Dimensional Fredholm Research Articles

Infinite Geraghty type extensions and their applications on integral equations

In this article, we discuss about a series of infinite dimensional extensions of some theorems given in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018), (Fisher in Math. Mag. 48(4):223–225, 1975), and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). We also prove a similar Geraghty type construction for Fisher (Math. Mag. 48(4):223–225, 1975) in an infinite dimension using similar techniques as in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018) and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). As an application, we ensure the existence of solutions for infinite dimensional Fredholm integral equation and Uryshon type integral equation.

Numerical Solution of Two Dimensional Fredholm Integral Equations of the Second Kind by the Barycentric Lagrange Function*

This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by this thought, we convert the equations into the associated algebraic equations. The results of the numerical examples are given to illustrate that the approximated method is feasible and efficient.

Padé-type approximation method for two dimensional Fredholm integral equations of the second kind

Two dimensional Fredholm integral equation of the second kind (2DF-II) on a bounded domain D is regarded as the problem with characteristic values. A two dimensional function-valued Padé-type approximation(2DFPTA) is defined. Its error formulas and convergence theorems are presented. To obtain higher order 2DFPTA, a determinantal expression and its recursive algorithm are given. In the end three numerical examples are tested, where one on the unit triangle of vertices (0,0),(0,1),(1,0) and the other two on the square. The testing results show that 2DFPTA method is more accurate.

FREDHOLM PROPERTY OF COMPOSITE TWO-DIMENSIONAL INTEGRAL OPERATORS WITH HOMOGENEOUS SINGULAR-TYPE KERNELS IN pL SPACE

The authors have previously studied two - dimensional Fredholm integral operators with homogeneous kernels of fiber - singular type. For this class of operators, the symbolic calculus is built using the theory of biloc al operators by V. Pilidi, and Fredholm criterion is formulated through the inversibility of t wo families: the family of one - dimensional convolution operators, and the family of one - dimensional singular integral operators with continuous coefficients. The aim of this work is to study composite two - dimensional integral operators with homogeneous ker nels of fiber singular type analogous to Simonenko’s continual convolution integral operators. This investigation is a part of a more general study of algebra of operators with homogeneous kernels which layers are singular operat ors with piecewise continuo us coefficients. For the considered operators, the symbolic calculus and the necessary and sufficient Fredholm conditions are obtained.

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.

Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind

In this paper, an efficient method is presented for solving two dimensional Fredholm and Volterra integral equations of the second kind. Chebyshev polynomials are applied to approximate a solution for these integral equations. This method transforms the integral equation to algebraic equations with unknown Chebyshev coefficients. The high accuracy of this method is verified through some numerical examples. Mathematical subject classification: 65R20, 41A50, 41A55, 65M70.

Two-dimensional wavelets for numerical solution of integral equations

Purpose: In this paper, we shall investigate the numerical solution of two-dimensional Fredholm integral equations (2D-FIEs). Methods: In this work, we apply two-dimensional Haar wavelets, to solve linear two dimensional Fredholm integral equations (2D-FIEs). Using 2D Haar wavelets and their properties, 2D-FIEs of the second kind reduce to a system of algebraic equations. Results: The numerical examples illustrate the efficiency and accuracy of the method. Conclusions: In comparison with other bases (for example, polynomial bases), one of the advantages of this method is, although the involved matrices have a large dimension, they contain a large percentage of zero entries, which keeps computational effort within reasonable limits.

A fast numerical solution method for two dimensional Fredholm integral equations of the second kind based on piecewise polynomial interpolation

In this paper we consider fast numerical solution methods for two dimensional Fredholm integral equation of the second kind f ( x , y ) - ∫ α β ∫ α β a ( x , y , u , v ) f ( u , v ) du dv = g ( x , y ) , ( x , y ) ∈ [ α , β ] × [ α , β ] , where a( x, y, u, v) is smooth and g( x, y) is in L 2[ α, β] 2. Discretizing the integral equation by certain quadrature rule, we get a linear system. To deduce fast approximate solution methods for the resulted linear system, we study the approximation of the four-variable kernel function a( x, y, u, v) by piecewise polynomial: partition the domain [ α, β] 4 into subdomains of the same size and interpolate the kernel function a( x, y, u, v) in each subdomain. Fast matrix–vector multiplication algorithms and efficient iterative methods are derived. Numerical results are given to illustrate the efficiency of our methods.

A fast numerical solution method for two dimensional Fredholm integral equations of the second kind

In this paper we consider numerical solution methods for two dimensional Fredholm integral equation of the second kind f ( x , y ) − ∫ − 1 1 ∫ − 1 1 a ( x , y , u , v ) f ( u , v ) d u d v = g ( x , y ) , ( x , y ) ∈ [ − 1 , 1 ] × [ − 1 , 1 ] , where a ( x , y , u , v ) is smooth and g ( x , y ) is in L 2 [ − 1 , 1 ] 2 . We discuss polynomial interpolation methods for four-variable functions and then use the interpolating polynomial to approximate the kernel function a ( x , y , u , v ) . Based on the approximation we deduce fast matrix-vector multiplication algorithms and efficient preconditioners for the above two dimensional integral equations. The residual correction scheme is used to solve the discretization linear system.

A numerical method for solving a class of functional and two dimensional integral equations

In this paper, the Chebyshev collocation method has been used to solve the functional integral equation of the first and second kind. Also the numerical solution of the two dimensional Fredholm–Volterra integral equations (F–VIE) of the second kind is considered. The Chebyshev collocation method transforms any integral equation into a system of linear algebraic equations. In this method the Chebyshev expansion coefficients of the solution is obtained. Finally some examples show the accuracy of this method.

Transversality for equivariant exact 1-forms and gauge theory on 3-manifolds

An equivariant jet transversality framework is developed for the study of critical sets of invariant functions on G manifolds. Techniques are developed to extend transversality results to the infinite dimensional Fredholm setting. As an application, the generic structure of the SU ( 4 ) perturbed flat moduli space of an integral homology three-sphere is described, as well as the generic structure of the parameterized moduli space for a path of perturbations. A similar analysis of the U ( 3 ) moduli space for rational homology three-spheres is also carried out.