Titlebook: Distributions in the Physical and Engineering Sciences, Volume 2; Linear and Nonlinear Alexander I. Saichev,Wojbor A. Woyczynski Textbook 2

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Diffusions and Parabolic Evolution Equationsf a linear parabolic partial differential equations. Well-posedness of an initial value problem with periodic data is then discussed. Subsequently, the exposition switches to the complex domain, and we introduce a simple version of the general Schr?dinger equation. This makes it possible to study th

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First-Order Nonlinear PDEs and Conservation Lawsrstanding of some of them, it is necessary to move beyond the linear “universe” and consider nonlinear models, which in the case of continuous media, means nonlinear partial differential equations. Even today, their theory is far from complete and is the subject of intense study. On closer inspectio

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Other Standard Nonlinear Models of Higher Ordere begin with the model equations of gas dynamics, expand the Burgers–KPZ model to the multidimensional case and the related concentration fields, study in detail the Korteweg–de Vries (KdV) equations, in which one can observe the creation of solitary waves (solitons), and finally, discuss nonlinear

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Textbook 2013s. It is written from the unifying viewpoint of distribution theory and enriched with many modern topics which are important for practitioners and researchers. The goal of the books is to give the reader, specialist and non-specialist, useable and modern mathematical tools in their research and anal

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Potential Theory and Fundamental Solutions of Elliptic Equations Helmholtz equation with a variable coefficient and an analysis of waves in waveguides. The latter can be reduced to the celebrated Sturm–Liouville problem, and we study properties of its eigenvalues and eigenfunctions.

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Diffusions and Parabolic Evolution Equationse diffraction problem and the so-called Fresnel zones. Multidimensional parabolic equation follow, and the general reflection method is explained. The chapter concludes with a study of the moving boundary problem and the standard physical problem of particle motion in a potential well.

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Generalized Solutions of First-Order Nonlinear PDEsluence of factors not taken into account sometimes is gradual, and does not affect the qualitative picture of the physical phenomenon, but sometimes it is abrupt, and the simplified model is unable to describe the real course of events.