Numerical treatment of the initial value problem for systems of quasilinear partial differential equations of first order

1962 ◽  
Vol 4 (1) ◽  
pp. 253-261 ◽  
Author(s):  
Rudolf Albrecht ◽  
Wolfram Urich
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Numerical Treatment ◽  
Initial Value ◽  
Partial Differential ◽  
First Order ◽  
Quasilinear Partial Differential Equations

The initial-value problem for long waves of finite amplitude

1964 ◽  
Vol 20 (1) ◽  
pp. 161-170 ◽  
Author(s):  
Robert R. Long
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Integration ◽  
Initial Value Problem ◽  
Finite Amplitude ◽  
Long Waves ◽  
Initial Value ◽  
Partial Differential ◽  
Long Wave ◽  
Constant Slope

Derived herein is a set of partial differential equations governing the propagation of an arbitrary, long-wave disturbance of small, but finite amplitude. The equations reduce to that of Boussinesq (1872) when the assumption is made that the disturbance is propagating in one direction only. The equations are hyperbolic with characteristic curves of constant slope. The initial-value problem can be solved very readily by numerical integration along characteristics. A few examples are included.

scholarly journals CONVERGENCE SPEED OPTIMIZATION IN METHODS OF APPROXIMATE SOLVING OF INITIAL VALUES PROBLEM FOR THE SYSTEM OF QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

2018 ◽  
pp. 47-51
Author(s):  
A. I. Kazmerchuk
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Generalized Solution ◽  
Convergence Speed ◽  
Partial Differential ◽  
Approximate Methods ◽  
First Order ◽  
Initial Values ◽  
Quasilinear Partial Differential Equations ◽  
Uniqueness Of The Solution

In the theory of systems of quasilinear partial differential equations of the first order, the main questions are the solvability of initial values problem and justification of the approximate methods. This is due to problems in gas dynamics and hydromechanics. In the second half of the previous century attempts were made to construct a correct theory of solvability of problems or the systems of quasilinear partial differential equations of the first order. The necessity of the correct way of introductions the nothions of a generalized solution of initial values problems is connected with this. In this paper a class of systems of quasilinear partial differential equations of the first order is singled out for which the concept of a generalized solution is introduced. A method for constructing approximate methods for solving initial values problem is proposed. We obtained estimates of the convergence speed in approximate methods and proved the existence and uniqueness of the solution of initial values problem for systems of quasilinear partial differential equations of the first order of a certain form.

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