Abstract
In this paper, the existence and uniqueness of solution of singular Hammerstein-Volterra integral equation (H-VIE) are considered. Toeplitz matrix (TMM) and product Nystrom method (PNM) to solve the H-VIE with singular logarithmic kernel are used. The absolute error is calculated.
Highlights
In this paper, the existence and uniqueness of solution of singular Hammerstein-Volterra integral equation (H-VIE) are considered
Great attention must be considered for the numerical solution of these equations
We prove that the solution exists using the successive approximation method, called the Picard method, that we pick up any real continuous function φ0 ( x,t ) in L2 [−a, a]× C [0,T ], we assume φ0 ( x,t ) = f ( x,t ), construct a sequence φn defined by a φn= ( x,t ) f ( x,t ) + λ ∫ K ( x, y)γ ( y,t,φn−1 ( y,t )) dy
Summary
We prove that the solution exists using the successive approximation method, called the Picard method, that we pick up any real continuous function φ0 ( x,t ) in L2 [−a, a]× C [0,T ] , we assume φ0 ( x,t ) = f ( x,t ) , construct a sequence φn defined by a φn= ( x,t ) f ( x,t ) + λ ∫ K ( x, y)γ ( y,t,φn−1 ( y,t )) dy. Since φ ( x,t ) −φ ( x,t ) is necessarily non-negative, and α < 1: φ ( x,t ) −φ ( x,t ) = 0 ⇒ φ ( x,t ) = φ ( x,t ) It follows that if (1) has a solution it must be unique. The formula (10) represents SHIEs ofj=t0he second kind, and we have N unknown φn ( x)
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