Related papers: Exponentially Fitted Finite Difference Approximati…
Over the past few decades, kernel-based approximation methods had achieved astonishing success in solving different problems in the field of science and engineering. However, when employing the direct or standard method of performing…
We show that MFDEs with infinite range discrete and/or continuous interactions admit exponential dichotomies, building on the Fredholm theory developed by Faye and Scheel for such systems. For the half line, we refine the earlier approach…
We consider the numerical approximation of general semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space-time noise. In contrast to the standard time stepping methods which uses basic increments of…
Building on our previous work on Fredholm Neural Networks (Fredholm NNs/ FNNs) for solving integral equations, we extend the framework to inverse problems for linear and nonlinear elliptic partial differential equations. The proposed scheme…
In this work, we develop and analyze a higher-order finite element method for the multidimensional fragmentation equation. To the best of our knowledge, this is the first study to establish a rigorous, conforming finite element framework…
We present a new fixed mesh algorithm for solving a class of interface inverse problems for the typical elliptic interface problems. These interface inverse problems are formulated as shape optimization prob- lems whose objective…
The Fredholm-Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical…
We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the…
For elliptic interface problems with discontinuous coefficients, the maximum accuracy order for compact 9-point finite difference scheme in irregular points is three [7]. The discontinuous coefficients usually have abrupt jumps across the…
We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layer-adapted piecewise…
An efficient linear solver plays an important role while solving partial differential equations (PDEs) and partial integro-differential equations (PIDEs) type mathematical models. In most cases, the efficiency depends on the stability and…
In this paper, we investigate approximate solutions for nonlinear Fredholm integro-differential equations of fractional order. We present an operational Tau method by obtaining the Tau matrix representation. We solve a special class of…
Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…
A crucial part of successful wave propagation related inverse problems is an efficient and accurate numerical scheme for solving the seismic wave equations. In particular, the numerical solution to a multi-dimensional Helmholtz equation can…
An efficient finite-difference time-domain (FDTD) algorithm is built to solve the transverse electric 2D Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface.…
In this article, a numerical scheme is introduced for solving the fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM) by using an efficient class of finite difference methods. The…
For the numerical solution of the Lippmann-Schwinger equation, while the pre-corrected trapezoidal rule converges with high-order for smooth compactly supported densities, it exhibits only the linear convergence in the case of discontinuity…
In this paper, an efficient algorithm is presented by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with non-smooth solution. Two popular finite…
A nonlinear Helmholtz (NLH) equation with high frequencies and corner singularities is discretized by the linear finite element method (FEM). After deriving some wave-number-explicit stability estimates and the singularity decomposition for…