Solution Of Integral Equation Research Articles

Multi-level Power Series Solution for Large Surface and Volume Electric Field Integral Equation

In this paper, we propose a new multi-level power series solution method for solving a large surface and volume electric field integral equation-based H-Matrix. The proposed solution method converges in a fixed number of iterations and is solved at each level of the H-Matrix computation. The solution method avoids the computation of a full matrix, as it can be solved independently at each level, starting from the leaf level. Solution at each level can be used as the final solution, thus saving the matrix computation time for full H-Matrix. The paper shows that the leaf level matrix computation and solution with power series gives as accurate results as the full H-Matrix iterative solver method. The method results in considerable savings time and memory savings compared to the H-Matrix iterative solver. Further, the proposed method retains the O(NlogN) solution complexity.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

We have four kinds of solutions for Cauchy type integral equations: by expanding on known functions and using the Maclaurin series, we can convert these four kinds of solutions into linear combinations of some elements that are basis of these solutions. Using these bases gives exact solutions for polynomials and, for some other functions, a high-accuracy approximate solution.

An iterative Nyström-based method to solve nonlinear Fredholm integral equations of the second kind

In the current paper, an iterative numerical scheme based upon the quasilinearization technique and Nyström method to find the approximate solution of the nonlinear Fredholm integral equations of the second kind is provided. First, using the quasilinearization method, the nonlinear Fredholm integral equation is reduced to a sequence of linear Fredholm integral equations. Under some suitable assumptions, the exact solutions of this sequence of linear integral equations converge to the unique solution of the original problem quadratically. Then in each iteration, utilizing the linear barycentric rational quadrature, the linear Fredholm integral equation is approximated with the Nyström method. Convergence analysis of the method is investigated thoroughly. Numerical examples and comparisons with other existing methods are provided to confirm the theoretical results and demonstrate the performance and validity of the numerical method.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

Existence of solution of Erdelyi-Kober fractional integral equations using measure of non-compactness

Existence of solution of Erdelyi-Kober fractional integral equations using measure of non-compactness

Approximate Analytical Solution of Fuzzy Linear Volterra Integral Equation via Elzaki ADM

In this paper, the fuzzy Volterra integral equations’ solutions are calculated using a hybrid methodology. The combination of the Elzaki transform and Adomian decomposition method results in the development of a novel regime. The precise fuzzy solutions are determined using Elzaki ADM after the fuzzy linear Volterra integral equations are first translated into two crisp integral equations utilizing the fuzzy number in parametric form. Three instances of the considered equations are solved to show the established scheme’s dependability, efficacy, and application. The results have a substantial impact on the fuzzy analytical dynamic equation theory. The comparison of the data in a graphical and tabular format demonstrates the robustness of the defined regime. The lower and upper bound solutions’ theoretical convergence and error estimates are highlighted in this paper. A tolerable order of absolute error is also obtained for this inquiry, and the consistency of the outcomes that are approximated and accurate is examined. The regime generated effective and reliable results. The current regime effectively lowers the computational cost, and a faster convergence of the series solution to the exact answer is signaled.

Finite temperature properties of an integrable zigzag ladder chain

We consider the interaction-round-a-face version of the six-vertex model for arbitrary anisotropy parameter, which allow us to derive an integrable one-dimensional quantum Hamiltonian with three-spin interactions. We apply the quantum transfer matrix approach for the face model version of the six-vertex model. The integrable quantum Hamiltonian shares some thermodynamical properties with the Heisenberg XXZ chain, but has different ordering and critical exponents. Two gapped phases are the dimerized antiferromagnetic order and the usual antiferromagnetic (Néel) order for positive nearest neighbor Ising coupling. In between these, there is an extended critical region, which is a quantum spin-liquid with broken parity symmetry inducing an oscillatory behavior at the long distance <σ1zσℓ+1z> correlation. At finite temperatures, the numerical solution of the non-linear integral equations allows for the determination of the correlation length as well as for the momentum of the oscillation.

Numerical solution of stochastic Volterra integral equations based on uniform Haar wavelets by using direct method

Nowadays, wavelets have played a crucial role in approximating the solution of a wide range of problems arising in science and engineering. Wavelets have been used in numerous areas of applied mathematics as diverse as signal analysis, statistics, computer-aided geometric, image processing and numerical analysis. The uniform Haar wavelets are most widely used wavelets for solving stochastic integral equations, outperforming Legendre, Daubechies, etc. The uniform Haar wavelets are generated from pairs of piecewise constant functions and can be simply integrated. Furthermore, the uniform Haar functions are orthogonal and form a good transform basis. In this paper, we have proposed an efficient numerical method to solve a class of stochastic linear Volterra integral equations (SLVIEs) based on the uniform Haar wavelets. The aim of this paper is to apply numerical method to obtain approximate solution of SLVIEs. By using this method, the SLVIEs can be reduced to a linear system of algebraic equations. Also, the error analysis of this method is performed. Finally, some numerical examples are given to demonstrate pertinent features of this method. Furthermore, we compare obtained results from this method with the achieved results from relevant studies.

A numerical investigation of crack behavior near a fixed boundary using singular integral equation and finite element methods

In this paper, we present the method of integral transforms and the Gauss-Chebyshev quadrature methods to solve the problem of a crack parallel to a fixed boundary under remote tension. We derive a system of singular integral equations of the first kind, specific to the problem at hand, which we numerically solve using Gauss-Chebyshev integration. We specialize our results to the problem of a crack in an infinite plate under remote tension, and show that the relative error in our numerically derived solutions is within machine precision of the closed-form analytical solutions. Stress intensity factors are calculated that are in excellent agreement with those derived by others using different methods. We also demonstrate that both the stress intensity factors and normal σyy(x,y) and shear σxy(x,y) stress fields derived via numerical solution of the singular integral equations, compare well with those determined using the commercially available Abaqus finite element code where the crack is modeled using the eXtended Finite Element Method.

New fixed point theorems via measure of noncompactness and its application on fractional integral equation involving an operator with iterative relations

In this paper, Darbo’s fixed point theorem is generalized and it is applied to find the existence of solution of a fractional integral equation involving an operator with iterative relations in a Banach space. Moreover, an example is provided to illustrate the results.

A Class of Relational Functional Contractions with Applications to Nonlinear Integral Equations

In this article, we investigate some fixed-point results under certain functional contractive mappings in a relation metric space. In the process, we utilize more general contraction condition which must be verified for comparative elements only. Our results enrich, modify, refine, unify and sharpen several existing fixed-point results. We construct some examples in support of our results. To attest to the applicability of our results, we establish the existence and uniqueness of theorems regarding the solutions of certain nonlinear integral equations.

Numerical solution of the two-dimensional first kind Fredholm integral equations using a regularized collocation method

Numerical solution of the two-dimensional first kind Fredholm integral equations using a regularized collocation method

Superconvergent Nyström Method Based on Spline Quasi-Interpolants for Nonlinear Urysohn Integral Equations

Integral equations play an important role for their applications in practical engineering and applied science, and nonlinear Urysohn integral equations can be applied when solving many problems in physics, potential theory and electrostatics, engineering, and economics. The aim of this paper is the use of spline quasi-interpolating operators in the space of splines of degree d and of class Cd−1 on uniform partitions of a bounded interval for the numerical solution of Urysohn integral equations, by using a superconvergent Nyström method. Firstly, we generate the approximate solution and we obtain outcomes pertaining to the convergence orders. Additionally, we examine the iterative version of the method. In particular, we prove that the convergence order is (2d+2) if d is odd and (2d+3) if d is even. In case of even degrees, we show that the convergence order of the iterated solution increases to (2d+4). Finally, we conduct numerical tests that validate the theoretical findings.

Distributed source scheme for Poisson equation using finite element method

The finite-element method (FEM) is one of the most versatile numerical techniques to solve partial differential equations (PDEs). The approximations in FEM result in numerous errors and various techniques are available to reduce the errors. Distributed Source Scheme (DSS) is a recently proposed method to reduce error around a single source of field which obeys the Poisson equation using the 3-D finite-difference method (FDM). DSS achieves this feat by introducing pseudo sources around the singular source. In this article, the proposed method is studied for single and bulk sources using 2-D FEM along with experimental validation. The numerical solution from FEM is compared with the approximate solution from integral equations over the whole computational domain for the error analysis. The error due to boundary values are isolated by considering the solution of integral equations on boundary elements in FEM. The maximum error in the numerical solution of a single source is reduced from 4.96% to 3.28×10−4% using DSS and to 0.55% using Truncated Distributed Source Scheme (TDSS) with a truncation number of 5. The mean difference in numerical and approximate potential at the boundaries of bulk sources is reduced from 1.21×105 Volt to 2.69×104 Volt using DSS.

Approximate solutions of fuzzy delay integral equations with weakly singular kernels by piecewise fuzzy polynomial interpolation

In this paper, a piecewise fuzzy polynomial interpolation method is proposed to approximate the solutions of fuzzy delay integral equations with weakly singular kernels. Firstly, the existence and uniqueness of the solutions of these fuzzy delay integral equations are proved by using the Gronwall inequality and an iterative method. Then, the fuzzy weakly singular delay integral equations are discretized into related algebraic equations by using piecewise fuzzy polynomial interpolation and the fuzzy Gauss-Jacobi quadrature formula. Moreover, the convergence of the numerical method is investigated and an error estimate is provided. Finally, the accuracy and efficiency of the proposed method are verified by some numerical experiments.

Solving a singular integral equation for the one-dimensional Coulomb problem

A new integral equation that describes the behavior of the momentum space wave function for the one-dimensional Coulomb potential is proposed. The obtained result turned out to be a homogeneous Fredholm integral equation of the second kind and a singular integral equation, because its kernel has a singularity at some point in the momentum space. A nontriviality of the method of solving this singular integral equation lies in the application of the integral representation for its integral kernel. The technique applied in this paper made it possible to show that the wave function in the momentum representation is simultaneously a solution of the homogeneous Fredholm integral equation of the second kind and of the linear Volterra integral equation of the second kind. Since a linear Volterra integral equation of the second kind was easily transformed into a second order linear inhomogeneous differential equation with constant coefficients, the eigenfunctions and eigenvalues in the one-dimensional Coulomb problem were found without any difficulties. Such a circumstance may indicate the validity of the new integral equation and the proposed method of its solving.

On the approximation of highly oscillatory Volterra integral equations of the first kind via Laplace transform

This paper focuses on approximation of Volterra integral equations of the first kind with highly oscillatory Bessel kernel. We first give a lemma about the coefficient relations among several different interpolation polynomials of continuous functions g(x), and present the existence and uniqueness theorem of the solution of the Volterra integral equation. Based on the Laplace transform and inverse Laplace transform, we derive the explicit formulas for the solution of the first kind integral equation. Furthermore, based on the asymptotic of the solution for large values of the parameters, we deduce some simpler formulas for approximating the solution. Finally, we present two numerical methods that is the efficient quadrature rule and the Clenshaw–Curtis-type method, which can efficiently calculate highly oscillatory integrals in the solution of the Volterra integral equation.

Singularity resolving in solution of the boundary integral equation in two-dimensional vortex methods

The problem of 2D incompressible flow simulation around airfoils with sharp edges and corner point is considered. The solution of the boundary integral equation with respect to vortex sheet intensity arising in Lagrangian vortex method has weak singularity that cannot be resolved correctly in the framework of the existing Galerkin-type numerical schemes. It is shown that for piecewise-smooth bounded solutions the known schemes allow for solution reconstruction with high quality and provide the 2-nd order of accuracy, while for singular solution their order of accuracy goes down to the 1-st. A numerical scheme is suggested that allows for solution singularity resolving and provides the 2-nd order of accuracy. As a model problem, the added mass tensor components computation is considered, since its exact value is known for the Joukowsky wing airfoil with sharp edge (cusp point).

Hybrid Barnes – Hut/Multipole algorithm application to vortex particles velocities calculation and integral equation solution

The approximate fast algorithm is developed that makes it possible to calculate velocities of the vortex particles in two-dimensional flow simulations by using vortex methods. It is adapted to efficient solution of the boundary integral equation arising at each time step of simulation. The proposed method can be considered as generalization of the classical Barnes – Hut method, taking into account some ideas of the Fast Multipole Method. Some model problems are considered that can be solved by using vortex methods, the developed fast algorithm is implemented for multi-cores CPUs and (with some modifications) for GPUs. The scalability of the code is rather high; the developed algorithms can run with millions of vortex particles taking only tens or hundreds of milliseconds per time step; linear systems with dimension of about 104 can be solved on multicore CPU in tens milliseconds.

Common fixed point result for multivalued mappings with applications to systems of integral and functional equations

In this paper, we establish a common fixed point theorem for two multivalued mappings satisfying some dominated conditions on a complete metric space. This new rational type contractive inequality refines various results in the literature. The main theorem is illustrated with examples. As application, we found the existence of solutions of system nonlinear integral equations and functional equations.