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The Bogacki–Shampine method is implemented in the ode3 for fixed step solver and ode23 for a variable step solver function in MATLAB (Shampine & Reichelt 1997). Low-order methods are more suitable than higher-order methods like the Dormand–Prince method of order five, if only a crude approximation to the solution is required. Bogacki and ...
Multigrid methods can be generalized in many different ways. They can be applied naturally in a time-stepping solution of parabolic partial differential equations, or they can be applied directly to time-dependent partial differential equations. [12] Research on multilevel techniques for hyperbolic partial differential equations is underway. [13]
MFEM is a free, lightweight, scalable C++ library for finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretizations, and emphasis on usability, generality, and high-performance computing efficiency.
Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. [1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
Cnoidal wave solution to the Korteweg–De Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9). Numerical solution of the KdV equation u t + uu x + δ 2 u xxx = 0 (δ = 0.022) with an initial condition u(x, 0) = cos(πx). Time evolution was done by the Zabusky–Kruskal scheme. [1]
The inverse scattering transform arose from studying solitary waves. J.S. Russell described a "wave of translation" or "solitary wave" occurring in shallow water. [5] First J.V. Boussinesq and later D. Korteweg and G. deVries discovered the Korteweg-deVries (KdV) equation, a nonlinear partial differential equation describing these waves. [5]
Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation. The method is to reduce a partial differential equation (PDE) to a family of ordinary differential equations (ODE) along which the solution can be integrated from some initial data ...
Optimal control is the use of mathematical optimization to obtain a policy that is constrained by differential (=), equality (() =), or inequality (()) equations and minimizes an objective/reward function (()). The basic optimal control is solved with GEKKO by integrating the objective and transcribing the differential equation into algebraic ...