Lie symmetries of a generalized Fisher equation in cylindrical coordinates

This paper employs the Lie symmetry method to identify specific source functions that allow a generalized Fisher equation with exponential diffusion in cylindrical coordinates to possess symmetries beyond time translation, and subsequently derives the corresponding reduced ordinary differential equations.

The Gist

Imagine you are watching a crowd of people spreading out in a circular room (like a cylinder). Some people are moving randomly (diffusion), while others are being influenced by a rule that makes them multiply or stop based on how crowded they are (reaction). This is the basic idea behind the Fisher equation, a famous math model used to describe how things like populations, heat, or chemicals spread and change over time.

In this paper, the authors, Bayarjargal Batsukh and Uuganbayar Zunderiya, decided to look at this problem in a cylindrical room (like a pipe or a silo) rather than a flat line. They also made the rules more complex by allowing the "crowd" to behave in different ways depending on how many people are already there. They call this the Generalized Fisher Equation.

Here is the simple breakdown of what they did, using some everyday analogies:

1. The Goal: Finding the "Secret Patterns"

The authors used a powerful mathematical tool called Lie Symmetry. Think of this like looking for a secret "magic trick" in the math.

• The Magic Trick: Usually, if you wait a little bit longer (time passes), the math changes. But sometimes, the math has a hidden symmetry where you can stretch time, stretch space, or shift the crowd's behavior, and the underlying pattern stays exactly the same.
• The Goal: They wanted to find out: "Under what specific rules (functions) does this complex equation have these special, hidden patterns?"

2. The Setup: The "Diffusion" and the "Source"

The equation has two main parts:

• The Diffusion ($g(u)$): How easily the crowd moves. The authors focused on a specific, tricky type of movement where the ease of moving changes exponentially (like a crowd that moves much faster if it gets slightly denser).
• The Source ($f(u)$): The rule that makes the crowd grow or shrink. This is the variable they were trying to solve for.

3. The Discovery: Three Special "Recipes"

The authors found that for the equation to have these special "magic patterns" (symmetries) beyond just waiting for time to pass, the "Source" rule ($f(u)$) must be one of exactly three specific types.

Think of it like baking a cake. You have a specific type of flour (the diffusion). You can only get a perfect, symmetrical cake if you use one of three specific recipes for the sugar (the source):

• Recipe A: The sugar grows exponentially at a specific rate.
• Recipe B: The sugar grows exponentially but has a constant "base" amount added to it.
• Recipe C: The sugar is just a constant amount (no growth or decay, just a steady push).

If you use any other recipe, the "magic symmetry" disappears, and the math becomes much harder to solve exactly.

4. The Result: Simplifying the Puzzle

Once they identified these three special recipes, they used the symmetry to simplify the problem.

• The Analogy: Imagine you have a complex 3D video game level that is impossible to beat. Suddenly, you realize that if you only move in a straight line, the game simplifies into a 2D puzzle that is easy to solve.
• The Math: They took the complicated equation (which depends on space and time) and turned it into a simpler Ordinary Differential Equation (ODE). This is like turning a complex 3D map into a simple 1D line.
• The Solution: For two of the three recipes, they found that the solution involves Bessel functions. You can think of Bessel functions as the "standard shapes" that waves or ripples take in circular environments (like ripples in a pond). They even drew 3D pictures of what these solutions look like, showing how the "crowd" spreads out over time.

Summary

In short, this paper is a detective story about a complex math equation. The authors asked, "What specific rules make this equation behave in a perfectly symmetrical way?" They found that there are only three specific rulebooks that allow this to happen. Once those rules are identified, the authors showed how to turn the difficult, multi-dimensional problem into a simpler, solvable one, revealing the exact shapes these patterns take in a cylindrical space.

They did not discuss real-world applications like cancer treatment or forest fires; they strictly focused on the mathematical structure and finding the exact solutions for these specific cases.

Technical Summary

Technical Summary: Lie Symmetries of a Generalized Fisher Equation in Cylindrical Coordinates

Problem Statement
This work investigates a generalized Fisher equation formulated in cylindrical coordinates, described by the partial differential equation (PDE):
$$u_t = f(u) + \frac{1}{x}(x g(u) u_x)_x$$
where $u(x,t)$ represents a concentration, and $f(u)$ and $g(u)$ are sufficiently smooth functions. While the equation possesses a trivial time-translation symmetry ($X = \partial/\partial t$) for arbitrary $f(u)$ and $g(u)$, the authors seek to determine specific functional forms of the source term $f(u)$ and the diffusion function $g(u)$ that admit additional Lie point symmetries. Specifically, the study focuses on the case where the diffusion function is exponential, $g(u) = k_1 e^{k_2 u}$, to identify corresponding source functions $f(u)$ that allow for non-trivial symmetry reductions.

Methodology
The authors employ the Lie symmetry method to analyze the invariance properties of the PDE. The procedure involves:

1. Infinitesimal Generator: Defining a vector field $X = \xi(t, x, u)\partial_x + \tau(t, x, u)\partial_t + \eta(t, x, u)\partial_u$.
2. Prolongation: Calculating the second prolongation $X^{(2)}$ of the vector field to account for derivatives up to second order.
3. Determining Equations: Applying the invariance condition $X^{(2)}(u_t - f(u) - \frac{1}{x}(x g(u) u_x)_x) = 0$ on the solution manifold. By substituting the prolongation formulas and equating coefficients of various partial derivatives of $u$ to zero, a system of determining equations is derived.
4. Classification: Solving this overdetermined system under the constraint $g(u) = k_1 e^{k_2 u}$ to classify the allowable forms of $f(u)$.
5. Symmetry Reduction: For each identified symmetry, constructing the characteristic equations to find similarity variables and transforming the original PDE into reduced ordinary differential equations (ODEs).

Key Contributions and Results
The primary contribution is the complete classification of source functions $f(u)$ that yield Lie symmetries beyond time translation when $g(u) = k_1 e^{k_2 u}$. The authors prove that only three specific types of $f(u)$ satisfy this condition:

• Case (a): $f(u) = k_3 e^{k_4 u}$ (with $k_3, k_4 \neq 0$).
  • Symmetry: The equation admits an additional symmetry $X_2 = (k_4 - k_2)x \partial_x + 2k_4 t \partial_t - 2 \partial_u$.
  • Reduction: This leads to a similarity variable $z = x^2 t^{\frac{k_2}{k_4 - 1}}$ and reduces the PDE to a second-order nonlinear ODE involving $h(z)$. The authors note that if $k_2 = k_4$, this recovers a known case from previous literature, but the case $k_2 \neq k_4$ is a new finding.
• Case (b): $f(u) = k_3 e^{k_2 u} + k_5$ (with $k_3, k_5 \neq 0$).
  • Symmetry: The equation admits $X_2 = e^{-k_2 k_5 t} \partial_t + k_5 e^{-k_2 k_5 t} \partial_u$.
  • Reduction: The similarity variable is $z = x$, and the invariant solution takes the form $u = k_5 t + \frac{1}{k_2} \ln(p(x))$. The reduced equation is a linear ODE solvable in terms of Bessel functions of the first and second kind ($J_0$ and $Y_0$).
• Case (c): $f(u) = k_5$ (with $k_5 \neq 0$).
  • Symmetry: The equation admits two additional symmetries: the same $X_2$ as in Case (b) and $X_3 = k_2 x \partial_x + 2 \partial_u$.
  • Reduction: Using $X_3$, the reduction yields an invariant solution $u(x,t) = \frac{1}{k_2} \ln(x^2/p(t))$, where $p(t)$ satisfies a first-order linear ODE with an exponential solution.

The paper also provides explicit invariant solutions for these cases. For specific parameter choices (e.g., $k_2 = k_4$ or specific integer ratios), the reduced ODEs are solved analytically using Bessel functions. For other cases (e.g., Example 2 where $k_4 = 2k_2$), the authors demonstrate the ability to visualize the solution surface using computational tools (Maple), even when an exact closed-form solution is not immediately available.

Significance and Claims
The authors claim that their work extends previous studies (cited as [7, 16, 17]) which had only examined particular combinations of power-law or exponential functions for $f(u)$ and $g(u)$. By systematically solving the determining equations, this paper:

1. Identifies that the exponential diffusion function $g(u) = k_1 e^{k_2 u}$ restricts the admissible source terms $f(u)$ to exactly three specific forms for non-trivial symmetries to exist.
2. Completes the classification for the exponential diffusion case, including scenarios (such as $k_2 \neq k_4$ in Case a) that were not covered in prior literature.
3. Provides the corresponding reduced ordinary differential equations and invariant solutions for all identified cases, thereby enlarging the set of known solvable forms for the generalized Fisher equation in cylindrical coordinates.

The work concludes that while the Lie symmetry method does not guarantee exact solutions for all PDEs, it successfully transforms this generalized Fisher equation into solvable or analyzable ODEs for the identified specific functional forms.