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MATLAB Partial Differential Equation Toolbox: ... see step-64 for matrix-free GPU+MPI example Preliminary API for sparse linear algebra Solver: Dimension:
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.. The function is often thought of as an "unknown" that solves the equation, similar to how x is thought of as an unknown number solving, e.g., an algebraic equation like x 2 − 3x + 2 = 0.
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. [1] In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then ...
For such problems, to achieve given accuracy, it takes much less computational time to use an implicit method with larger time steps, even taking into account that one needs to solve an equation of the form (1) at each time step. That said, whether one should use an explicit or implicit method depends upon the problem to be solved.
Spectral methods can be used to solve differential equations (PDEs, ODEs, eigenvalue, etc) and optimization problems. When applying spectral methods to time-dependent PDEs, the solution is typically written as a sum of basis functions with time-dependent coefficients; substituting this in the PDE yields a system of ODEs in the coefficients ...
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. [1] The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the ...