The Riccati System and a Diffusion-Type Equation

We discuss a method of constructing solution of the initial value problem for duffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered.


Introduction
A goal of this note, complementary to our recent paper [37], is to elaborate on the Cauchy initial value problem for a class of nonautonomous and inhomogeneous diffusion-type equations on R. A corresponding nonautonomous Burgers-type equation is also analyzed as a by-product. Here, we use explicit transformations to the standard forms and emphasize natural relations with certain Riccati and Ermakov-type systems, which seem are missing in the available literature. Similar methods are applied to the corresponding Schrödinger equation (see, for example, [6], [7], [8], [9], [11], [24], [25], [26], [27], [36], [38], [39] and references therein). A group theoretical approach to a similar class of partial differential equations is discussed in Refs. [15], [28] and [34].
For an introduction to fundamental solutions for parabolic equations, see chapter one of the book by Friedman [14]. Among numerous applications, we only elaborate here on an important role of fundamental solutions in probability theory [10], [21]. Consider an Itô diffusion X = {X t : t ≥ 0} which satisfies the stochastic differential equation are solutions of the Cauchy problem This PDE is known as Kolmogorov forward equation [10], [21]. Thus if p (x, y, t) is the appropriate fundamental solution of (1.3), then one can compute the given expectations according to In this context, the fundamental solution is known as the probability transition density for the process and Ω p (x, y, t) dy = 1. (1.5) See also Refs. [1] and [20] for applications to stochastic differential equations related to Fokker-Planck and Burgers equations.

Transformation to the Standard Form
We present the following result.
Lemma 1. The nonautonomous and inhomogeneous diffusion-type equation where a, b, c, d, f, g are suitable functions of time t only, can be reduced to the standard autonomous with the help of the following substitution: Here, µ, α, β, γ, δ, ε, κ are functions of t that satisfy µ ′ 2µ + 2aα + d = 0 (2.4) Equation (2.5) is called the Riccati nonlinear differential equation [32], [40], [42] and we shall refer to the system (2.5)-(2.10) as a Riccati-type system.
The substitution (2.4) reduces the nonlinear Riccati equation (2.5) to the second order linear equation which shall be referred to as a characteristic equation [37].
A group theoretical approach to a similar class of partial differential equations is discussed in Refs. [15], [28] and [34].

Fundamental Solution
By the superposition principle one can solve (formally) the Cauchy initial value problem for the diffusion-type equation ( with the fundamental solution (heat kernel) [37]: where a particular solution of the Riccati-type system (2.5)-(2.10) is given by Here, µ 0 and µ 1 are the so-called standard solutions of the characteristic equation (2.11) subject to the following initial data Lemma 2. The Riccati-type system (2.4)-(2.10) has the following (general) solution: , , and Remark 1. It is worth noting that our transformation (2.3), combined with the standard heat kernel [29]: and

26)
which gives the following asymptotics (The proof is left to the reader.) These formulas allows to establish a required asymptotic of the fundamental solution (3.2): .
(Here, f ∼ g as t → 0 + , if lim t→0 + (f /g) = 1. The proof is left to the reader.) By a direct substitution one can verify that the right hand sides of (3.11)-(3.17) satisfy the Riccati-type system (2.4)-(2.10) and that the asymptotics (3.27)-(3.31) result in the continuity with respect to initial data: The transformation property (3.11)-(3.17) allows one to find solution of the initial value problem in terms of the fundamental solution (3.3)-(3.9) and may be referred to as a nonlinear superposition principle for the Riccati-type system.

Eigenfunction Expansion and Ermakov-type System
With the help of transformation (2.3) one can reduce the diffusion equation (2.1) to another convenient form ∂v which allows to find solution of the Cauchy initial value problem in terms of an eigenfunction expansion similar to the case of the corresponding Schrödinger in Refs. [24] and [38]. This method requires an extension the Riccati-type system (2.5)-(2.10) to a more general Ermakov-type system [24], which is integrable in quadratures once again in terms of solutions of the characteristic equation (2.11). Further details are left to the reader.

Lemma 4. The following identity holds
and a, b, c, d, f, g are functions of t only).
where the heat kernel is given by (3.2), for suitable initial data v (z, 0) on R.

Traveling Wave Solutions of Burgers-type Equation
Looking for solutions of our equation (5.1) in the form v = β (t) F (β (t) x + γ (t)) = βF (z) , z = βx + γ (6.1) (β and γ are functions of t only), one gets provided that (c 0 , c 1 , c 2 , c 3 are constants). From (6.2): where c 4 is a constant of integration. The substitution transforms the Riccati equation (6.6) into a special case of generalized equation of hypergeometric type: which can be solved in general by methods of Ref. [30]. Elementary solutions are discussed, for example, in [22] and [23].

Some Examples
Now we consider from a united viewpoint several elementary diffusion and Burgers equations that are important in applications.

Example 1 For the standard diffusion equation on R :
∂u ∂t = a ∂ 2 u ∂x 2 , a = constant > 0 (7.1) the heat kernel is given by , t > 0. (7.2) (See [4], [29] and references therein for a detailed investigation of the classical one-dimensional heat equation.)

Example 2
In mathematical description of the nerve cell a dendritic branch is typically modeled by using cylindrical cable equation [18]: The fundamental solution on R is given by (See also [16] and references therein.)
according as the + or − sign is taken. In the first case there is no definite value of v when a tends to zero, while in the second case the limiting value of v is either A − V or A + V according as x + V t is less or greater than c. The limiting form of the solution is thus discontinuous [2].

Conclusion
In this note, we have discussed connections of certain nonautonomous and inhomogeneous diffusiontype equation and Burgers equation with solutions of the Riccati and Ermakov-type systems that seem are missing in the available literature. Traveling wave solutions of the Burgers-type equations are also discussed.