Abstract
The equations of mathematical physics, which describe some actual processes, are defined on manifolds (tangent, a companying or others) that are not integrable. The derivatives on such manifolds turn out to be inconsistent, i.e. they don’t form a differential. Therefore, the solutions to equations obtained in numerical modelling the derivatives on such manifolds are not functions. They will depend on the commutator made up by noncommutative mixed derivatives, and this fact relates to inconsistence of derivatives. (As it will be shown, such solutions have a physical meaning). The exact solutions (functions) to the equations of mathematical physics are obtained only in the case when the integrable structures are realized. So called generalized solutions are solutions on integrable structures. They are functions (depend only on variables) but are defined only on integrable structure, and, hence, the derivatives of functions or the functions themselves have discontinuities in the direction normal to integrable structure. In numerical simulation of the derivatives of differential equations, one cannot obtain such generalized solutions by continuous way, since this is connected with going from initial nonintegrable manifold to integrable structures. In numerical solving the equations of mathematical physics, it is possible to obtain exact solutions to differential equations only with the help of additional methods. The analysis of the solutions to differential equations with the help of skew-symmetric forms [1,2] can give certain recommendations for numerical solving the differential equations.
Highlights
The exact solution can be obtained in the case when the derivatives obeying the equations made up the differential
The solutions to equations obtained in numerical modelling the derivatives on such manifolds are not functions. They will depend on the commutator made up by noncommutative mixed derivatives, and this fact relates to inconsistence of derivatives. (As it will be shown, such solutions have a physical meaning)
In numerical simulation of the derivatives of differential equations, one cannot obtain such generalized solutions by continuous way, since this is connected with going from initial nonintegrable manifold to integrable structures
Summary
The exact solution (the solution that depends only on variables, i.e. it is a function) can be obtained in the case when the derivatives obeying the equations made up the differential. This means that the solution u to Equation (1) obtained from such derivatives is not a function of variables xi only. This solution will depend on the commutator Kij with nonzero value related to inconsistence of derivatives. The equations of conservation laws are differential (or integral) equations that describe the variation of functions corresponding to physical quantities like the particle velocity (of elements), temperature or energy, pressure and density. Since these physical quantities relate to one material system, the connection between them has to exist. From the equations of conservation law, it follows the evolutionary relation for state functional, which enables one to disclose the properties and peculiarities of the solutions to the equations of mechanics and the physics of continuous media
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