Kind Integral Equations Research Articles

Numerical Solution of Two-Dimensional Fredholm–Volterra Integral Equations of the Second Kind

The paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm–Volterra integral equations of the second kind. As these equations arise in many applications, there is a constant need for accurate, but fast and simple to use numerical approximations to their solutions. The method proposed here uses successive approximations of the Mann type and a suitable cubature formula. Mann’s procedure is known to converge faster than the classical Picard iteration given by the contraction principle, thus yielding a better numerical method. The existence and uniqueness of the solution is derived under certain conditions. The convergence of the method is proved, and error estimates for the approximations obtained are given. At the end, several numerical examples are analyzed, showing the applicability of the proposed method and good approximation results. In the last section, concluding remarks and future research ideas are discussed.

Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

On the Smoothness of the Solution of Fuzzy Volterra Integral Equations of the Second Kind with Weakly Singular Kernels

In this paper. we consider fuzzy Volterra integral equation of the second kind with weakly singular kernel. We prove existence and uniqueness of the solution. When analyzing the convergence of a numerical method for a given integral equation one needs information about the smoothness of the exact solution. We describe the smoothness of the solution, assuming that the sign of the kernel can change only along the horizontal and vertical lines.

Regularization in a functional reproducing kernel Hilbert space

We consider solving a system of semi-discrete first kind integral equations with a right-hand-side being a finite dimensional vector of sampling values and propose a regularization method for the system in a functional reproducing kernel Hilbert space (FRKHS), where the linear functionals that define the semi-discrete integral operator are continuous. A representer theorem for the regularization method is established, which reduces the infinite dimensional problem to a finite dimensional linear system and expresses its solution as a linear combination of the FRKHS kernel sessions. We construct specific FRKHSs and their associated kernels for reconstruction of a function from Radon data, and develop related regularization methods for the reconstruction. We present numerical results which demonstrate that the proposed regularization method outperforms either the traditional Tikhonov regularization in the L2 space or the regularization in the classical reproducing kernel Hilbert space.

Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind

The present paper is devoted to the application of local polynomial and non-polynomial interpolation splines of the third order of approximation for the numerical solution of the Volterra integral equation of the second kind. Computational schemes based on the use of the splines include the ability to calculate the integrals over the kernel multiplied by the basis function which are present in the computational methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra integral equations is also discussed. The results of the numerical experiments are presented.

About Power Absolute Integrated of the Solution of Volterra System of Linear Second Kind Integral Equations on Half-axis

About Power Absolute Integrated of the Solution of Volterra System of Linear Second Kind Integral Equations on Half-axis

Product Block for Volterra Integral Equations of The Second Kind with Singular Kernel.(Dept.P)

Product Block for Volterra Integral Equations of The Second Kind with Singular Kernel.(Dept.P)

Non-Polynomial Spline Function for a Class of Weakly Singular Volterra Integral Equations of the Second Kind

Non-Polynomial Spline Function for a Class of Weakly Singular Volterra Integral Equations of the Second Kind

Application of Splines of the Second Order Approximation to Volterra Integral Equations of the Second Kind. Applications in Systems Theory and Dynamical Systems

This paper discusses the application of local interpolation splines of the second order of approximation for the numerical solution of Volterra integral equations of the second kind. Computational schemes based on the use of polynomial and non-polynomial splines are constructed. The advantages of the proposed method include the ability to calculate the integrals which are present in the computational methods. The application of splines to the solution of nonlinear Volterra integral equations is also discussed. The results of numerical experiments are presented

Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions

Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This paper will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piece-wise polynomials of degree <i>n</i> joined together at the break points with <i>n</i>-1 continuous derivatives. The break points of splines are called Knot, spline function can be integrated and differentiated due to being piece wise polynomials and can easily store and implemented on digital computer, non-polynomial spline function apiece wise is a blend of trigonometric, as well as, polynomial basis function, which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this paper is to compare between Adomain decomposition method and numerical solution to solve Volterra Integral Equations of second kind using the fifth order non-polynomial Spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.

The hypothesis testing statistics in linear ill-posed models

In the geodetic community, an adjustment framework is established by the four components of model choice, parameter estimation, variance component estimation (VCE), and quality control. For linear ill-posed models, the parameter estimation and VCE have been extensively investigated. However, at least to the best of our knowledge, nothing is known about the quality control of hypothesis testing in ill-posed models although it is indispensable and rather important. In this paper, we extend the theory of hypothesis testing to ill-posed models. As the Tikhonov regularization is typically applied to stabilize the solution of ill-posed models, the solution and its associated residuals are biased. We first derive the statistics of overall-test, w-test and minimal detectable bias for an ill-posed model. Due to the bias influence, both overall-test and w-test statistics do not follow the distributions as those used in well-posed models. Then, we develop the bias-corrected statistics. As a result, the bias-corrected w-test statistic can sufficiently be approximated by a standard normal distribution; while the bias-corrected overall-test statistic can be approximated by two non-central chi-square distributions, respectively. Finally, some numerical experiments with a Fredholm first kind integral equation are carried out to demonstrate the performance of our designed hypothesis testing statistics in an ill-posed model.

A Numerical Method for Weakly Singular Nonlinear Volterra Integral Equations of the Second Kind

This paper presents a numerical iterative method for the approximate solutions of nonlinear Volterra integral equations of the second kind, with weakly singular kernels. We derive conditions so that a unique solution of such equations exists, as the unique fixed point of an integral operator. Iterative application of that operator to an initial function yields a sequence of functions converging to the true solution. Finally, an appropriate numerical integration scheme (a certain type of product integration) is used to produce the approximations of the solution at given nodes. The resulting procedure is a numerical method that is more practical and accessible than the classical approximation techniques. We prove the convergence of the method and give error estimates. The proposed method is applied to some numerical examples, which are discussed in detail. The numerical approximations thus obtained confirm the theoretical results and the predicted error estimates. In the end, we discuss the method, drawing conclusions about its applicability and outlining future possible research ideas in the same area.

A Numerical Method for Solving Fredholm Integral Equations of the First Kind with Logarithmic Kernels and Singular Unknown Functions

In this paper, we present a numerical method for solving singular Fredholm integral equations of the first kind. The method is based on the application of the shifted Chebyshev polynomials of the second kind using two techniques. By using the first technique, we solve singular Fredholm integral equations of the first kind with singular logarithmic kernels and singular unknown functions, while the second technique determines the associated data function for a specified exact solution. The unknown function is factorized into two functions; the first is badly-behaved and the second an unknown regular function. By expanding these two functions into shifted Chebyshev polynomials of the second kind through monomial basis functions, the unknown function’s singularity is completely removed. We develop an algebraic formula for evaluating the Chebyshev coefficients instead of using the Chebyshev integral formula. This minimizes the steps of the solution’s procedure and reduces the roundoff errors. The singularity of the kernel is treated by numerical integration. Two boundary integral equations related to the radar, electromagnetism, and scattering engineering problems are solved. The obtained results ensure the efficiency and the high accuracy of the presented method compared with other methods.

A Highly Efficient and Accurate Finite Iterative Method for Solving Linear Two-Dimensional Fredholm Fuzzy Integral Equations of the Second Kind Using Triangular Functions

This work introduces a computational method for solving the linear two-dimensional fuzzy Fredholm integral equation of the second form (2D-FFIE-2) based on triangular basis functions. We have used the parametric form of fuzzy functions and transformed a 2D-FFIE-2 with three variables in crisp case to a linear Fredholm integral equation of the second kind. First, a method based on the use of two m-sets of orthogonal functions of triangular form is implemented on the integral equation under study to be changed to coupled algebraic equation system. In order to solve these two schemes, a finite iterative algorithm is then applied to evaluate the coefficients that provided the approximate solution of the integral problems. Three examples are given to clarify the efficiency and accuracy of the method. The obtained numerical results are compared with other direct and exact solutions.

On the Laguerre fractional integro-differentiation

A fractional power interpretation of the Laguerre derivative is discussed. The corresponding fractional integrals are introduced. Mapping and semigroup properties, integral representations and Mellin transform analysis are presented. A relationship with the Riemann–Liouville fractional integrals is demonstrated. Finally, a second kind integral equation of the Volterra type, involving the Laguerre fractional integral is solved in terms of the double hypergeometric type series as the resolvent kernel.

On One Class of Nonclassical Linear Volterra Integral Equations of the First Kind

On the basis of a new approach, we prove the uniqueness theorem and construct Lavrent’ev’s regularizing operators for the solution of nonclassical linear Volterra integral equations of the first kind with nondifferentiable kernels.

An Accurate and Efficient Technique for Approximating Fuzzy Fredholm Integral Equations of the Second Kind Using Triangular Functions

An Accurate and Efficient Technique for Approximating Fuzzy Fredholm Integral Equations of the Second Kind Using Triangular Functions

An integral equation method for the Cahn-Hilliard equation in the wetting problem

We present an integral equation approach to solving the Cahn-Hilliard equation equipped with boundary conditions that model solid surfaces with prescribed Young's angles. The discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step. To solve it, we split the solution into a volume potential computed with free space kernels, plus the solution to a second kind integral equation (SKIE). The volume potential is evaluated with the help of a box-based volume-FMM method. For non-box domains, the source density is extended by solving a biharmonic Dirichlet problem. The near-singular boundary integrals are computed using quadrature by expansion (QBX) with FMM acceleration. Our method has linear complexity in the number of surface/volume degrees of freedom and can achieve high order convergence in space with adaptive refinement to manage error from function extension.

Численное решение нелинейныхинтегра льных уравнений второго рода типа Урысона методом последовательныхквадра тур с использованием погруженной схемы Дормана-Принса 5(4)

Представлен итерационный алгоритм, который численно решает нелинейные одномерные несингулярные интегральные уравнения Фредгольма и Вольтерры второго рода типа Урысона. Показано, что метод последовательных приближений Пикара может быть использован при численном решении такого типа уравнений. Сходимость числовой схемы гарантируется теоремами о неподвижной точке. При этом квадратурный алгоритм основан на явной форме встроенного правила Рунге–Кутты пятого порядка с адаптивным контролем размера шага. Возможность контроля локальных ошибок квадратур позволяет создавать очень точные автоматические числовые схемы и значительно уменьшить основной недостаток итераций Пикара, а именно чрезвычайно большое количество вычислений с увеличением глубины рекурсии. Наш алгоритм организован так, что по сравнению с большинством подходов нелинейность интегральных уравнений не вызывает каких-либо дополнительных вычислительных трудностей, его очень просто применять и реализовывать в программе. Наш алгоритм демонстрирует практически важные черты универсальности. Во-первых, следует подчеркнуть, что метод столь же прост в применении к нелинейным, как и к линейным уравнениям типа Фредгольма и Вольтерры. Во-вторых, алгоритм снабжен правилами останова, по которым вычисления могут в значительной степени контролироваться автоматически. Представлен компактный C++-код описанного алгоритма. Реализация нашей программы является самодостаточной: она не требует никаких предварительных вычислений, никаких внешних функций и библиотек и не требует дополнительной памяти. Приведены числовые примеры, показывающие применимость, эффективность, надежность и точность предложенного подхода.

Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels

In this paper, we consider product integration method based on orthogonal polynomials to solve mixed system of Volterra integral equations of the first and second kind with weakly singular kernels. For investigation of the theoretical and numerical analysis of the mixed systems, the notions of the tractability index and ν-smoothing property are extended for a weakly singular Volterra integral operator. Convergence analysis of the product integration method is derived. Finally, the proposed method is illustrated by two examples, which confirm the theoretical prediction of the error estimation.