Numerical solutions for nonlinear mixed integro-differential equation and the neutron transport equation
Source abstract
Abstract
In this work, I will discuss the solution of the new in a nonlinear integro-differential equation (NI-DE) form. The NI-DE is established from the nuclear transport equation, which is a linearized derivative of the equation developed by Boltzmann for the kinetic theory of gases. In this study, I will introduce the basic equations and boundary conditions for the formulation of the problem. Then, the NI-DE is established from the planar geometry problem of the neutron transport equation. In addition, I will prove the existence of a unique solution to the problem. In addition, I will use the numerical method to obtain a system of NI-DEs. Therefore, I will prove, under certain conditions, the existence of a unique solution to this system. Finally, I will present two different effective methods to solve the problem numerically, and I will discuss the results.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Numerical methods in inverse problems · Fractional Differential Equations Solutions · Advanced Mathematical Physics Problems
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
Affiliations
Department of Mathematics and Statistics, Collage of Sciences, Taif University , P.O. Box 11099, Taif 21944, Saudi Arabia
References (38)
Show source-provided references
1. 1960
2. A Chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in a discrete random medium10.1006/jcph.1999.6247 ↗
3. New development in the discrete ordinates approximation for the dimensional transport equation
4. 1987
5. M. Asadzadeh and A.Kadem, “Spectral method for the transport equation, university of setif faculty of sciences, in computers and mathematics with applications,” Ph.D. thesis (Doctroate Detat, 2006).
6. Fredholm–Volterra integral equation of the first kind and contact problem10.1016/s0096-3003(00)00118-1 ↗
7. Fredholm–Volterra integral equation and generalized potential kernel10.1016/s0096-3003(01)00128-x ↗
8. On a symptotic methods for Fredholm–Volterra integral equation of the second kind in contact problems10.1016/s0377-0427(02)00862-2 ↗
9. Fredholm–Volterra integral equation with singular kernel10.1016/s0096-3003(02)00046-2 ↗
10. Numerical solution for Fredholm–Volterra integral equation of the second kind by using collocation and Galerkin methods10.1016/j.jksus.2009.12.006 ↗
11. On the numerical solutions of Fredholm-Volterra integral equation10.1016/s0096-3003(02)00615-x ↗
12. Numerical solution for Fredholm integral equation with Hilbert kernel
13. Asymptotic error expansion for the Nystrom method for a nonlinear Volterra–Fredholm integral equation
14. A method of solving nonlinear mixed Volterra–Fredholm integral equation
15. A fast iterative method for discretized Volterra–Fredholm integral equations10.1016/j.cam.2005.05.018 ↗
16. Modification of Block Pulse Functions and their application to solve numerically Volterra integral equation of the first kind10.1016/j.cnsns.2010.09.032 ↗
17. New approach to solve two-dimensional Fredholm integral equations10.1016/j.cam.2018.12.029 ↗
18. Approximate solution of two-dimensional Fredholm integral equation of the first kind using wavelet base method10.1007/s40819-019-0717-9 ↗
19. Numerical solution of two dimensional IE of the first kind by multi-step method
20. On solution of Fredholm integro differential equations using composite Chebyshev finite difference method10.1155/2013/694043 ↗
21. Solution of Fredholm integro-differential equations system by modified decomposition method10.22436/jmcs.05.04.02 ↗
22. Solving Fredholm integro–differential equations using reproducing kernel Hilbert space method10.1016/j.amc.2013.03.006 ↗
23. Numerical solution of linear Fredholm integro-differential equations by non-standard finite difference method
24. Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases10.1142/s1793557119500554 ↗
25. Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients10.1016/j.camwa.2010.02.018 ↗
26. Nonlinear Fredholm integro-differential equation in two-dimensional and its numerical solutions10.3934/math.2021602 ↗
27. Two dimensional Fredholm integral equation with time10.20454/jmmnm.2012.339 ↗
28. Nonlinear Fredholm-Volterra integral equation and its numerical solutions with quadrature methods
29. Mixed integral equation with Cauchy kernel and contact problem
30. Singular Hammerstein–Volterra integral equation and its numerical processing10.4236/jamp.2021.92026 ↗
31. Numerical solution of nonlinear 2D volterra–fredholm integro-differential equations by two-dimensional triangular function10.1007/s40819-015-0079-x ↗
32. Bivariate Jacobi polynomials for solving Volterra partial integro-differential equations with the weakly singular kernel10.1002/mma.7662 ↗
33. A physical phenomenon for the fractional nonlinear mixed integro-differential equation using a general discontinuous kernel10.3390/fractalfract7020173 ↗
34. Nonlinear system of impulsive integro-differential equations with hilfer fractional operator and mixed maxima10.47475/2500-0101-2022-17305 ↗
35. Convergence of the adomian method applied to a class of nonlinear integral equations10.1016/j.aml.2007.05.008 ↗
36. Convergence of adomian’s method applied to a class of Volterra type integro-differential equations
37. Approximate solution for the system of non-linear Volterra integral equations of the second kind by using Block-by-block Method
38. Runge-Kutta method and Bolck by Block method to solve nonlinear Fredholm–Volterra integral equation with continuous kernel10.4236/jamp.2020.89152 ↗