Abstract
In this paper, we prove the existence of at least one solution for Volterra- Hammerstein integral equation (V-HIE) of the second kind, under certain conditions, in the space , Ω is the domain of integration and T is the time. The kernel of Hammerstein integral term has a singularity, while the kernel of Volterra is continuous in time. Using a quadratic numerical method with respect to time, we have a system of Hammerstein integral equations (SHIEs) in position. The existence of at least one solution for the SHIEs is considered and discussed. Moreover, using Toeplitz matrix method (TMM), the SHIEs are transformed into a nonlinear algebraic system (NAS). Many theorems related to the existence of at least one solution for this system are proved. Finally, numerical results and the estimate error of it are calculated and computed using Mable 12. Key words: Volterra- Hammerstein integral equation, nonlinear algebraic system (NAS), singular kernel, Toeplitz matrix method, Holder inequality.
Highlights
Linear and nonlinear singular integral equations have received considerable interest the mathematical applications in different areas of sciences
The different numerical methods play an important role in solving the nonlinear integral equations (NIE)
Kummer and Sloan (2003) used a new collection type method to discuss the solution of HIE with continuous kernel
Summary
Faculty of Applied Science, Umm Al– Qurah University, Makah, Kingdom of Saudi Arabia. We prove the existence of at least one solution for Volterra- Hammerstein integral equation (V-HIE) of the second kind, under certain conditions, in the space Lp ( ) C[0,T ],T 1 , Ω is the domain of integration and T is the time. The kernel of Hammerstein integral term has a singularity, while the kernel of Volterra is continuous in time. Using a quadratic numerical method with respect to time, we have a system of Hammerstein integral equations (SHIEs) in position. The existence of at least one solution for the SHIEs is considered and discussed. Using Toeplitz matrix method (TMM), the SHIEs are transformed into a nonlinear algebraic system (NAS). Many theorems related to the existence of at least one solution for this system are proved.
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