Related papers: A fully diagonalized spectral method using general…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
In this paper, we develop an efficient spectral-Galerkin-type search extension method (SGSEM) for finding multiple solutions to semilinear elliptic boundary value problems. This method constructs effective initial data for multiple…
The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler-Lagrange…
In this paper we show that we can use a modified version of the h-p spectral element method proposed in \cite{duttora1,duttom,duttora2,tomarth} to solve elliptic problems with general boundary conditions to exponential accuracy on polygonal…
We derive level set version of partial uniform ellipticity for symmetric concave functions. This suggests an effective approach to investigate second order fully nonlinear equations of elliptic and parabolic type.
Many nonlinear differential equations arising from practical problems may permit nontrivial multiple solutions relevant to applications, and these multiple solutions are helpful to deeply understand these practical problems and to improve…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
We extend the result on the spectral projected gradient method by Birgin et al. in 2000 to a log-determinant semidefinite problem (SDP) with linear constraints and propose a spectral projected gradient method for the dual problem. Our…
We study the Fourier orthogonal expansions with respect to the Laguerre type weigh functions on the conic surface of revolution and the domain bounded by such a surface. The main results include a closed form formula for the reproducing…
We derive well-posed boundary conditions for the linearized Serre equations in one spatial dimension by utilizing the energy method. An energy stable and conservative discontinuous Galerkin spectral element method with simple upwind…
The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation…
We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three-dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin-Bubnov method takes as…
The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall…
Based on a new approximation method, namely pseudospectral method, a solution for the three order nonlinear ordinary differential laminar boundary layer Falkner-Skan equation has been obtained on the semi-infinite domain. The proposed…
The Jacobian elliptic functions are generalized and applied to a nonlinear eigenvalue problem with $p$-Laplacian. The eigenvalue and the corresponding eigenfunction are represented in terms of common parameters, and a complete description…
We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions…
In this paper, based on the two-step discretization scheme proposed by Dahlquist, Liniger and Nevanlinna (DLN), we develop a semi-implicit Galerkin finite element method for solving the coupled generalized Ginzburg-Landau equations. By…
Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often…
Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses…
We study the application of generalized symmetry for reducing nonlinear partial differential equations. We construct the ansatzes for dependent variable $u$ which reduce the scalar partial differential equation with two independent…