A New Matrix Approach For Solving Second-Order Linear Matrix Partial Differential Equations

A new matrix method for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions

“…By solving the Sylvester equation (23) via a robust Krylov subspace iterative method (i.e., generalized BICGSTAB), that are provided in [31] , one can obtain an approximate value for ˜ U and replacing it instead of U in (16) . Therefore, a numerical solution of (1) -(3) in terms of Bernoulli polynomials can be achieved.…”

“…Similar to the BMM, we first assume that According to the properties of the kronecker product, which were used in [30,31] , one can deduce that Since the computational domain of this example is standard, one can consider the following uniform mesh for both of the spatial and temporal variables:…”

“…As it is mentioned earlier, the corresponding algebraic generalized Sylvester equations are solved via the generalized BICGSTAB algorithm that are introduced in [31] . All computations are carried out in MATLAB 7:12:0 software with double precision in which the Digits environment variable assigned to be 32.…”

“…In this paper, in the light of the methods [30][31][32] and by means of the matrix relations between the Bernoulli polynomials and their derivatives and integrals, we extend a new method, namely, the Bernoulli matrix method for solving two-dimensional diffusion equations in the form of ∂u ∂t…”

Two‐dimensional shifted Legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices

“…The main aim of using an orthogonal basis is that our proposed problem reduces to a system of linear or nonlinear algebraic equations. This can be performed by the solution of the proposed problem using the operational matrices (e.g., ). An important property of operational matrix method is that it simplifies the problem and also speeds up the computation.…”