numerical methods for fractional advection-diffusion equation with heredity. J. Math. Sci. (N. Y.), 230(5), 737-741.

. Shen, J., Sheng, Ch. and Wang, Zh. (2015). Generalized Jacobi spectral-galerkin method for nonlinear volterra integral equations with weakly singular kernels. J. of Math. Study, 48(4), 315-329.

. Eshkuvatov, Z. K., Zulkarnain, F. S., Nik Long, N. M. A. and Muminov, Z. (2016). Modified homotopy perturbation method for solving hypersingular integral equations of the first kind. Springer Plus, 5(1473), 1-21.

. Alhawamda, H., Taib, B. M., Eshkuvatov, Z. K., Ibrahim, R. I. (2020). A new class of orthogonal polynomials for solving logarithmic singular integral equations. Ain Shams Engineering Journal, 11, 489-494.

. Behiry, S. H., Abd-Elmonem, R. A., Gomaa A. M. (2010). Discrete adomian decomposition solution of nonlinear Fredholm integral equation. Ain Shams Engineering Journal, 1, 97-101.

. Adomian, G. (1990). A review of the decomposition method and some recent results for nonlinear equations. Mathematical and Computer Modelling, 13(7), 17-43.

. Adomian, G. (1994). Solving frontier problems of physics: The decomposition method, vol. 60 of fundamental theories of physics. Kluwer Academic, Dordrecht, The Netherlands.

. Cherruault, Y., Saccomandi, G. and Some, B. (1992). New results for convergence of Adomian’s method applied to integral equations. Mathematical and Computer Modelling, 16(2), 85-93.

. Wazwaz, A.-M. (1999). A reliable modification of Adomian decomposition method. Applied Mathematics and Computation, 102(1), 77-86.

. Wazwaz, A.-M. and El-Sayed, S. M. (2001). A new modification of the Adomian decomposition method for linear and nonlinear operators, Applied Mathematics and Computation, 122(3), 393-405.

. Hosseini, M. M. (2006). Adomian decompositionmethod with Chebyshev polynomials. Applied Mathematics and Computation, 175(2), 1685-1693.

. Liu, Y. (2009). Adomian decompositionmethod with orthogonal polynomials: Legendre polynomials, Mathematical and Computer Modelling, 49(5-6), 1268-1273.

. Hadamard, J. (1923). Lectures on Cauchy’s problem in linear partial differential equations. Yale University Press, New Haven.

. Phillips, D. L. (1962). A technique for the numerical solution of certain integral equations of the first kind, J. Ass. Comput. Mach, 9, 84-96.

. Lavrent’ev, M. M., Romanov, V. G., Shishatskii, S. R. (1980). Noncorrect problems of mathematical physics and analysis. Moscow, Science, 1980 [in Russian].

. Yuldashev, T. K., Saburov, Kh. Kh. (2021). On Freholm integral equations of the first kind with nonlinear deviation. Azerbaijan Journal of Mathematics, 11(2), 137-152.

. Yuldashev, T. K., Eshkuvatov Z. K., Nik Long N. M. A. (2022). Nonlinear the first kind Fredholm integro-differential first-order equation with degenerate kernel and nonlinear maxima. Mathematical Modeling and Computing, 9(1), 74-82.

. Imanaliev, M. I., Asanov A. (1988). Regularization, uniqueness and existence of solution for Volterra integral equations of first kind. Studies by Integro-Diff. Equations, 21. Frunze. 3-38 [in Russian].

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