A fictitious domain spectral method for time-dependent Sobolev equations in complex domains

We consider a weakly nonlinear singularly perturbed equation in complex domains. The problem is posed about the possibility of splitting the equation into several components. By introducing new unknown functions, a system of two equations is obtained. Next, the asymptotic behavior of solutions of the resulting equations in complex domains is studied. It has been proven that the solution to each of these equations is dominant in certain parts of the areas under consideration. The solution of one of these equations determines the boundary lines and regions, and the solution of the other system determines the regular region.

ABSTRACTFredholm integral equations are an important class of integral equations widely used in science and engineering. This paper aims to numerically solve Fredholm integral equations of the second kind in two‐dimensional complex domains using a combination of spectral methods and mapping techniques. First, we map the computational domain onto a rectangular domain through coordinate transformation. Then, within this rectangular domain, we employ the classical spectral method for numerical simulation. Our analysis focuses on discussing the existence, uniqueness, and convergence of numerical solutions. Numerical results demonstrate that the proposed method achieves high‐order accuracy.

A high-order embedded domain method combining a Predictor–Corrector-Fourier-Continuation-Gram method with an integral Fourier pseudospectral collocation method for solving linear partial differential equations in complex domains

Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) —nonlinear ODEs in the complex plane—by generalizing Prelle–Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.

To solve the shallow shell equations with the help of the plate's equation solution (lowering the order of partial differential equation from 8 to 2)

We consider an integro-differential equation model for traffic flow, which extends the Burgers equation model. To derive exact solutions to the equation, we first examine the features of integrable integro-differential equations and find that they are obtained through the residue formula from integrable equations in complex domains. Using this approach, we construct several new integrable equations with double singular integrals and elliptic singular integrals. Then, we discuss the traffic model and show that it exhibits interactions between free-flow and congested regions, as well as the deadlock phenomenon, depending on the non-locality parameter.

Moving Frames for the Numerical Solution of Partial Differential Equations in Complex Domains

In recent years, a variety of meshless methods have been developed to solve partial differential equations in complex domains. Meshless methods discretize the partial differential equations over scattered points instead of grids. Radial basis functions (RBFs) have been popularly used as high accuracy interpolants of function values at scattered locations. In this paper, we apply the polyharmonic splines (PHS) as the RBF together with appended polynomial and solve the heat conduction equation in several geometries using a collocation procedure. We demonstrate the expected exponential convergence of the numerical solution as the degree of the appended polynomial is increased. The method holds promise to solve several different governing equations in thermal sciences.

In this paper the structure of integral surfaces near a higher order focus and saddle, of ordinary differential equations in complex domain is studied by reducing to the normal form system. The formulae of the normal form system are also given.

The Projection Extension Method: A Spectrally Accurate Technique for Complex Domains

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrodinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.

An efficient implicit spectral element method for time-dependent nonlinear diffusion equations by evaluating integrals at one quadrature point

Simulation of environmental flow problems in geometrically complex domains. Part I: A general coordinate transformation

SPHEREPACK 3.0 is a collection of FORTRAN programs that facilitates computer modeling of geophysical processes. The package contains programs for computing certain common differential operators including divergence, vorticity, gradients, and the Laplacian of both scalar and vector functions. Programs are also available for inverting these operators. For example, given divergence and vorticity, the package can be used to compute the velocity components. The Laplacian can also be inverted and therefore the package can be used to solve both the scalar and vector Helmholtz equations. Use of the package is illustrated by three sample programs that solve the scalar Helmholtz equation, the time-dependent linear advection equation, and the time-dependent nonlinear shallow-water equations. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms that are available to the user. The package also contains utility programs for computing the associated Legendre functions, Gauss points and weights, and multiple fast Fourier transforms. Programs are provided for both equally spaced and Gauss distributed latitudinal points as well as programs that transfer data between these grids.