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.
A goal of this work, complementary to our recent paper , is to elaborate on the Cauchy initial value problem for a class of nonautonomous and inhomogeneous diffusion-type equations on 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 to be missing in the available literature. Similar methods are applied to the corresponding Schrödinger equation (see, for example, [2,3,4,5,6,7,8,9,10,11,12,13] and references therein). A group theoretical approach to a similar class of partial differential equations is discussed in Refs. [14,15,16].
For an introduction to fundamental solutions for parabolic equations, see chapter one of the book by Friedman . Among numerous applications, we only elaborate here on an important role of fundamental solutions in probability theory [18,19]. Consider an Itô diffusion which satisfies the stochastic differential equation
in which is a standard Wiener process. The existence and uniqueness of solutions of Equation (1) depends on the coefficients b and (See Ref.  for conditions of unique strong solution to Equation (1).) If the Equation (1) has a unique solution, then the expectations
are solutions of the Cauchy problem
This PDE is known as Kolmogorov forward equation [18,19]. Thus if is the appropriate fundamental solution of Equation (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
See also Refs. [20,21] for applications to stochastic differential equations related to Fokker–Planck and Burgers equations. The Black-Scholes model of financial markets is discussed in [22,23,24,25,26,27] (see also  for the one-factor Gaussian Hull-White model and  for a connection with quantum mechanics).
The paper is organized as follows: We present the main result and sketch the proof in the next section. In Section 3 and Section 4, a solution of the Cauchy initial value problem and the symmetry group of diffusion equations are revisited from a new perspective. An extension to a Ermarov-type system is discussed in Section 5. Section 6 and Section 7 deal with nonautonomous Burgers-type equations. In the last section, we discuss some examples.
2. Transformation to the Standard Form
We present the following result.
The nonautonomous and inhomogeneous diffusion-type equation
where are suitable functions of time t only, can be reduced to the standard autonomous form
with the help of the following substitution:
Here, are functions of t that satisfy
Let and . To reduce Equation (6) into the standard nonautonomous form Equation (7) we need the following derivatives:
Direct substitution of Equations (16)–(18) into Equation (6) leads to the desired reduced form (7) subject to Equations (9)–(15). Further details are left to the reader. ☐
Equation (10) is called the Riccati nonlinear differential equation [30,31,32] and we shall refer to the system (10)–(15) as a Riccati-type system.
The substitution (9) reduces the nonlinear Riccati Equation (10) to the second order linear equation
which shall be referred to as a characteristic equation .
It is also known  that the diffusion-type equation (6) has a particular solution of the form
provided that the time dependent functions satisfy the Riccati-type system (9)–(15) (our original interpretation of this system).
A group theoretical approach to a similar class of partial differential equations is discussed in [14,15,16].
3. Fundamental Solution
By the superposition principle one can solve (formally) the Cauchy initial value problem for the diffusion-type equation (6) subject to initial data on the entire real line in an integral form
For the derivations of the corresponding asymptotics consider first the Taylor expansion of , and centered at 0 as follow:
as . In order to obtain the corresponding asypmtotic for we can follow similar procedure as for Equation (54) resulting in
Consider now Equation (45). Then, for the asymptotic of we have that:
as . Finally, considering the Taylor expansion of and centered at the zero, the asymptotic for results in
as . In the case of the corresponding expansion is given by
Notice that Equations (41)–(47) are inversions of Equations (32)–(38). These formulas allows to establish a required asymptotic of the fundamental solution (23):
We have corrected a typo in  (Here, as if The proof is left to the reader.)
By a direct substitution one can verify that the right hand sides of Equations (32)–(38) satisfy the Riccati-type system (9)–(15) and that the asymptotics (48)–(52) result in the continuity with respect to initial data:
The transformation property (32)–(38) allows one to find solution of the initial value problem in terms of the fundamental solution (24)–(30) and may be referred to as a nonlinear superposition principle for the Riccati-type system see also [34,35].
4. Symmetries of the Autonomous Diffusion Equation
In the simplest case , when our Lemma 1 provides the following general transformation
with of the diffusion equation into itself (see [15,16,36]). It includes the familiar Galilei transformations:
when and supplemented by dilatations:
with and and expansions:
with . The symmety group of the corresponding Schrödinger equations is discussed in [14,15,36,37,38,39].
5. Eigenfunction Expansion and Ermakov-Type System
With the help of transformation (8) one can reduce the diffusion equation (6) to another convenient form
It is important to notice that with the substitution (9) Equation (68) can be transformed into a Ermakov type equation of the form
However, in this paper, we work with the module of the solution; because we are considering real solutions, the complex solution has been used in the case of linear and nonlinear Schrödinger equation see  for more details.
We present the following result.
The Ermakov-type system with has the following solution:
in terms of the fundamental solution of the Riccati-system (24)–(30) subject to arbitrary initial data
(See Appendix A for a sketch of the proof for this Lemma. The corresponding integral can be found in [41,42].)
Then a particular solution of the diffusion equation (6) has the form
where are the Hermite polynomials . The solution of the Ermakov-type system (68)–(73) is provided by our Lemma 4.
The solution of the Cauchy initial value problem can be found in terms of an eigenfunction expansion by the superposition principle, similar to the case of the corresponding Schrödinger equationin [7,12]:
One can choose and when the eigenfunctions are orthonormal
in view of the orthogonality property of Hermite polynomials . Then the expansion coefficients are given by
Equations (85) and (88) provide the solution of the initial value problem.
6. Nonautonomous Burgers Equation
The nonlinear equation
when and is known as Burgers’ equation [44,45,46,47,48,49,50] and we shall refer to Equation (89) as a nonautonomous Burgers-type equation; see also [16,51].
The following identity holds (Occationally is custom to use the shorten notation , etc.)
Substitution of space and time derivatives of Equation (104) into Equation (105) leads to
from which we can establish the desired commutative relation. Additional details are left to the reader. ☐
Diagram 1. A diagram summarizing the relations between the non-autonomous and inhomogeneous diffusion-type equation (6), the linear heat equation (7) and the Burgers-type equations (102) and (113).
Following the same strategy, if we consider the transformation
with and , the corresponding space and time derivatives will be given by
Susbtitution of Equations (107)–(110) into Equation (102) yields
After dividing by and using Equation (12), Equation (112) reduces to the Burgers equation
We refer the reader to the existent literature for further details regarding forced Burgers equations. The connections between the inhomogeneous diffusion equation (6), the linear heat equation (7), the Burgers equation (113) and the non-autonomous and inhomogeneous Burgers equation (102) is portrayed in Diagram 1. Equations (89) and (102) may be useful generalizations in a diverse of Physical contexts, and could be used to test certain numerical schemes.
7. Traveling Wave Solutions of Burgers-Type Equation
Looking for solutions of our Equation (89) in the form
(β and γ are functions of t only), one gets
( are constants). Integration of Equation (115) yields
where is a constant of integration. The substitution
transforms the Riccati equation (119) into a special case of generalized equation of hypergeometric type:
which can be solved in general by methods of . Elementary solutions are discussed, for example, in [53,54].
Now we consider from a united viewpoint several elementary diffusion and Burgers-type equations that are important in applications.
Example 1 For the standard diffusion equation on
the heat kernel is given by
(See [33,55] and references therein for a detailed investigation of the classical one-dimensional heat equation.)
Example 2 In a mathematical description of the nerve cell a dendritic branch is typically modeled by using cylindrical cable equation :
The solution of the Cauchy initial value problem for the Fokker-Planck equation on can also be given in terms of eigenfunction expansion with the aid of the superposition principle. In the corresponding eigenfunction expansion:
and after choosing , , and , the corresponding eigenfunction is given by
and the expansion coefficients are
Further details are left to the reader.
Example 4 Equation
corresponds to the heat equation with linear drift when . In stochastic differential equations this equation corresponds the Kolmogorov forward equation for the regular Ornstein–Uhlenbech process . The fundamental solution is given by
Example 5 The Black-Scholes model provides a mathematical description of financial markets and derivative investment instruments [22,24]. If S is the price of the stock, is the price of a derivative as a function of time and stock price, r is the annualized risk-free interest rate, continuously compound, σ is the volatility of stock’s returnes; this is the square root of the quadratic variation of the stock’s log price process, the celebrated Black-Scholes equation is given by [22,23,24,25,26,27]:
The substitution where (due to Euler) and (the time to maturity), results in the diffusion-type equation
which can be transformed into the standard heat equation for variable r and σ with the help of Lemma 1.
The corresponding characteristic equation,
can be solved explicitly when σ and r are constants. The standard solutions are given by
and the corresponding fundamental solution can be obtain in a closed form :
from our Equations (24)–(30). Then, by using initial conditions, V can be computed explicitly in terms of the error function, leading to Black-Scholes formula .
It is worth adding, in conclusion, that by our Lemma 1 the following transformation:
The characteristic equation, has two standard solutions:
and the corresponding Green function:
for can be found by the method of this paper.
Example 7 The viscous Burgers equation [44,45,48,50,54]:
can be linearized by the Cole–Hopf substitution [46,47]:
which turns it into the diffusion equation (122). Solution of the initial value problem has the form:
for and suitable initial data on
Example 8 Equation (154) possesses a solution of the form:
(we follow the original Bateman paper  with slightly different notations), if
where A is a positive constant. The solution is thus either
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 or according as is less or greater than The limiting form of the solution is thus discontinuous .
Example 9 According to Ref. , the propagation of nonlinear magnetosonic waves is governed by a modified Burgers equation,
where is the amplitude of the wave, and is the coordinate streched by a smallness parameter
If the following substitution
transforms the nonautonomous equation (162) into the Burgers equation
that is completely integrable.
Example 10 Assuming (formally) in the Black-Scholes equation (133), one gets a nonlinear equation of the form
This modification of the Black-Scholes equation can be used for a mathematical description of market collapse. The substitution where and transforms Equation (166) into the generalized Burgers-Huxley equation [53,60].
In this paper, we have discussed connections of certain nonautonomous and inhomogeneous diffusion-type equations, and the Burgers equation with solutions of the Riccati-type system, which seem to be missing in the available literature in general. Traveling wave solutions of the Burgers-type equations are also discussed. Examples include, but are not limited to, the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance.
We are grateful to the referees for their useful recommendations to improve our presentation. We thank Professor Carlos Castillo-Chávez, Professor Carl Gardner and Professor Ambar Sengupta for support, valuable discussions and encouragement.
Lemmas 1 to 6 of this paper are a natural extension of the results presented in . Lemmas 1 to 6 are the result of a continuous collaboration for the last two years of E. Suazo, S. K. Suslov and J. Vega-Guzman. Similarly, the selection of the examples, diagram and extended bibliography is the result of a continuous long interaction between the authors.
The standard oscillatory wave functions for Equation (67) can be given by
where are the Hermite polynomials, provided that solution for the Ermakov-type system is available. Considering the heat kernel given by Equation (23) with corresponding coefficients given by Equations (24)–(30), we have that the corresponding Cauchy initial value problem can be solve formally once again by the superposition principle
for certain initial data . Particularly, using the eigenfunction (167) we get
Uniqueness of the Cauchy initial value problem allows one to find the desired solution. Thus, the solution of the Ermakov-type system can be obtain by evaluating Equation (169) with the help of
The Z arise when completing the square. Extensive and tedious calculations are left to the reader.
Conflicts of Interest
The authors declare no conflict of interest.
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