Diffusions and Parabolic Evolution Equations

  • Alexander I. Saichev
  • Wojbor A. Woyczyński
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We begin with a study of the classic 1-D diffusion equation (also called heat equation) and its self-similar solutions. This is the simplest example of 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 Schrödinger equation. This makes it possible to study the 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.


Parabolic Equation Diffusion Equation Fundamental Solution Fresnel Zone Reflection Method 
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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexander I. Saichev
    • 1
    • 2
  • Wojbor A. Woyczyński
    • 3
  1. 1.Department of Management, Technology, and EconomicsETH ZürichZürichSwitzerland
  2. 2.Department of Radio PhysicsUniversity of Nizhniy NovgorodNizhniy NovgorodRussia
  3. 3.Department of Mathematics, Applied Mathematics and Statistics, and Center for Stochastic and Chaotic Processes in Science and TechnologyCase Western Reserve UniversityClevelandUSA

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