A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

In this paper, we discuss a new model to obtain the answer to the following question: how can we establish the different types of mixed integral equations from the Fredholm integral equation? For this, we consider three types of mixed integral equations (MIEs), under certain conditions. The existence of a unique solution of such equations is guaranteed. Using analytic and numerical methods, the three MIEs formulas yield the same Fredholm integral equation (FIE) formula of the second kind. For continuous kernel, the solution of these three MIEs, via the FIEs, is discussed analytically. In addition, for a discontinuous kernel, the Toeplitz matrix method (TMM) and Product Nyström method (PNM) are used to obtain, in each method, a linear algebraic system (LAS). Then, the numerical results are obtained, the error is computed in each case, and compared as well.

A Novel Method for Solving Second Kind Volterra Integral Equations with Discontinuous Kernel

Load leveling problems and energy storage systems can be modeled in the form of Volterra integral equations (VIE) with a discontinuous kernel. The Lagrange–collocation method is applied for solving the problem. Proving a theorem, we discuss the precision of the method. To control the accuracy, we apply the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library. For this aim, we apply discrete stochastic mathematics (DSA). Using this method, we can control the number of iterations, errors and accuracy. Additionally, some numerical instabilities can be identified. With the aid of this theorem, a novel condition is used instead of the traditional conditions.

Nyström Method to Solve Two-Dimensional Volterra Integral Equation with Discontinuous Kernel

In this paper, a linear two-dimensional Volterra integral equation of the second kind with the discontinuous kernel is considered. The conditions for ensuring the existence of a unique continuous solution are mentioned. The product Nystrom method, as a well-known method of solving singular integral equations, is presented. Therefore, the Nystrom method is applied to the linear Volterra integral equation with the discontinuous kernel to convert it to a linear algebraic system. Some formulas are expanded in two dimensions. Weights’ functions of the Nystrom method are obtained for kernels of logarithmic and Carleman types. Some numerical applications are presented to show the efficiency and accuracy of the proposed method. Maple18 is used to compute numerical solutions. The estimated error is calculated in each case. The Nystrom method is useful and effective in treating the two-dimensional singular Volterra integral equation. Finally, we conclude that the time factor and the parameter v have a clear effect on the results.

Jumping with variably scaled discontinuous kernels (VSDKs)

In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new basis we provide error bounds and numerical results that support our claims. The method is also effectively tested for approximating satellite images.