This paper presents a numerical method for a class of singularly perturbed parabolic partial differential equations with integral boundary conditions (IBC). The solution to the considered problem exhibits pronounced boundary layers on both the left and right sides of the spatial domain. To address this challenging problem, we propose the use of the implicit Euler method for time discretization and a finite difference method on a well-designed piecewise uniform Shishkin mesh for spatial discretization. The integral boundary condition is approximated using Simpson's $\frac{1}{3}$ rule. The presented method demonstrates almost second-order uniform convergence in the discretization of the spatial derivative and first-order convergence in the discretization of the time derivative. To validate the applicability and accuracy of the proposed method, two illustrative examples are employed. The computational results not only accurately reflect the theoretical estimations but also highlight the method's effectiveness in capturing the intricate features of singularly perturbed parabolic partial differential equations with integral boundary conditions.

In this article, a linear inverse problem for approximating the right hand side of a fourth order parabolic equation is studied. In this problem, it is assumed that the homogeneous boundary conditions along with an integral condition on the time domain and a local condition at a point of the space domain are known. In the first step, we show that this problem has a unique classical solution. Then, we convert the initial problem into a new problem by using suitable transformations, in which the time-dependent unknown function is transferred to the boundary conditions, and then we provide a spectral approximation based on the Ritz method to detect the unknown functions. The discretization of the problem using the presented technique leads to a system of linear algebraic equations which is solved by employing the Tikhonov's regularization method. The numerical simulation results confirm the acceptable accuracy and stability of the approximate solution.

In this paper, we are concerned with positive solutions for higher order m{point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m{point nonlinear fractional boundary value problems with integral boundary conditions by using some results from the theory of  xed point index, Avery{Henderson  xed point theorem and the Legget{Williams  xed point theorem, respectively.

A numerical model based on potential theory is developed for a two dimensional nonlinear water waves using the Cauchy boundary integral equation together with a Lagrangian time marching method for free water surface. This model can deal with stable deformations of free water surface.Applying this model, the forces exerted by waves on fixed or mobile objects can be computed

2 Abstract One of the principal criteria for development of the boundary element method (BEM) in porous media is derivation of the required fundamental solutions in the boundary integral equations (BIE). Furthermore, setting up the governing BIEs based on the governing partial differential equations (PDE) is another challenge in solving a physical phenomenon using BEM. In this regard, the governing BIEs for unsaturated porous media have been developed using the available derived fundamental solutions. In this research, a perturbation type approximation is exploited for developing a system of BIEs for the quasi-static unsaturated porous media with moderate variations in its properties. Nevertheless, the fundamental solutions of the medium with constant properties are applied. The method produces two sets of equations with constant parameters instead of the original equations. Besides, the required boundary conditions have been formulated. This type of BIEs is essential to be used in the BEM for unsaturated porous media as the fundamental solutions for a medium with coordinates dependent properties is not available so far. The resulted introduced BIEs may be used directly in a BEM numerical model for an unsaturated porous media in one, two or three dimensional conditions.

In this paper, using orthogonally of Tchebychev polynomials, we present an orthonormal wavelet basis for L2[0, 1]. We use this basis for solving Neumann problems with Galerkin method. The property of this basis is that a variety of integral operators is represented in this basis as sparse matrices, to high precision. Some examples are solved to illustrate the efficiency and accuracy of this method.

A boundary integral method is used to simulate spilling breakers. The bottom is also included in the introduced closed boundary and the problem is directly solved in the physical plane. The method has been shown to be remarkably stable, and no numerical instability has occurred in any of the calculations. The results reveal that both the momentum and total energy are almost constant in time during the simulation period. As a result, the breaking process of a spilling breaker is fairly well simulated.

The main aim of this paper is to study a kind of boundary value problem with an integral boundary condition including Hadamard-type fractional differential equations. To do this, upper and lower solutions are used to guarantee their existence, and Schauder’s fixed point theorem is used to prove the uniqueness of the positive solutions to this problem. An illustrated example is presented to explain the theorems that have been proved.

This paper presents an efficient numerical method to solve two versions of the Duffing equation by the hybrid functions based on the combination of Block-pulse functions and Legendre polynomials. This method reduces the solution of the considered problem to the solution of a system of algebraic equations. Moreover, the convergence of the method is studied. Some examples are given to demonstrate the applicability and effectiveness of the proposed method. Also, the obtained results are compared with some other results.

A new numerical method has been proposed by combining the boundary integral and spectral methods. In this method, similar to the boundary element method, shape functions are used to solve the integral equation. However, for the functions, an orthogonal basis called wavelet functions is used. Combining these two methods, together with the use of wavelet functions, leads to a modified. Through this method, the Poisson equation is solved for the torsion of prismatic bars and the distribution of the normal derivative of stress function is extracted.