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2 2017

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Dynamic projection operator method in the theory of hyperbolic systems of partial differential equations with variable coefficients

Leble S., Vereshchagina I.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

2017

Vol. 21, No 2

109--120

We consider a generalization of the projection operator method for the case of the Cauchy problem in 1D space for systems of evolution differential equations of first order with variable coefficients. It is supposed that the dependence of coefficients on the only variable χ is weak, that is described by the introduction of a small parameter. Such problem corresponds, for example, to the case of wave propagation in a weakly inhomogeneous medium. As an example, we specify the problem to adiabatic acoustics in waveguides with a variable cross-section. Projection operators are constructed for the Cauchy problem to fix unidirectional modes. The method of successive approximations (perturbation theory) is developed and based on the pseudodifferential operators theory. The application of projection operators adapted for the case under consideration allows deriving approximate evolution equations corresponding to the separated directed waves.

A New Family of the Local Fractional PDEs

Yang X.-J., Machado J. A. T., Nieto J. J.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

63--75

A new family of the local fractional PDEs is investigated in this article. The linear, quasilinear, semilinear and nonlinear local fractional PDEs are presented. Furthermore, three types of the local fractional PDEs are discussed, namely, parabolic, hyperbolic and elliptic. Several examples illustrate the corresponding models in nonlinear mathematical physics.