Semiorthogonal B-spline Wavelet for Solving 2D- Nonlinear Fredholm-Hammerstein Integral Equations

This work is concerned with the study of the second order (linear) semiorthogonal B-spline wavelet method to solve one-dimensional nonlinear Fredholm-Hammerstein integral equations of the second kind. Proof of the existence and uniqueness solution for the two-dimensional Fredholm-Hammerstein nonlinear integral equations of the second kind was introduced. Moreover, generalization the second order (linear) semiorthogonal B-spline wavelet method was achieved and then using it to solve two-dimensional nonlinear Fredholm-Hammerstein integral equations of the second kind. This method transform the one-dimensional and two-dimensional nonlinear Fredholm-Hammerstein integral equations of the second kind to a system of algebraic equations by expanding the unknown function as second order (linear) semiorthogonal B-spline wavelet with unknown coefficients. The properties of these wavelets functions are then utilized to evaluate the unknown coefficients. Also some of illustrative examples which show that the second order (linear) semiorthogonalB-spline wavelet method give good agreement with the exact solutions

Solution of Integral Equation using Second and Third Order B-Spline Wavelets

It was proven that semi-orthogonal wavelets approximate the solution of integral equation very finely over the orthogonal wavelets Here we used the compactly supported semi-orthogonal B-spline wavelets generated in our paper Compactly Supported B-spline Wavelets with Orthonormal Scaling Functions satisfying the Daubechies conditions to solve the Fredholm integral equation The generated wavelets satisfies all the properties on the bounded interval The method is computationally easy which is illustrated with two examples whose solution closely resembles the exact solution as the order of wavelet increase

Superconvergence of Iterated Solutions for Linear and Nonlinear Integral Equations: Wavelet Applications

In this dissertation, we develop the Petrov-Galerkin method and the iterated Petrov-Galerkin method for a class of nonlinear Hammerstein equation. We also investigate the superconvergence phenomenon of the iterated Petrov-Galerkin and degenerate kernel numerical solutions of linear and nonlinear integral equations with a class of wavelet basis. The Fredholm integral equations and the Hammerstein equations are considered in linear and nonlinear cases respectively. Alpert demonstrated that an application of a class of wavelet basis elements in the Galerkin approximation of the Fredholm equation of the second kind leads to a system of linear equations which is sparse. The main concern of this dissertation is to address the issue of how this sparsity manifests itself in the setting of nonlinear equations, particularly for Hammerstein equations. We demonstrate that sparsity appears in the Jacobian matrix when one attempts to solve the system of nonlinear equations by the Newton\u27s iterative method. Overall, the dissertation generalizes the results of Alpert to nonlinear equations setting as well as the results of Chen and Xu, who discussed the Petrov-Galerkin method for Fredholm equation, to nonlinear equations setting

A numerical solution of fredholm integral equations of the second kind based on tight framelets generated by the oblique extension principle

© 2019 by the authors. In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the integral equation to a system of linear equations. We provide an example of the construction of quasi-affine tight framelet systems. We also give some numerical evidence to illustrate our method. The numerical results confirm that the method is efficient, very effective and accurate

Solving Linear Volterra – Fredholm Integral Equation of the Second Type Using Linear Programming Method

في هذا البحث تم عرض تقنية جديدة لإيجاد حل لثلاثة أنواع من المعادلات التكاملية الخطية من النوع الثاني المتضمنة: معادلة فولتيرا-فريدهولم التكاملية (LVFIE) ( الحالة العامة), معادلة فولتيرا التكاملية (LVIE) و معادلة فريدهولم التكامليىة (LFIE) (كحالتين خاصتين). التقنية الجديدة تعتمد على تقريب الحل الى متعددة حدود من الدرجة (m-1) وبعد ذلك تحويل المسألة الى مسألة برمجة خطية (LPP) والتي سوف تحل لايجاد الحل التقريبي لمعادلة فولتيرا- فريدهولم التكاملية الخطية من النوع الثاني(LVFIE) . علاوة على ذلك تم استخدام الطرق التربيعية التي تضم: قاعدة شبه المنحرف (TR), قاعدة سمبسون 3/1 (SR), قاعدة بول (BR) وصيغة رومبرك للتكامل (RI) لتقريب التكامل الموجود في LVFIE , كما تم عمل مقارنات بين هذه الطرق. واخيرا لزيادة التوضيح تم اعطاء الخوارزمية المتبعة في الحل وتم تطبيقها على امثلة اختبارية لتوضيح فعالية التقنية الجديدة.In this paper, a new technique is offered for solving three types of linear integral equations of the 2nd kind including Volterra-Fredholm integral equations (LVFIE) (as a general case), Volterra integral equations (LVIE) and Fredholm integral equations (LFIE) (as special cases). The new technique depends on approximating the solution to a polynomial of degree  and therefore reducing the problem to a linear programming problem(LPP), which will be solved to find the approximate solution of LVFIE. Moreover, quadrature methods including trapezoidal rule (TR), Simpson 1/3 rule (SR), Boole rule (BR), and Romberg integration formula (RI) are used to approximate the integrals that exist in LVFIE. Also, a comparison between those methods is produced. Finally, for more explanation, an algorithm is proposed and applied for testing examples to illustrate the effectiveness of the new technique

Fast wavelet collocation methods for second kind integral equations on polygons

In this thesis we develop fast wavelet collocation methods for integral equations of the second kind with weakly singular kernels over polygons . For this purpose, we construct multiscale wavelet functions and collocation functionals having vanishing moments. Moreover, we propose several truncation strategies, which lead to fast algorithms, for the coefficient matrix of the corresponding discrete system. Critical issues for numerical implementation of such methods are considered, such as choices of practical truncation strategies, numerical integration of weakly singular integrals, error controls of numerical quadrature and numerical solutions of resulting compressed linear systems. Numerical experiments are given to demonstrate proposed ideas and methods. Finally, parallel computing using developed methods is investigated.;That this work received partial support from the US NSF grant EPSCoR-0132740

Numerical Methods for Integral Equations

We first propose a multiscale Galerkin method for solving the Volterra integral equations of the second kind with a weakly singular kernel. Due to the special structure of Volterra integral equations and the ``shrinking support property of multiscale basis functions, a large number of entries of the coefficient matrix appearing in the resulting discrete linear system are zeros. This result, combined with a truncation scheme of the coefficient matrix, leads to a fast numerical solution of the integral equation. A quadrature method is designed especially for the weakly singular kernel involved inside the integral operator to compute the nonzero entries of the compressed matrix so that the quadrature errors will not ruin the overall convergence order of the approximate solution of the integral equation. We estimate the computational cost of this numerical method and its approximate accuracy. Numerical experiments are presented to demonstrate the performance of the proposed method. We also exploit two methods based on neural network models and the collocation method in solving the linear Fredholm integral equations of the second kind. For the first neural network (NN) model, we cast the problem of solving an integral equation as a data fitting problem on a finite set, which gives rise to an optimization problem. In the second method, which is referred to as the NN-Collocation model, we first choose the polynomial space as the projection space of the Collocation method, then approximate the solution of the integral equation by a linear combination of polynomials in that space. The coefficients of the linear combination are served as the weights between the hidden layer and the output layer of the neural network. We train both neural network models using gradient descent with Adam optimizer. Finally, we compare the performances of the two methods and find that the NN-Collocation model offers a more stable, accurate, and efficient solution

Harmonic wavelet method towardssolution of the Fredholm type integral equations of the secondkind

Periodic harmonic wavelets (PHW) were applied as basis functions in solution of the Fredholm integral equations of the second kind. Two equations were solved in order to find out advantages and disadvantages of such choice of the basis functions. It is proved that PHW satisfy the properties of the multiresolution analysis