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On a Boundary-Initial Value Problem for Fractional Differential Equation with Sequential Caputo derivatives

This paper investigates a boundary-initial value problem for a fractional differential equation with sequential Caputo derivatives by deriving its exact analytic solution in terms of the bivariate Mittag-Leffler function and developing a corresponding numerical scheme using the L1-finite element method.

Original authors: Fayziev Yusuf, Jumaeva Shakhnoza

Published 2026-04-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Fayziev Yusuf, Jumaeva Shakhnoza

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A System with "Super-Memory"

Imagine you are pushing a heavy swing. In the real world, if you stop pushing, the swing slows down and stops because of friction. This is how standard physics works: the past doesn't really matter once you stop; only the current state counts.

But in the fractional world described in this paper, the swing has a super-memory. It doesn't just remember where it is now; it remembers every single push and pull it has ever felt, and that memory influences how it moves right now. The "heavier" the memory, the slower and more sluggish the swing becomes.

The authors of this paper are studying a specific type of equation that models this kind of "super-memory" system, but with a twist: two different types of memory interacting with each other.

The Problem: Two Layers of Memory

The equation they are solving looks like this:
$D^\beta (D^\alpha u) + D^\beta u - u_{xx} = f$

Let's break this down with a metaphor:

The "Sequential" Twist ( $D^\beta(D^\alpha u)$ ):
Imagine you are trying to measure the speed of a car, but your speedometer is broken and only works if you first measure the car's position, and then measure how that position changes.
- Standard Math: You just measure the change in position once.
- This Paper: You measure the change in position ( $\alpha$ ), and then you measure the change of that result ( $\beta$ ).
- Why it matters: In normal math, doing two steps is the same as doing one big step. In this "fractional" world, doing two steps in a row is not the same as doing one big step. The order matters, and the "memory" gets complicated. It's like trying to bake a cake by first mixing the flour, then letting the mixer sit, and then adding the eggs. The result is different than if you just mixed everything at once.
The Boundary Conditions (The Walls):
The problem takes place in a box (from $x=0$ to $x=1$ ). The walls are fixed. If this were a guitar string, the ends are tied down tight. The string can vibrate, but it can't move at the very edges.
The Goal:
The authors want to know: If we know how the string started (initial conditions) and how it was pushed (the force $f$ ), can we predict exactly where the string will be at any future time?

The Solution: The "Magic Recipe" (Mittag-Leffler Function)

To solve this, the authors didn't just guess; they found an exact mathematical "recipe" for the answer.

The Fourier Method: Imagine the vibrating string is made up of many different pure musical notes (frequencies) playing at once. The authors broke the complex problem down into these individual notes.
The Bivariate Mittag-Leffler Function: This is the fancy name for the "Magic Recipe."
- In normal calculus, the answer to a vibrating string is usually a sine wave or an exponential decay (like a ball rolling to a stop).
- In this fractional world, the answer is a Bivariate Mittag-Leffler function. Think of this as a "super-sine wave" that can stretch, shrink, and remember its past in complex ways. It's the specific shape the solution takes when it has this double-layered memory.

They proved that this recipe always works (existence) and that there is only one correct answer (uniqueness) as long as the memory parameters ( $\alpha$ and $\beta$ ) follow certain rules (specifically, their sum must be greater than 1).

The Computer Experiment: Teaching the Computer to "Feel" the Singularity

Solving this on a computer is tricky. Why? Because right at the very beginning ( $t=0$ ), the math gets "spiky" or "singular." It's like trying to take a photo of a lightning strike; if you take the picture at regular intervals, you might miss the most important part.

The Problem: Standard computer methods use equal time steps (like a clock ticking every second). This is bad for this problem because the action happens so fast at the start that a standard clock misses the details.
The Fix (Graded Mesh): The authors used a "smart clock."
- At the start ( $t=0$ ), the clock ticks very, very fast (tiny time steps) to catch all the rapid changes.
- As time goes on and things settle down, the clock slows down to larger steps.
- Analogy: It's like driving a car. You drive very slowly and carefully when leaving a crowded parking lot (the start), but once you are on the open highway, you can speed up and check your speed less often.

They tested this method with a "manufactured solution" (a fake problem where they already knew the answer). The computer's answer matched the known answer almost perfectly, proving their method works.

What Did They Discover?

By running simulations, they found some fascinating behaviors:

The "Rebound" Effect: Usually, energy in a system just fades away to zero. But with these specific memory settings, the system's energy drops to zero, hits a "floor," and then bounces back up slightly before settling.
- Analogy: Imagine a pendulum that, instead of just stopping at the bottom, swings slightly past the center, stops, and then wiggles back. It has too much "inertia" from its memory to stop cleanly.
The Trade-Off: They found that you can swap the two memory parameters ( $\alpha$ and $\beta$ ) to get the same result.
- Analogy: It's like baking a cake. You can use a little more sugar and a little less flour, or a little less sugar and a little more flour, and end up with a cake that tastes the same. This gives engineers flexibility in real-world modeling.

Summary

This paper is about solving a very difficult math problem involving a system with two layers of memory.

They found the exact mathematical formula (using a special function called the Bivariate Mittag-Leffler) to predict how the system behaves.
They built a smart computer algorithm that zooms in on the start of the process to get accurate results.
They discovered that this system behaves strangely: it can bounce back after losing energy, and the two memory settings can trade places to produce the same outcome.

This work helps scientists and engineers better model real-world phenomena like heat flow in complex materials, electrical circuits with memory, or biological systems where the past heavily influences the present.

1. Problem Statement

The paper investigates a specific Boundary-Initial Value Problem (BIVP) for a multi-term fractional partial differential equation (FPDE) defined on the domain $\Omega = \{(x, t) : 0 < x < 1, 0 < t < T\}$ .

The governing equation is:
$D^\beta_t (D^\alpha_t u(x, t)) + D^\beta_t u(x, t) - u_{xx}(x, t) = f(x, t)$
where:

$D^\alpha_t$ and $D^\beta_t$ are sequential Caputo fractional derivatives with orders $0 < \alpha < 1$ and $0 < \beta < 1$ .
The term $D^\beta_t (D^\alpha_t u)$ represents the sequential derivative, which does not satisfy the classical index law (i.e., $D^\beta_t D^\alpha_t \neq D^{\alpha+\beta}_t$ ) due to memory effects and non-commutativity.
Initial Conditions:
- $u(x, 0) = \phi(x)$
- $D^\alpha_t u(x, 0) = \psi(x)$
Boundary Conditions:
- $u(0, t) = u(1, t) = 0$ (Homogeneous Dirichlet conditions).

The authors define a regular solution as a function $u(x,t)$ where the sequential derivative, spatial second derivative, and the first sequential derivative are continuous in the appropriate domains.

2. Methodology

The paper employs a dual approach combining analytical theory and numerical implementation.

A. Analytical Approach

Fourier Method: The solution is expanded into a sine series: $u(x, t) = \sum U_k(t) \sin(k\pi x)$ . This transforms the PDE into a sequence of ordinary differential equations (ODEs) for the coefficients $U_k(t)$ .
Auxiliary ODE Problem: The authors solve a supplementary Cauchy problem for the ODE:
$D^\beta_t (D^\alpha_t y) + D^\beta_t y + \lambda y = f(t)$
using the Laplace transform.
Special Functions: The exact analytic solution is derived in terms of the Bivariate Mittag-Leffler function ( $E_2$ ). The authors utilize properties of this function (convergence, integral representations, and asymptotic bounds) to prove the existence and uniqueness of the solution.
Convergence Analysis: Using the Weierstrass M-test and properties of Hölder spaces ( $\mathring{C}^a$ ), the authors prove that the infinite series solution and its derivatives converge uniformly on $\bar{\Omega}$ , provided specific conditions on the parameters and initial data are met.

B. Numerical Approach

Sequential Reformulation: To handle the nested sequential derivative numerically, an auxiliary variable $v(x, t) = D^\alpha_t u(x, t)$ is introduced. This converts the single high-order equation into a coupled system of two first-order sequential equations.
Spatial Discretization: A Finite Element Method (FEM) with piecewise linear elements (P1) is used for the spatial variable $x$ .
Temporal Discretization:
- L1 Scheme: The Caputo derivatives are approximated using the L1 formula.
- Graded Mesh: To address the weak singularity of the solution near $t=0$ (typical in fractional problems), a graded time mesh is employed: $t_n = T(n/N)^r$ , where $r \geq 1$ . The optimal grading parameter is chosen as $r = 2 - \min\{\alpha, \beta\}$ .
Algorithm: A fully discrete scheme is formulated, resulting in a linear system $A_n U^n = b_n$ at each time step, where the matrix $A_n$ depends on the stiffness and mass matrices, and the right-hand side $b_n$ incorporates history terms.

3. Key Contributions

Exact Analytic Solution: The paper derives the first exact analytic solution for this specific class of sequential fractional diffusion-wave equations in terms of the Bivariate Mittag-Leffler function.
Well-Posedness Proof: The authors rigorously prove the existence and uniqueness of the regular solution under the conditions:
- $\alpha + \beta > 1$
- $\alpha > \beta$
- Initial data $\phi, \psi$ and source $f$ belong to specific Hölder spaces.
Numerical Scheme Development: They propose a robust L1-FEM scheme utilizing a sequential reformulation and graded temporal mesh. This approach effectively handles the non-local nature of sequential derivatives and the initial singularity.
Property Analysis: Several new properties and bounds for the Bivariate Mittag-Leffler function are formulated and utilized to support the convergence proofs.

4. Results

Theoretical Validation: The series solution is proven to converge uniformly. The uniqueness is established via the completeness of the sine-Fourier series in $L^2(0,1)$ .
Numerical Accuracy:
- A "manufactured solution" test confirmed the method's accuracy.
- Error Analysis: The use of a graded mesh ( $r=1.6$ ) significantly outperformed uniform meshes. For $N=400$ , the $L^2$ error was reduced by a factor of $\approx 7.75$ compared to a uniform mesh.
- Convergence Rates: The graded mesh restored the convergence order to $>1.1$ , whereas the uniform mesh yielded suboptimal rates ( $\approx 0.75$ ).
Physical Insights (Parametric Study):
- Memory Effects: Smaller values of $\alpha$ and $\beta$ correspond to "stronger memory," slowing down the process.
- Oscillatory Behavior: As $\alpha$ and $\beta$ increase, the system transitions from purely diffusive decay to a damped oscillatory regime.
- Energy Rebound: For higher fractional orders, the system's energy drops to zero and then slightly "rebounds" before stabilizing, indicating an inertial effect where the system crosses the equilibrium point.
- Parameter Trade-off: Heatmaps revealed that different combinations of $(\alpha, \beta)$ can produce identical solution profiles, suggesting a trade-off useful for inverse problems and parameter identification.

5. Significance

This work is significant for several reasons:

Theoretical Advancement: It extends the theory of fractional differential equations to sequential operators with mixed boundary-initial conditions, a complex area where standard semigroup properties often fail.
Modeling Capability: The derived model is relevant for physical systems exhibiting multiple memory effects or multiple time scales (e.g., viscoelasticity, anomalous diffusion in complex media) where standard single-term fractional models are insufficient.
Computational Efficiency: The proposed numerical scheme provides a reliable tool for simulating these complex systems, demonstrating that graded meshes are essential for capturing the correct dynamics near the initial time.
Foundation for Future Research: The explicit solution and the identified "rebound" energy behavior provide a rigorous basis for future studies on inverse problems, control theory, and more complex identification problems involving sequential fractional derivatives.