c 2011 Modern Science Publishers ⃝ www.m-sciences.com
Journal of Modern Methods in Numerical Mathematics ISSN: 2090-4770 online Vol. 2, No. 1-2, 2011, 1–15
RESEARCH ARTICLE Numerical Solution for Volterra-Ferdholm Integral Equation with A Generalized Singular Kernel M. A. Abdou ∗ , I. L. El- Kalla
†
and A. M. Al-Bugami ‡
∗
†
Department of Mathematics, Faculty of Education, Alexandria University, Egypt. Physics and Engineering Mathematics Department, Faculty of Engineering, Mansoura University, P.O. Box 35516, Mansoura, Egypt. ‡ Department of Mathematics, Faculty of Applied Sciences, Taif University, Saudi Arabia. (Received: 2 Jan 2011, Accepted: 8 April 2011)
In this paper, the existence of a unique solution of Volterra-Fredholm integral equation of the second kind (V-FIESK) is discussed. The Volterra integral term (VIT) is considered in time with a continuous kernel, while the Fredholm integral term (FIT) is considered in position with a generalized singular kernel. Using a numerical technique, V-FIESK is reduced to a system of Fredholm integral equations ( SFIEs ). Using Toeplitz matrix method and Product Nystrm method we have a linear algebraic system of equations ( LAS ). Finally, some numerical examples when the kernel takes the logarithmic, Carleman, Cauchy and Hilbert forms, are considered. Keywords: Singular integral equation; Volterra –Fredholm integral equation; Toeplitz matrix; product Nystrm method; Cauchy kernel; Hilbert kernel. AMS Subject Classification: 47J30; 26A33.
1.
Introduction
Many authors have interested in solving the linear and nonlinear integral equation, Diogo and Lima, in [1], discussed the application of spline collocation methods to a certain class of weakly integral equations. Also, in [2], Kangro and Oja discussed the convergence of spline for Volterra integral equations. In [3], Anastassiou and Aral introduced the multivariate of the Picard singular integral to obtain the rate of convergence and to define a new modulus of continuity. Abdou and Salama, in [4], obtained the solution in one, two and three dimensional for the V–FIT of the first kind using spectral relationships. In [5], EL-Borai et al. studied the existence and uniqueness of solution of nonlinear integral equation of the second kind of type V–FIE. Maleknejad and Sohrabi, in [6], solved the nonlinear V–F–Hammerstein integral equations in terms of Legendre polynomials. In [7], Matar studied the fractional semi linear mixed V–F integro differential equation. Also, the same author, in [8], studied the controllability problem of fractional semi linear mixed V–F integro differential equations with nonlocal conditions. In this work, we consider the linear V–FIESK with a new generalized singular kernel ∫ t∫ µϕ(x, t) = λ 0
F (t, τ )k(|g(x) − g(y)|)ϕ(y, τ )dydτ + f (x, t)
Ω
∗ Corresponding author. Email adresses:
[email protected] (A. M. Al-Bugami).
(1)
M. A. Abdou et al.
2
The integral equation (1) is considered in time, for VIT and position for FIT. The functions k(|g(x) − g(y)|), F (t, τ ) and f (x, t) are given and called the kernel of FIT, VIT and the free term, respectively. The constant µ defines the kind of the integral equation and λ is a real parameter (may be complex and has physical meaning). Also, Ω is the domain of integration with respect to position, and the time t ∈ [0, T ], T < ∞. While ϕ(x, t) is the unknown function to be determined in the space L2 (Ω) × C[0, T ]. In this work, the existence and uniqueness of solution of V-FIESK, when the kernel of position has a singular term, under certain conditions, are considered in the space L2 (Ω)×C[0, T ]. Using a numerical method, the equation of V-FIE reduces to SFIEs then, the existence of a unique solution of this system is discussed. Using the Toeplitz matrix and product Nystrm methods we obtain the numerical solution. Finally, some numerical examples are considered and discussed when the generalized kernel of position term takes the following forms: logarithmic form, Carleman function, Cauchy kernel, and Hilbert kernel.
2.
Existence of A Unique Solution of The V-FIESK
In this section, the existence of a unique solution of the integral equation (1) will be proved, using Picard method. So, we assume the following conditions: (a) The kernel of FIT k(|g(x) − g(y)|) satisfies the discontinuity condition: ∫ ∫ 1 [ k 2 (|g(x) − g(y)|)dxdy] 2 = N , Ω
( N is a constant )
Ω
(b) The kernel of VIT F (t, τ ) ∈ C[0, T ], 0 ≤ τ ≤ t ≤ T < ∞, and satisfies |F (t, τ )| ≤ M, ∀t, τ ∈ [0, T ], (M is a constant) (c) The given function f (x, t) with its partial derivatives with respect to x and t are continuous in the space L2 (Ω) × C[0, T ] where, ∫ ∥f (x, t)∥ = max
0≤t≤T
0
t
∫ 1 [ |f (x, t)|2 dx] 2 dτ = L, ( L is a constant ) Ω
The unknown function ϕ(x, t) ∈ L2 (Ω) × C[0, T ] is called the potential function and it behaves, in this space, as the known function f (x, t). Theorem 2.1 The V-FIE (1) is exist and has a unique solution in the space L2 (Ω) × C[0, T ], under the condition |µ| > |λ| (N M T )
(2)
Proof The idea of using Picard method is constructing the solution of the FIE as a sequence of functions {ϕn (x, t)} as n tends to ∞. So, we assume the solution of (1) in the form ϕ(x, t) = lim ϕn (x, t) n→∞
where, ϕn (x, t) =
n ∑ i=0
ψi (x, t), t ∈ [0, T ],
n = 1, 2, ...
(3)
Numerical Solution for Volterra-Ferdholm Integral Equation with A Generalized Singular Kernel
3
And ψi (x, t) are continuous functions, takes the form ψn (x, t) = ϕn (x, t) − ϕn−1 (x, t),
(4)
ψ0 (x, t) = f (x, t).
Now, we must prove the following lemmas: ∑ Lemma 2.2 The series ni=0 ψi (x, t) is uniformly convergent to a continuous solution function ϕ(x, t). Proof We construct a sequence ϕn defined by ∫ t∫ µϕn (x, t) = λ 0
F (t, τ )k(|g(x) − g(y)|)ϕn−1 (y, τ )dydτ + f (x, t),
n = 1, 2, ...
(5)
Ω
Then, we get µ[ϕn (x, t) − ϕn−1 (x, t)] = λ
∫ t∫ 0
F (t, τ )k(|g(x) − g(y)|)[ϕn−1 (y, τ ) − ϕn−2 (y, τ )]dydτ
(6)
Ω
By using (4) in (5), we get
ϕn (x, t) =
n ∑
ψi (x, t)
i=0
Hence, we have ∫ t ∫
λ
∥ψn (x, t)∥ ≤ |F (t, τ )| k(|g(x) − g(y)|)ψn−1 (y, τ )dydτ
µ 0 Ω For n = 1 and using Cauchy-Schwarz inequality with the condition (b), we get
∫ t ∫
λ
∥ψ1 (x, t)∥ ≤ [M k(|g(x) − g(y)|)ψ0 (y, τ )dydτ
] µ 0 Ω By using the conditions (a) and (c), we get λ ∥ψ1 (x, t)∥ ≤ L(N M T ) µ where T = max0≤t≤T t. So, we can prove that λ ∥ψn (x, t)∥ ≤ Lσ , σ = (N M T ) < 1, n = 1, 2, ... µ n
This bound makes the sequence ψn (x, t) converges if |λ|
