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Mathematics 2014, 2(2), 96-118; doi:10.3390/math2020096
Article

The Riccati System and a Diffusion-Type Equation

1,2,* , 1
 and
3
1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287–1804, USA 2 Department of Mathematical Sciences, University of Puerto Rico, Mayagüez, Call Box 9000,Puerto Rico 00681–9018, USA 3 Department of Mathematics, Howard University, 223 Academic Support Building B, Washington,DC 20059, USA
* Author to whom correspondence should be addressed.
Received: 31 December 2013 / Revised: 27 February 2014 / Accepted: 14 April 2014 / Published: 15 May 2014
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
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Abstract

We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance.
Keywords: diffusion-type equations; Green’s function; fundamental solution; autonomous and nonautonomous Burgers equations; Fokker-Planck equation; Black-Scholes equation; the Hull-White model; Riccati equation and Riccati-type system; Ermakov equation and Ermakov-type system diffusion-type equations; Green’s function; fundamental solution; autonomous and nonautonomous Burgers equations; Fokker-Planck equation; Black-Scholes equation; the Hull-White model; Riccati equation and Riccati-type system; Ermakov equation and Ermakov-type system
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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Suazo, E.; Suslov, S.K.; Vega-Guzmán, J.M. The Riccati System and a Diffusion-Type Equation. Mathematics 2014, 2, 96-118.

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