Variants of the hodograph method for solving a system of two quasilinear equations

T. F. Dolgikh   ; M. Yu. Zhukov

Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 2, pp. 34-50

Résumé

The solution of the Cauchy problem for a system of two quasilinear homogeneous first-order partial differential equations is constructed using the hodograph method, which allows us transform the solution of quasilinear first-order partial differential equations to the solution of some second-order linear partial differential equation with variable coefficients. It is shown that various variants of the hodograph method (standard method, method based on the conservation law, and generalized hodograph method) to construct a solution to the Cauchy problem in implicit form, ultimately lead to the same result and differ only in the amount of technical work. The proof is given by calculating the Laplace invariants of the second-order linear partial differential equation in the canonical form. In the case when the equations permit an explicit connection of the initial variables with Riemann invariants and the corresponding linear equation of the hodograph method allows us to specify the explicit form of the Riemann-Green function, a method for constructing an explicit solution on the level-lines of the implicit solution is described. The Cauchy problem for a system of two quasilinear first-order partial differential equations reduces to the Cauchy problem for a certain system of ordinary differential equations. An exact implicit solution for a system of the linear degenerate equations is given as an example. All the methods presented and the method of constructing an explicit solution can be used for hyperbolic and elliptic equations. In the case of hyperbolic equations, it is possible to construct self-similar and discontinuous solutions (after adding discontinuity conditions), as well as multi-valued solutions in the spatial coordinate (if such solutions are allowed by the problem statement). Despite the fact that at the final stage of the method, the Cauchy problem for ordinary differential equations has to be solved numerically, no approximations of partial differential equations typical for the finite difference method, the finite element method, the finite volume method, etc. are used. The method is accurate in the sense that the error of calculations is related only to the accuracy of integration of ordinary differential equations.

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