Fredholm Integral Term Research Articles

Solvability of quadratic integral equations with singular kernel

In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic integral equation (SQIE). The Fredholm integral term is assumed in position with singular kernel. Under certain conditions and new discussions, the singular kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main of spectral relationships are stated and used to obtain the solution of the singular Quadratic integral equation with the logarithmic kernel and a smooth kernel. Finally, the Fredholm integral equation of the second kind is established and its solution is discussed, also numerical results are obtained and the error, in each case, is computed.

Hierarchical‐matrix method for a class of diffusion‐dominated partial integro‐differential equations

AbstractA hierarchical matrix approach for solving diffusion‐dominated partial integro‐differential problems is presented. The corresponding diffusion‐dominated differential operator is discretized by a second‐order accurate finite‐volume scheme, while the Fredholm integral term is approximated by the trapezoidal rule. The hierarchical matrix approach is used to approximate the resulting algebraic problem and includes the implementation of an efficient preconditioned generalized minimum residue (GMRes) solver. This approach extends previous work on integral forms of boundary element methods by taking into account inherent characteristics of the diffusion‐dominated differential operator in the resultant algebraic problem. Numerical analysis estimates of the accuracy and stability of the finite‐volume and the trapezoidal rule approximation are presented and combined with estimates of the hierarchical‐matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical accuracy and convergence estimates, and demonstrate the almost optimal computational complexity of the proposed solution procedure.

Solution of mixed integral equation in position and time using spectral relationships

In this article, the existence of a unique solution of Fredholm–Volterra integral equation of the second kind is guaranteed. The Fredholm integral term is assumed in position with bad kernel, while the Volterra integral term is considered in time with continuous kernel. Under certain conditions and new discussions, the bad kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main theorem of spectral relationships of Fredholm integral equation of the first kind with logarithmic kernel multiplying by a smooth kernel is stated and used to obtain numerically the Fredholm–Volterra integral equation of the second kind. Finally, numerical results are obtained and the error, in each case, is computed.

Numerical treatments for solving nonlinear mixed integral equation

We consider a mixed type of nonlinear integral equation (MNLIE) of the second kind in the space C[0,T]×L2(Ω),T<1. The Volterra integral terms (VITs) are considered in time with continuous kernels, while the Fredholm integral term (FIT) is considered in position with singular general kernel. Using the quadratic method and separation of variables method, we obtain a nonlinear system of Fredholm integral equations (NLSFIEs) with singular kernel. A Toeplitz matrix method, in each case, is then used to obtain a nonlinear algebraic system. Numerical results are calculated when the kernels take a logarithmic form or Carleman function. Moreover, the error estimates, in each case, are then computed.

Nonlinear Fredholm-Volterra Integral Equation and its Numerical Solutions with Quadrature Methods

In this work, the existence and uniqueness of solution of the nonlinear Fredholm-Volterra integral equation ( NF - VIE ), with continuous kernels, are discussed and proved in the space L 2 ( Ω) ×C(0,T). The Fredholm integral term ( FIT ) is considered in position while the Volterra integral term ( VIT ) is considered in time. Using a numerical technique we have a nonlinear system of Fredholm integral equations ( SFIEs ). This system of integral equations can be reduced, using quadrature methods, to a nonlinear algebraic system ( NAS ). Then, the NAS can be solved, using two numerical methods. These methods are: Trapezoidal rule method and Simpson's rule method. Finally, some numerical examples are considered and the error estimate, in each case, is computed.

Toeplitz Matrix Method and Volterra-Hammerstien Integral Equation with a Generalized Singular Kernel

In this work, the existence of a unique solution of Volterra-Hammerstein integral equation of the second kind (V-HIESK) 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-HIESK is reduced to a nonlinear system of Fredholm integral equations (SFIEs). Using Toeplitz matrix method we have a nonlinear algebraic system of equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, some numerical examples when the kernel takes the logarithmic, Carleman, and Cauchy forms, are considered.

Two Dimensional Fredholm Integral Equation with Time

In this paper, the existence and uniqueness of solution of the twodimensional Fredholm integral equation with time, with singular kernel, arediscussed and proved. The Fredholm integral term (FIT) is considered inposition while the Volterra integral term (VIT) is considered in time. Usinga numerical technique we have a system of Fredholm integral equations(SFIEs) in two dimensions . This system of integral equations can be reducedto a linear algebraic system (LAS) of equations by using two differentmethods. These methods are: Toeplitz matrix method and Product Nystrommethod. A numerical examples are considered when the kernel takes thefollowing forms: Carleman function, logarithmic form, Cauchy kernel.

Numerical Solution for Volterra-Ferdholm Integral Equation with A Generalized Singular Kernel

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 Nystr�m 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.

Mixed integral equation in position and time with weakly kernels

Abstract: In this paper, the existence of a unique solution of a mixed integral equation are discussed and proved. The Fredholm integral term is considered in position with weakly kernels, while the Volterra integral terms in time with continuous kernels. Using a numerical method, the mixed integral equation leads us to a linear system of Fredholm integral equations. Then, after using Toeplitz matrix method, we have a linear algebraic system. Many important theorems about the solutions and the estimate errors of the produced linear system are discussed and proved. Finally, numerical examples are considered, and the error, in each case, is calculated.

On the solution of linear and nonlinear integral equation

The purpose of this paper is to establish the solution of Fredholm–Volterra integral equation of the second kind in the space L 2(Ω)×C[0,T] considering the following when, Fredholm integral term in Li 2(Ω) and Volterra integral term, in [0, T], are linear. Also, each of Fredholm or Volterra integral term is linear while the other term is nonlinear. Here, Ω is defined as the domain of integration with respect to position, while t is consider as the time, t∈[0, T], T<∞. Also the Fredholm–Volterra integral equation of the first kind is established as special case of these work. Many special in one, two and three dimensional problems of integral equation are considered. Representing the kernel of Fredholm integral term, in the form of logarithmic function, Carleman function, potential and generalized potential function and Macdonald function are considered.

On a symptotic methods for Fredholm–Volterra integral equation of the second kind in contact problems

A method is used to obtain the general solution of Fredholm–Volterra integral equation of the second kind in the space L 2(Ω)×C(0,T), 0⩽t⩽T<∞;Ω is the domain of integrations. The kernel of the Fredholm integral term belong to C([Ω]×[Ω]) and has a singular term and a smooth term. The kernel of Volterra integral term is a positive continuous in the class C(0, T), while Ω is the domain of integration with respect to the Fredholm integral term. Besides the separation method, the method of orthogonal polynomials has been used to obtain the solution of the Fredholm integral equation. The principal (singular) part of the kernel which corresponds to the selected domain of parameter variation is isolated. The unknown and known functions are expanded in a Chebyshev polynomial and an infinite algebraic system is obtained.

Fredholm–Volterra integral equation with singular kernel

In this paper, under certain conditions, a series in the Legendre polynomials form is used to obtain the solution of Fredholm–Volterra integral equation of the second kind in L 2(−1,1)× C(0, T), T<∞. The uniqueness of the solution is considered. The kernel of Fredholm integral term is established in a logarithmic form in position, while the kernel of Volterra integral term is a positive continuous function in time and belongs to the class C(0, T), T<∞. The solution by series leads us to obtain an infinite system of linear algebraic equations, where the convergence of this system is studied. In the end of this paper, the Fredholm–Volterra integral equation of the first kind is established and its solution is discussed.

Fredholm–Volterra integral equation and generalized potential kernel

A method is used to solve the Fredholm–Volterra integral equation of the first kind in the space L 2(Ω)×C(0,T) , Ω= (x,y)∈ Ω: x 2+y 2 ⩽a, z=0 and T<∞. The kernel of the Fredholm integral term considered in the generalized potential form belongs to the class C([Ω]×[Ω]), while the kernel of the Volterra integral term is a positive and continuous function which belongs to the class C[0, T). Also in this work the solution of the Fredholm integral equation of the first and second kind with a generalized potential kernel is discussed. Many interesting cases are derived and established from the work.

Integral equation of mixed type and integrals of orthogonal polynomials

A method is used to obtain the general solution of the Fredholm–Volterra integral equation of the first kind (FVIEFK) in the space L 2(0,a)×C(0,T), 0⩽x⩽a<∞, 0⩽t⩽T<∞ . The kernel of the Fredholm integral term is considered in a generalized discontinuous form which belongs to C([0, a]×[0, a]), while the kernel of Volterra integral term is a positive continuous function in the class C(0, T). Many interesting cases of integrals of orthogonal polynomials and spectral relationships are obtained and established from this work.

Fredholm-Volterra Integral Equation with Potential Kernel

A method is used to solve the Fredholm-Volterra integral equation of the first kind in the space L2(Ω) × C(O, T), Ω = {(x, y) \(\in\) Ω : \(\sqrt{{x^2}+{y^2}}\) ≤ a, z = 0} and T &lt; ∞. The kernel of the Fredholm integral term is considered in the generalized potential form belongs to the class C ([Ω] × [Ω]), while the kernel of Volterra integral term is a positive and continuous function belongs to the class C[0,T). Also in this work the solution of Fredholm integral equation of the second and first kind with a potential kernel is discussed. Many interesting cases are derived and established from the wok.

Fredholm‐Volterra integral equation with potential kernel

A method is used to solve the Fredholm‐Volterra integral equation of the first kind in the space L2(Ω) × C(0, T), , z = 0, and T &lt; ∞. The kernel of the Fredholm integral term considered in the generalized potential form belongs to the class C([Ω] × [Ω]), while the kernel of Volterra integral term is a positive and continuous function that belongs to the class C[0, T]. Also in this work the solution of Fredholm integral equation of the second and first kind with a potential kernel is discussed. Many interesting cases are derived and established in the paper.