On solution of Diffusion Equation using Conformable Laplace Transform

Technical Summary: Solution of Fractional Diffusion Equation Using Conformable Laplace Transform

Problem Statement
The paper addresses the analytical treatment of fractional differential equations, specifically focusing on time-fractional diffusion equations. While classical fractional derivatives (such as Riemann–Liouville and Caputo) are mathematically rigorous, they often fail to satisfy fundamental properties of ordinary calculus, such as the standard product and chain rules. This limitation complicates the analytical solution of fractional differential equations. Although the conformable fractional derivative, introduced by Khalil et al., preserves many properties of classical calculus, the systematic development of its associated Laplace transform—specifically regarding inversion and convolution theorems—remains an area requiring further formalization to handle initial-boundary value problems effectively.

Methodology
The authors employ the Conformable Fractional Laplace Transform (CFLT), defined for a function $f(t)$ on $[a, \infty)$ as:
$$L^\alpha_a(f(t))(s) = \int_a^\infty e^{-s \frac{(t-a)^\alpha}{\alpha}} (t-a)^{\alpha-1} f(t) \, dt$$
where $\alpha \in [0, \infty)$.

The methodology proceeds through the following logical steps:

1. Theoretical Foundation: The paper establishes the relationship between the CFLT and the classical Laplace transform via a change of variables $\phi_a(t) = \frac{(t-a)^\alpha}{\alpha}$. This mapping allows the authors to extend elementary properties of the classical transform (linearity, shifting, scaling, differentiation, and integration) to the conformable fractional domain.
2. Derivation of Key Theorems:
   • Derivative and Integral Rules: The authors derive the CFLT of conformable derivatives and integrals, providing general formulas for $n$-th order derivatives and specific cases where the transform reduces to classical forms.
   • Inversion Theorems: The paper develops the inversion theorem, expressing the inverse CFLT in terms of the classical inverse Laplace transform evaluated at the transformed variable. It further provides the complex inversion formula (Bromwich integral) and the residue form for meromorphic functions.
   • Convolution Theorem: A new definition for fractional convolution, denoted as $*_\alpha$, is introduced:
     $$(f *_\alpha g)(t) = \int_0^t f(p) g\left((t^\alpha - p^\alpha)^{1/\alpha}\right) p^{\alpha-1} \, dp$$
     The authors prove that this operator satisfies commutativity, associativity, and distributivity. They also establish boundedness in weighted $L^1_\alpha$ spaces and prove a Young-type inequality for $L^n_\alpha$ spaces.
3. Application to Diffusion Equations: The developed framework is applied to solve initial-boundary value problems (IBVPs) for the time-fractional diffusion equation in both semi-infinite and finite domains. The process involves transforming the partial differential equation (PDE) into an ordinary differential equation (ODE) in the spatial domain, solving the ODE, and applying the inverse CFLT (often utilizing the convolution theorem) to retrieve the time-domain solution.

Key Contributions and Results

• Extension of Classical Properties: The paper successfully extends all elementary properties of the classical Laplace transform to the conformable fractional setting, demonstrating that the CFLT behaves consistently with classical calculus under the conformable derivative definition.
• Inversion and Convolution Formulas: The primary theoretical contribution is the rigorous derivation of the inversion theorem and the convolution theorem for the CFLT. These tools are essential for solving differential equations analytically.
• Analytical Solutions: The paper provides explicit analytical solutions for several specific problems:
  • A first-order conformable fractional IBVP.
  • A time-fractional diffusion equation in a semi-infinite medium with a general boundary condition $u(0,t)=f(t)$, resulting in a solution expressed as a fractional convolution.
  • A time-fractional diffusion equation in a finite medium with mixed boundary conditions, solved using the residue theorem to yield a series solution.
  • A finite domain problem with Dirichlet boundary conditions, yielding a closed-form solution involving an exponential decay term dependent on $t^\alpha/\alpha$.
• Functional Analysis: The paper establishes the boundedness of the fractional convolution operator in weighted Lebesgue spaces, providing a functional-analytic foundation for the method.

Significance and Claims
The authors claim that the Conformable Fractional Laplace Transform serves as a powerful analytical tool that bridges the gap between classical and fractional models. By preserving the structural simplicity of the classical Laplace transform (such as the product and chain rules) while extending applicability to fractional-order systems, the method simplifies the solution process for fractional diffusion equations.

The paper concludes that the developed methodology offers a consistent framework for solving fractional initial-boundary value problems. The results demonstrate that the conformable fractional model can yield explicit analytical solutions for diffusion phenomena in both finite and semi-infinite domains, validating the CFLT as a viable alternative to more complex fractional calculus approaches. The authors suggest that this framework can be extended to investigate more complex fractional partial differential equations in physics and engineering, though they do not propose specific new experimental applications beyond the mathematical examples provided.