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The split-step Padé approach is an extremely efficient tool for the integration of pseudodifferential parabolic equations, which are widely used for the modeling of acoustic wave propagation. In this study, a generalization of this method to the case of parabolic equations with unknown vector functions is presented. Such generalization requires an algorithm for efficient numerical evaluation of a function of a matrix with differential operators as its elements. After a finite-difference discretization this algorithm reduces to the solution of several Sylvester-like problems. The generalized split-step Padé method presented here is an attractive tool for solving 3D problems of sound propagation in the ocean within the framework of coupled mode parabolic equations theory. • A method is proposed for the numerical solution of the one-way counterpart of a system of coupled Helmholtz equations. • The method is based on a generalization of the split-step Padé algorithm to the case of unknown vector functions. • The method is an attractive tool for handling many practical problems where the propagation of coupled guided modes takes place. [ABSTRACT FROM AUTHOR]

This short note provides an explicit description of the Fréchet derivatives of the principal square root matrix function at any order. We present an original formulation that allows to compute sequentially the Fréchet derivatives of the matrix square root at any order starting from the first order derivative. A Taylor expansion at any order with an integral remainder term is also provided, yielding the first result of this type for this class of matrix function. [ABSTRACT FROM AUTHOR]

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lamé's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems. [ABSTRACT FROM AUTHOR]

We develop a new matrix-based approach to the numerical solution of partial differential equations (PDE) and apply it to the numerical solution of partial fractional differential equations (PFDE). The proposed method is to discretize a given PFDE as a Sylvester Equation, and parameterize the integral surface using matrix algebra. The combination of these two notions results in an algorithm which can solve a general class of PFDE efficiently and accurately by means of an O ( n 3 ) algorithm for solving the Sylvester Matrix Equation (over an m × n grid with m ∼ n ). The proposed parametrization of the integral surface allows for the solution with the more general Robin boundary conditions, and allows for high-order approximations to derivative boundary conditions. To achieve our ends, we also develop a new matrix-based approximation to fractional order derivatives. The proposed method is demonstrated by the numerical solution of the fractional diffusion equation with fractional derivatives in both the temporal and spatial directions. [ABSTRACT FROM AUTHOR]