a collocation method via the quasi affine biorthogonal systems for solving weakly singular type of volterra fredholm integral equations

Abstract Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B-spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples

Solution of systems of disjoint Fredholm-Volterra integro-differential equations using Bezier control points

Systems of disjoint Fredholm-Volterra integro-differential equations and the Bezier curves control-point-based algorithm are considered. Systems of two, three and four Fredholm-Volterra integro-differential equations are solved using a developed algorithm. The convergence analysis for the Bezier curves method proves that it is convergent. The examples considered agree with the convergence analysis. The method is more accurate and effective when compared to other existing methods

Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm

A Computational Method for Solving a Class of Fractional-Order Non-Linear Singularly Perturbed Volterra Integro-Differential Boundary-Value Problems

In this thesis, we present a computational method for solving a class of fractional singularly perturbed Volterra integro-differential boundary-value problems with a boundary layer at one end. The implemented technique consists of solving two problems which are a reduced problem and a boundary layer correction problem. The reproducing kernel method is used to the second problem. Pade’ approximation technique is used to satisfy the conditions at infinity. Existence and uniformly convergence for the approximate solution are also investigated. Numerical results provided to show the efficiency of the proposed method

Approximate Solution of Second-Order Integrodifferential Equation of Volterra Type in RKHS Method

Abstract In this paper, an application of reproducing kernel Hilbert space (RKHS) method is applied to solve second-order integrodifferential equation of Volterra type. The analytical solution is represented in the form of series in the reproducing kernel space. The n−truncation approximation u n (x) is obtained and proved to converge to the analytical solution u(x). Moreover, the presented method has an advantages that it is possible to pick any point in the interval domain and as well the approximate solution and its derivatives will be applicable Numerical experiments are displayed to illustrate the validity, accuracy, efficiency and applicability of the proposed method. Results indicates that our technique is simple, straightforward and effective. Mathematics Subject Classification: 47B32, 45J05, 34K2

Iterative Reproducing Kernel Method for Solving Second-Order Integrodifferential Equations of Fredholm Type

We present an efficient iterative method for solving a class of nonlinear second-order Fredholm integrodifferential equations associated with different boundary conditions. A simple algorithm is given to obtain the approximate solutions for this type of equations based on the reproducing kernel space method. The solution obtained by the method takes form of a convergent series with easily computable components. Furthermore, the error of the approximate solution is monotone decreasing with the increasing of nodal points. The reliability and efficiency of the proposed algorithm are demonstrated by some numerical experiments

A Novel Method for the Solution of the Schroedinger Eq. in the Presence of Exchange Terms

In the Hartree-Fock approximation the Pauli exclusion principle leads to a Schroedinger Eq. of an integro-differential form. We describe a new spectral noniterative method (S-IEM), previously developed for solving the Lippman-Schwinger integral equation with local potentials, which has now been extended so as to include the exchange nonlocality. We apply it to the restricted case of electron-Hydrogen scattering in which the bound electron remains in the ground state and the incident electron has zero angular momentum, and we compare the acuracy and economy of the new method to three other methods. One is a non-iterative solution (NIEM) of the integral equation as described by Sams and Kouri in 1969. Another is an iterative method introduced by Kim and Udagawa in 1990 for nuclear physics applications, which makes an expansion of the solution into an especially favorable basis obtained by a method of moments. The third one is based on the Singular Value Decomposition of the exchange term followed by iterations over the remainder. The S-IEM method turns out to be more accurate by many orders of magnitude than any of the other three methods described above for the same number of mesh points.Comment: 29 pages, 4 figures, submitted to Phys. Rev.

Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert space 1 2 [ , ] in order to formulate the analytical solutions in a rapidly convergent series form in terms of their -cut representation. The approximation solution is expressed by -term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations

New Challenges Arising in Engineering Problems with Fractional and Integer Order

Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem

Error Analyses for Nyström Methods for Solving Fredholm Integral and Integro-Differential Equations

This thesis concerns the development and implementation of novel error analyses for ubiquitous Nyström-type methods used in approximating the solution in 1-D of both Fredholm integral- and integro-differential equations of the second-kind, (FIEs) and (FIDEs). The distinctive contribution of the present work is that it offers a new systematic procedure for predicting, to spectral accuracy, error bounds in the numerical solution of FIEs and FIDEs when the solution is, as in most practical applications, a priori unknown. The classic Legendre-based Nyström method is extended through Lagrange interpolation to admit solution of FIEs by collocation on any nodal distribution, in particular, those that are optimal for not only integration but also differentiation. This offers a coupled extension of optimal-error methods for FIEs into those for FIDEs. The so-called FIDE-Nyström method developed herein motivates yet another approach in which (demonstrably ill-conditioned) numerical differentiation is bypassed by reformulating FIDEs as hybrid Volterra-Fredholm integral equations (VFIEs). A novel approach is used to solve the resulting VFIEs that utilises Lagrange interpolation and Gaussian quadrature for the Volterra and Fredholm components respectively. All error bounds implemented for the above numerical methods are obtained from novel, often complex extensions of an established but hitherto-unimplemented theoretical Nyström-error framework. The bounds are computed using only the available computed numerical solution, making the methods of practical value in, e.g., engineering applications. For each method presented, the errors in the numerical solution converge (sometimes exponentially) to zero with N, the number of discrete collocation nodes; this rate of convergence is additionally confirmed via large-N asymptotic estimates. In many cases these bounds are spectrally accurate approximations of the true computed errors; in those cases that the bounds are not, the non-applicability of the theory can be predicted either a priori from the kernel or a posteriori from the numerical solution