Solution Of Integral Equation Research Articles

Picard and Adomian decomposition methods for a fractional quadratic integral equation via generalized fractional integral

The primary focus of this paper is to thoroughly examine and analyze a class of a fractional quadraticintegral equation via generalized fractional integral. To achieve this, we introduce an operator that possessesfixed points corresponding to the solutions of the fractional quadratic integral equation, effectively transforming thegiven equation into an equivalent fixed-point problem. By applying the Banach fixed-point theorems, we prove theuniqueness of solutions to fractional quadratic integral equation. Additionally, The Adomian decomposition methodis used, to solve the resulting fractional quadratic integral equation. This technique rapidly provides convergentsuccessive approximations of the exact solution to the given fractional quadratic integral equation, therefore, weinvestigate the convergence of approximate solutions, using the Adomian decomposition method. Finally, we providesome examples, to demonstrate our results. Our findings contribute to the current understanding of fractionalquadratic integral equation and their solutions and have the potential to inform future research in this area.

Low rank approximation in the computation of first kind integral equations with TauToolbox

Tau Toolbox is a mathematical library for the solution of integro-differential problems, based on the spectral Lanczos' Tau method. Over the past few years, a class within the library, called polynomial, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework.In this work we discuss how this class has been designed to solve linear ill-posed problems and we provide a description of the available methods, Tikhonov regularization and truncated singular value expansion. For the solution of the Fredholm integral equation of the first kind, which is built from a low-rank approximation of the kernel followed by a numerical truncated singular value expansion, an error estimate is given.Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems are used to evaluate the performance and reliability of the solvers.The final product of this paper is the numerical solution of a first-kind integral equation, which is constructed using only two inputs from the user: the kernel and the right-hand side.

On existence of extremal integrable solutions and integral inequalities for nonlinear Volterra type integral equations

We prove the existence of maximal and minimal integrable solutions of nonlinear Volterra type integral equations. Two basic integral inequalities are obtained in the form of extremal integrable solutions which are further exploited for proving the boundedness and uniqueness of the integrable solutions of the considered integral equation.

Pseudo grid-based physics-informed convolutional-recurrent network solving the integrable nonlinear lattice equations

Traditional discrete learning methods involve discretizing continuous equations using difference schemes, necessitating considerations of stability and convergence. Integrable nonlinear lattice equations possess a profound mathematical structure that enables them to revert to continuous integrable equations in the continuous limit, particularly retaining integrable properties such as conservation laws, Hamiltonian structure, and multiple soliton solutions. The pseudo grid-based physics-informed convolutional-recurrent network (PG-PhyCRNet) is proposed to investigate the localized wave solutions of integrable lattice equations, which significantly enhances the model’s extrapolation capability to lattice points beyond the temporal domain. We conduct a comparative analysis of PG-PhyCRNet with and without pseudo grid by investigating the multi-soliton solutions and rational solitons of the Toda lattice and self-dual network equation. The results indicate that the PG-PhyCRNet excels in capturing long-term evolution and enhances the model’s extrapolation capability for solitons, particularly those with steep waveforms and high wave speeds. Finally, the robustness of the PG-PhyCRNet method and its effect on the prediction of solutions in different scenarios are confirmed through repeated experiments involving pseudo grid partitioning.

Development of an innovative diffraction scattering theory of X-rays and electrons in imperfect crystals.

Fundamental equations describing the X-ray and electron diffraction scattering in imperfect crystals have been derived in the form of the matrix Fredholm-Volterra integral equation of the second kind. A theoretical approach has been developed using the perfect-crystal Green function formalism. In contrast, another approach utilizes the wavefield eigenfunctions related to the diagonalized matrix propagators of the conventional Takagi-Taupin and Howie-Whelan equations. Using the Liouville-Neumann-type series formalism for building up the matrix Fredholm-Volterra integral equation solutions, the general resolvent function solutions of the X-ray and electron diffraction boundary-valued Cauchy problems have been obtained. Based on the resolvent-type solutions, the aim is to reveal the features of the diffraction scattering onto the crystal lattice defects, including the mechanisms of intra- and interbranch wave scattering in the strongly deformed regions in the vicinity of crystal lattice defect cores. Using the two-stage resolvent solution of the second order, this approach has been supported by straightforward calculation of the electron bright- and dark-field contrasts of an edge dislocation in a thick foil. The results obtained for the bright- and dark-field profiles of the edge dislocation are discussed and compared with analogous ones numerically calculated by Howie & Whelan [Proc. R. Soc. A (1962), 267, 206].

Numerical Solution of Volterra's Integral Equations by Collocation and Taylor Method

In this article, two numerical methods (collocation method and Taylor method) are introduced to solve Volterra integral equation system. In the collocation method, using Bessel polynomials and collocation points, we convert the device into a matrix form and solve the device using the matrix form and obtain an approximate solution for the device. This answer is such that the larger the N becomes, the approximate answer is closer to the exact answer of the device. In Taylor method, With the use of the Taylor series, the system of integral equations is transformed into a matrix equation, and by integrating the output of this new system, a system of algebraic equations is created. By working through this device, we can obtain a rough solution for the Integral Equations device. Using these two methods when the answers are polynomial, we can get real answers.

Approximate solution of integral equations based on generalized sampling operators

Approximate solution of integral equations based on generalized sampling operators

ON THE PROPERTIES OF A FAST ALGORITHM FOR SOLVING A THREE-DIMENSIONAL INVERSE PROBLEM OF SCALAR ACOUSTICS

t Theinverse single-frequency problem of scalar acoustics in three-dimensional space is considered. It consists in determining the characteristics of acoustic inhomogeneities lying in a flat layer from the distribution of the complex amplitude of the acoustic field in a flat recording layer. To effectively solve it, a special numerical algorithm is used, previously proposed and justified by the authors. As its component part, the algorithm uses solutions to special one-dimensional Fredholm equations of the 1st kind, and in the case of ambiguity in the solution of the latter, their normal solutions (solutions with a minimum norm) are sought. To regularize this ill-posed inverse problem, Tikhonov regularization and the TSVD method are used. A systematic numerical study is carried out of the influence of various parameters in the data recording scheme on the accuracy of the approximate solutions obtained using the algorithm. In particular, we study the dependence of this accuracy on the position of point sources and composite sources that cause sound vibrations, its dependence on the position of the data recording layer, and on the number of planes in which the recording sensors lie. It is shown that acceptable accuracy of approximate solutions can be obtained even with two such layers. In addition, an approach to the numerical study of the ambiguity of the solution to the inverse problem under consideration is proposed. It is based on special theoretical statements given in the article, and on the same numerical algorithm for solving the inverse problem. The approach is focused on comparing several normal solutions of one-dimensional integral equations from the problem under consideration. These normal solutions are calculated with respect to various elements. An example of using this theory to numerically estimate the non-uniqueness of a solution is given.

Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function

The two-parameter Mittag–Leffler function Eα,β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α>1, the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α∈(0,1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for Eα,β(−t), t≥0, with α∈(1,2). This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants.

Generalizations of Riemann–Liouville fractional integrals and applications

Generalizations of Riemann–Liouville fractional integrals and applications

Radially transverse isotropic inclusions in isotropic elastic media: Local fields, neutral inclusions, effective elastic properties

Radially transverse isotropic inclusions in homogeneous isotropic elastic host media are considered. Mellin transform method is used for solution of the volume integral equation of the problem for an isolated inclusion subjected to a constant external stress (strain) field. The tensor structure of the solution is revealed with precision to three scalar functions of the radial coordinate, and the system of ordinary differential equations for these functions is derived. For multilayered radially transverse isotropic inclusions with constant elastic coefficients inside layers, explicit solution of these equations is obtained. An efficient numerical algorithm of solution for inclusions with an arbitrary number of the layers is proposed. Neutral inclusions that do not disturb homogeneous external fields applied to the medium are considered. It is shown that an inclusion with an isotropic core and radially transverse isotropic external layer can be weak neutral by appropriate choice of the layer elastic constants. The effective field method is used for determination of the effective elastic stiffness tensor of a homogeneous isotropic medium containing a random set of radially transverse isotropic inclusions.

Mohand Transform for Solution of Integral Equations and Abel's Equation

Mohand Transform for Solution of Integral Equations and Abel's Equation

Fixed Point Results in Complex Valued Neutrosophic b-Metric Spaces with Application

In this manuscript, we introduce the idea of complex-valued Neutrosophic b-metric spaces along with numerous significant illustrations. We provide fixed-point results for contraction maps. To support the main result, we establish the existence and uniqueness of solutions for nonlinear integral equations after the work.

Collocation methods for nonlinear Volterra integral equations with oscillatory kernel

This work is devoted to the numerical solution of second kind nonlinear Volterra integral equations with highly oscillatory kernel. We use a collocation approach by discretizing the oscillatory integrals in the collocation equation using a Filon-type quadrature rule. We investigate the convergence of the numerical method in terms of step length h and frequency ω. As h decreases, the suggested technique converges with order d, while its asymptotic order as the frequency increase, is at least 1 and may reach 2 in some cases. Numerical experiments validate theoretical results.

Numerical Solution of Volterra-Hammerstein Integral Equation of the First Kind by Finite Difference Method Decomposition with Nyström Method

Numerical Solution of Volterra-Hammerstein Integral Equation of the First Kind by Finite Difference Method Decomposition with Nyström Method

The Numerical Solutions of weakly singular Fredholm integral equations of the Second kind Using Chebyshev Polynomials of the Second Kind

In this study, the second kind Chebyshev Polynomials were utilized to acquire interpolated solutions for the second kind Fredholm integral equations with weakly singular kernel. To accomplish this, the data, unknown, and kernel functions were converted into matrix form, and consequently we completely isolated the singularity of the kernel. The primary benefit of this method is the ability to change the form of integral equation to an equivalent algebraic system, which is easier to solve. The effectiveness of our technique was evaluated by applying it to three illustrated examples, and it was observed that the solutions obtained exhibit strong convergence towards the exact solutions.

SOLUTION OF FUZZY VOLTERRA INTEGRAL EQUATIONS USING COMPOSITE METHOD

With the rapid development of of fuzzy Integral equations, the problem of finding its solution becomes increasingly important for knowledge propagation and putting researchers wisdom to work. A recent development trend of fuzzy integral equations is modeling many problems in the field of science and technology. If the unknown function in the considered equation has a solution in terms of infinite series expansion, this proposed method becomes more accurate to find the exact solution. According on the parametric form of a fuzzy number, a fuzzy integral equation converts to two systems of integral equations in the crisp case, and then proposed method is achieved to obtain the exact fuzzy solutions of fuzzy Volterra integral equations. To validate the effectiveness of proposed method, some examples of considered equations are solved and the results show that it achieved the best performance.

Fixed point results for Geraghty–Ćirić-type contraction mappings in b-metric space with applications

In this study, we manifest a new class of mappings that satisfy Geraghty–Ćirić-type contractive conditions in the context of b-metric spaces and prove a theorem on the existence and uniqueness of fixed points. Our results unify and generalize the results of Geraghty; Ćirić; Dukic, Kadelburg, and Radenović; and Shu-fang Li, Fei Hi, and Ning Lu in the setting of b-metric spaces. Furthermore, we provide examples to verify the correctness and applicability of our results. We also utilize our findings to show the existence of a unique solution for a nonlinear integral equation.

An Improvement Meshless Method for the Numerical Solution of Two-Dimensional Stochastic Fredholm Integral Equations

An Improvement Meshless Method for the Numerical Solution of Two-Dimensional Stochastic Fredholm Integral Equations

Novel results for separate families of fuzzy-dominated mappings satisfying advanced locally contractions in b-multiplicative metric spaces with applications

The objective of this research is to present new fixed point theorems for two separate families of fuzzy-dominated mappings. These mappings must satisfy a unique locally contraction in a complete b-multiplicative metric space. Also, we have obtained novel results for families of fuzzy-dominated mappings on a closed ball that meet the requirements of a generalized locally contraction. This research introduces new and challenging fixed-point problems for families of ordered fuzzy-dominated mappings in ordered complete b-multiplicative metric spaces. Moreover, we demonstrate a new concept for families of fuzzy graph-dominated mappings on a closed ball in these spaces. Additionally, we present novel findings for graphic contraction endowed with graphic structure. These findings are groundbreaking and provide a strong foundation for future research in this field. To demonstrate the uniqueness of our novel findings, we provide evidence of their applicability in obtaining the common solution of integral and fractional differential equations. Our findings have resulted in modifications to several contemporary and classical results in the research literature. This provides further evidence of the originality and impact of our work.