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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 .
This is an implicit relation that determines the solution of the inviscid Burgers' equation provided characteristics don't intersect. If the characteristics do intersect, then a classical solution to the PDE does not exist and leads to the formation of a shock wave. Whether characteristics can intersect or not depends on the initial condition.
The h-principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems. The method of characteristics can be used in some very special cases to solve nonlinear partial differential equations. [11]
Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics, e.g., the advection equation. If a family of solutions of a single first ...
The method of integrating linear ordinary differential equations with constant coefficients was discovered by Leonhard Euler, who found that the solutions depended on an algebraic 'characteristic' equation. [2]
Hybrid methods have also been introduced that take advantage of FMM's efficiency with FSM's simplicity. For example, the Heap Cell Method (HCM) decomposes the domain into cells and performs FMM on the cell-domain, and each time a "cell" is updated FSM is performed on the local gridpoint-domain that lies within that cell. [ 10 ]
The origin of the method of characteristics can be traced back to Heisenberg’s matrix mechanics. Suppose that we have solved in the matrix mechanics the evolution equations for the operators of the canonical coordinates and momenta in the Heisenberg representation. These operators evolve according to
Method of characteristics, a technique for solving partial differential equations; See also. Characteristic (disambiguation) This page was last edited on 30 April ...