Solution of variable order fractional differential equations using Homotopy Analysis Method

This paper demonstrates the reliability and effectiveness of the Homotopy Analysis Method in obtaining approximate solutions for variable-order fractional diffusion equations, including cases where the derivative order varies with space, time, or other parameters.

The Gist

Imagine you are trying to predict how a drop of ink spreads through a glass of water. In a perfect, calm world, this happens at a steady, predictable pace. Mathematicians have long used "constant order" equations to describe this smooth spreading.

But the real world is messy. Imagine that the water isn't just water; it's a sponge with holes of different sizes, or the temperature changes as the ink moves, or the ink itself gets thicker over time. In these scenarios, the speed of the spreading ink doesn't stay constant—it speeds up, slows down, or changes direction depending on where it is and when it is there.

This is what the paper by Vivek Mishra and S. Das is tackling. They are solving a very complex math puzzle: How do we model diffusion (spreading) when the rules of the game keep changing?

Here is a breakdown of their work using simple analogies:

1. The Problem: The "Shape-Shifting" Rulebook

Usually, math models for diffusion use a fixed rulebook. Think of it like a recipe that says, "Stir for exactly 5 minutes."

However, in complex environments (like blood flowing through a tangled web of veins, or pollution moving through a rocky aquifer), the "stirring time" isn't fixed. It might be 5 minutes at the start, but 10 minutes in the middle, and 2 minutes at the end.

• The Old Way: Scientists tried to force these changing speeds into a fixed rulebook by adding complicated "coefficients" (extra numbers that tweak the math). It was like trying to fit a square peg in a round hole.
• The New Way (Variable Order): The authors use Variable Order Fractional Calculus. Imagine a rulebook where the "5 minutes" instruction literally changes its number based on your location in the room. If you are near the door, it says 2 minutes; if you are near the window, it says 10. This captures the "memory" and changing nature of the system much better.

2. The Tool: The "Homotopy Analysis Method" (HAM)

Solving equations where the rules change constantly is incredibly hard. It's like trying to solve a Rubik's Cube where the colors on the faces keep shifting while you are twisting it.

The authors use a technique called the Homotopy Analysis Method (HAM). Here is how to visualize it:

• The Analogy of the Bridge: Imagine you have a problem that is too hard to solve directly (a deep canyon). You know the answer to a simple version of the problem (a small stream you can easily jump over).
• The Process: HAM builds a bridge between the simple problem and the hard problem. It starts with the easy solution and slowly, step-by-step, stretches and bends it toward the complex reality.
• The "Convergence Control Parameter" (The Steering Wheel): As they build this bridge, they use a special knob called the convergence control parameter (denoted by the Greek letter $\eta$ or h).
  • If you turn the knob too far one way, the bridge collapses (the math goes wild).
  • If you turn it the other way, the bridge doesn't reach the other side.
  • The authors' job is to find the perfect setting for this knob so the bridge is stable and leads exactly to the right answer.

3. What They Did

The authors applied this "bridge-building" method to two specific types of "shape-shifting" diffusion problems:

1. The Linear Case: A standard diffusion equation where the "order" (the rule of change) varies with time and space. They checked their math against existing computer simulations and found their bridge matched perfectly.
2. The Non-Linear Case: A more chaotic scenario where the diffusion interacts with itself (like a chemical reaction happening while the ink spreads). This is a "first of its kind" solution for this specific type of variable-order problem.

4. The Results: Why It Matters

The paper proves that HAM is a reliable, flexible, and powerful tool.

• No "Small Number" Requirement: Many math methods only work if the changes in the system are tiny. HAM works even when the changes are huge.
• Precision: By minimizing the "residual error" (the difference between their guess and the actual math), they found the perfect "steering wheel" setting ($\eta \approx -0.97$ for the first problem and $\eta \approx -0.13$ for the second).
• Real-World Impact: This method helps scientists better understand:
  • How drugs move through the human body (which is full of different tissues).
  • How oil moves through porous rock.
  • How heat spreads in materials that change properties as they heat up.

The Bottom Line

Think of this paper as a new, highly sophisticated GPS for navigating a city where the traffic laws change every block. Instead of getting stuck or taking a wrong turn, the Homotopy Analysis Method gives scientists a flexible, step-by-step map that adapts to the changing rules, ensuring they arrive at the correct solution for complex, real-world diffusion problems.

Technical Summary

1. Problem Statement

The paper addresses the challenge of solving Variable Order Fractional Differential Equations (VOFDEs), specifically focusing on variable order fractional diffusion equations.

• Context: While constant-order fractional calculus is widely used for modeling diffusion in homogeneous media, it fails to accurately describe processes in heterogeneous media (e.g., porous media) where diffusion properties change with space, time, or external disturbances.
• The Gap: In such complex systems, the order of the fractional derivative, denoted as $\alpha$, is not constant but varies as a function of space ($x$), time ($t$), or both ($\alpha(x,t)$).
• Objective: The authors aim to solve two specific types of variable order diffusion equations:
  1. A linear variable order time-fractional diffusion equation (without a source term).
  2. A nonlinear variable order diffusion equation involving a reaction term.
• Novelty: The authors claim that while numerical methods (like finite differences) have been used previously, no researcher had attempted to solve these specific variable order equations using the Homotopy Analysis Method (HAM) prior to this work.

2. Methodology: Homotopy Analysis Method (HAM)

The authors employ the Homotopy Analysis Method (HAM), a powerful semi-analytical technique proposed by S.J. Liao. Unlike perturbation methods, HAM does not rely on small physical parameters.

Key Steps in the Methodology:

1. Definitions: The study utilizes the Caputo-type variable order fractional derivative and the variable order Riemann-Liouville integral.
2. Operator Construction:
   • A Linear Operator ($L$) is chosen, typically based on the highest order fractional derivative term.
   • A Nonlinear Operator ($N$) is constructed to represent the entire differential equation, including the source terms and nonlinearities.
3. Zero-Order Deformation Equation: A homotopy is constructed to deform the initial guess into the exact solution.
4. $m$-th Order Deformation Equations: The problem is decomposed into a sequence of linear sub-problems. The $m$-th order deformation equation is derived as:
   $$L[u_m(x,t) - \chi_m u_{m-1}(x,t)] = \hbar R_m(\vec{u}_{m-1})$$
   Where:
   • $\chi_m$ is a coefficient (0 if $m \leq 1$, 1 if $m > 1$).
   • $\hbar$ is the convergence control parameter.
   • $R_m$ is the residual error of the $(m-1)$-th order approximation.
5. Series Solution: The solution is expressed as an infinite series $u(x,t) = \sum_{n=0}^{\infty} u_n(x,t)$.
6. Convergence Control: The critical step involves determining the optimal value of the convergence control parameter ($\hbar$). The authors minimize the averaged squared residual error ($E_m$) to ensure the series solution converges rapidly and accurately.

3. Key Contributions

• First Application of HAM to VOFDEs: The paper establishes the first successful implementation of HAM for solving variable order fractional diffusion equations, expanding the applicability of HAM beyond constant-order problems.
• Handling Variable Orders: The method effectively handles cases where the order $\alpha$ depends on time ($t$), space ($x$), or a combination ($\alpha = 0.8 + 0.2xt$).
• Nonlinear Dynamics: The authors successfully solve a nonlinear variable order diffusion equation with a reaction term, providing stability analysis for different cases, which is described as a first in the context of variable order fractional systems.
• Analytical Series Solution: Unlike purely numerical methods that provide discrete points, HAM provides a series solution that offers deeper insight into the functional behavior of the system.

4. Results and Discussion

The authors validated their method through two case studies:

Case 1: Linear Variable Order Diffusion

• Equation: $D_t^{\alpha(x,t)} u = K \frac{\partial^2 u}{\partial x^2}$ with $\alpha(x,t) = 0.8 + 0.2xt$.
• Validation: The results were compared with existing numerical solutions by Sun et al. [25].
• Convergence: By minimizing the residual error, the optimal $\hbar$ was found to be -0.975296.
• Accuracy: With just five terms in the series, the residual error dropped to $2.18 \times 10^{-16}$, demonstrating extremely high precision. The evolution curves matched existing literature perfectly.

Case 2: Nonlinear Variable Order Diffusion

• Equation: A nonlinear equation involving diffusion and a reaction term ($u(1-u)$).
• Convergence: The optimal $\hbar$ was determined to be -0.134256.
• Performance: The residual error decreased significantly as the number of terms increased (from 0.289 for 2 terms to 0.0056 for 4 terms).
• Visualization: The study provided 3D plots showing the evolution of $u(x,t)$ over space and time, confirming the stability and physical relevance of the solution.

5. Significance and Conclusion

• Reliability: The study demonstrates that HAM is a reliable and effective tool for variable order problems, capable of handling derivatives where the order varies with space, time, or other parameters.
• Flexibility: The method's ability to choose initial approximations and control convergence via $\hbar$ makes it superior to methods constrained by small/large parameters.
• Physical Relevance: The results confirm that variable order models are essential for accurately describing diffusion in heterogeneous media and systems with changing memory properties.
• Future Impact: This work opens the door for using HAM to solve a broader class of complex fractional differential equations in physics, engineering, and biology where constant-order models are insufficient.

In summary, the paper successfully bridges the gap between advanced fractional calculus theory and practical analytical solution techniques, proving that HAM is a robust method for tackling the complexities of variable order fractional diffusion.