Advanced Analytic Self-Similar Solutions of Regular and Irregular Diffusion Equations
Imre Ferenc Barna, László Mátyás
Abstract
We study the diffusion equation with an appropriate change of variables. This equation is, in general, a partial differential equation (PDE). With the self-similar and related Ansatz, we transform the PDE of diffusion to an ordinary differential equation. The solutions of the PDE belong to a family of functions which are presented for the case of infinite horizon. In the presentation, we accentuate the physically reasonable solutions. We also study time-dependent diffusion phenomena, where the spreading may vary in time. To describe the process, we consider time-dependent diffusion coefficients. The obtained analytic solutions all can be expressed with Kummer’s functions.
Topics & Concepts
Partial differential equationMathematicsAnsatzDiffusion equationDiffusionFirst-order partial differential equationOrdinary differential equationMathematical analysisDifferential equationDiffusion processPhysicsMathematical physicsInnovation diffusionComputer scienceThermodynamicsService (business)EconomicsKnowledge managementEconomyFractional Differential Equations SolutionsDifferential Equations and Numerical MethodsNumerical methods for differential equations