[Paper Review] The Riccati Differential Equation and a Diffusion-Type Equation

Erwin Suazo, Sergeĭ K. Suslov+1 more|ArXiv.org|Jul 28, 2008

Differential Equations and Numerical MethodsMathematics19 citations

TL;DR

This paper presents an explicit analytical solution for the Cauchy initial value problem of a class of one-dimensional diffusion-type equations with time-dependent coefficients. By reducing the problem to a Riccati equation and its associated second-order linear characteristic equation, the authors derive a closed-form expression for the Green function (heat kernel) in terms of elementary functions and integrals, enabling exact solutions for both homogeneous and non-homogeneous cases via the Duhamel principle.

ABSTRACT

We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of the second order linear differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding non-homogeneous equation is also found.

Motivation & Objective

• To construct an explicit solution for the Cauchy initial value problem of a general diffusion-type equation with variable coefficients on the real line.
• To identify the conditions under which the Green function (heat kernel) can be expressed in terms of elementary functions and integrals.
• To establish a connection between the solution of the diffusion equation and the solution of a Riccati equation derived from the characteristic function.
• To extend the method to non-homogeneous equations using the Duhamel principle.
• To provide explicit solutions for special cases, including equations with hyperbolic and trigonometric time-dependent coefficients.

Proposed method

• The solution is constructed via the ansatz $ u = A(t) e^{S(x,y,t)} $, where $ A(t) = 1 / \sqrt{2\pi\mu(t)} $ and $ S $ is a quadratic form in $ x $ and $ y $ with time-dependent coefficients.
• The coefficients in $ S $ are determined by solving a system of ODEs derived from substituting the ansatz into the PDE, leading to a Riccati equation for $ \alpha(t) $.
• The Riccati equation is transformed into a second-order linear ODE for the characteristic function $ \mu(t) $, given by $ \mu'' - \tau(t)\mu' - 4\sigma(t)\mu = 0 $, with $ \tau $ and $ \sigma $ defined in terms of the time-dependent coefficients.
• The Green function is explicitly given by $ K(x,y,t) = \frac{1}{\sqrt{2\pi\mu(t)}} \exp(S(x,y,t)) $, where $ S $ depends on the solutions of the ODE system.
• The method relies on solving the characteristic equation $ \mu'' - \tau(t)\mu' - 4\sigma(t)\mu = 0 $ with initial conditions $ \mu(0) = 0 $, $ \mu'(0) = 2a(0) \neq 0 $ to ensure correct initial behavior.
• For non-homogeneous equations, the Duhamel principle is applied, expressing the solution as a time integral involving the homogeneous solution operator and the forcing term.

Experimental results

Research questions

• RQ1 Under what conditions can the fundamental solution of a diffusion-type PDE with time-dependent coefficients be expressed in closed form using elementary functions and integrals?
• RQ2 How is the solution of the diffusion equation related to the solution of a Riccati differential equation with time-dependent coefficients?
• RQ3 What is the role of the characteristic function $ \mu(t) $, and how does it determine the structure of the Green function?
• RQ4 How can the Duhamel principle be applied to derive solutions for non-homogeneous diffusion equations with variable coefficients?
• RQ5 What are the explicit forms of the heat kernel for specific time-dependent coefficient functions, such as those involving hyperbolic and trigonometric functions?

Key findings

• The Green function for the diffusion equation is explicitly constructed as $ K(x,y,t) = \frac{1}{\sqrt{2\pi\mu(t)}} \exp(\alpha x^2 + \beta xy + \gamma y^2 + \delta x + \varepsilon y + \kappa) $, where the coefficients are determined by solving a system of ODEs.
• The coefficient $ \alpha(t) $ satisfies a Riccati equation: $ \frac{d\alpha}{dt} + b(t) - 2c(t)\alpha - 4a(t)\alpha^2 = 0 $, which is central to the solution method.
• The characteristic function $ \mu(t) $ satisfies the second-order linear ODE $ \mu'' - \tau(t)\mu' - 4\sigma(t)\mu = 0 $, with $ \tau(t) = \frac{a'}{a} + 2c - 4d $, $ \sigma(t) = ab + cd - d^2 + \frac{d}{2}\left(\frac{a'}{a} - \frac{d'}{d}\right) $.
• For the case $ a(t) = \sinh^2 t $, $ b(t) = \cosh^2 t $, $ c(t) = \frac{1}{2}\sinh 2t $, the characteristic equation becomes $ \mu'' - 2\tanh t\, \mu' + 2\mu = 0 $, with solution $ \mu_1(t) = \cos t \sinh t + \sin t \cosh t $, valid for $ 0 < t < T_1 \approx 0.93755 $.
• The heat kernel for this case is explicitly given with a convergence condition tied to the sign of $ \gamma(t) $, ensuring integrability for suitable initial data.
• A similar solution is derived for the equation with $ a(t) = \cos^2 t $, $ b(t) = \sin^2 t $, $ c(t) = -\frac{1}{2}\sin 2t $, yielding the same form of the heat kernel but with $ x $ and $ y $ interchanged, valid for $ 0 < t < T_2 \approx 2.347 $.