[Paper Review] The Laplace Transform Method for Linear Differential Equations of the Fractional Order

Igor Podlubný|ArXiv.org|Oct 30, 1997

Fractional Differential Equations SolutionsMathematics2 references173 citations

TL;DR

This paper introduces a systematic Laplace transform-based method for solving linear fractional differential equations with constant coefficients, leveraging the two-parameter Mittag-Leffler function and its Laplace transform. The key contribution is a general formula for the fractional Green’s function of n-term equations, enabling analytical solutions via series inversion and extending applicability to standard and sequential fractional derivatives.

ABSTRACT

The Laplace transform method for solving of a wide class of initial value problems for fractional differential equations is introduced. The method is based on the Laplace transform of the Mittag-Leffler function in two parameters. To extend the proposed method for the case of so-called "sequential" fractional differential equations, the Laplace transform for the ''sequential'' fractional derivative is also obtained. Besides that, tools necessary for testing candidate solutions by direct substitution in corresponding equations are introduced: fractional derivatives of the Mittag-Leffler function and the rule for the fractional differentiation of integrals depending on a parameter. Definition of the fractional Green's function is given and some of its properties, necessary for constructing solutions of initial-value problems for fractional linear differential equations, are presented. Explicit expressions for the fractional Green's function for the special cases of one-, two-, three- and four-term equations are given, as well as the explicit expression for an arbitrary n-term fractional linear ordinary differential equation with constant coefficients. Several examples of solution of various types of fractional differential equations, including fractional diffusion-wave equation. The bibliography (54 items) covers many field of possible application of fractional derivatives.

Motivation & Objective

• To develop a unified, effective method for solving linear fractional differential equations of arbitrary real order.
• To overcome limitations of existing methods such as series expansions or iteration, which are inefficient for complex equations.
• To provide a systematic approach using the Laplace transform and Mittag-Leffler functions for initial-value problems.
• To define and derive explicit expressions for the fractional Green’s function across various n-term equations.
• To extend the method to both standard and sequential fractional derivatives, ensuring consistency in solution structure.

Proposed method

• The method employs the Laplace transform of the two-parameter Mittag-Leffler function $ E_{\alpha,\beta}(z) $, derived from integral transforms.
• It uses the Laplace transform of fractional derivatives, particularly for sequential operators, to derive solution kernels.
• A general solution framework is established using the inverse Laplace transform of the resolvent $ g_n(p) = \left(\sum_{k=0}^n a_k p^{\beta_k}\right)^{-1} $, where $ \beta_n > \cdots > \beta_0 $.
• The solution is expressed as a series involving multinomial coefficients and generalized Mittag-Leffler functions, enabling term-by-term inversion.
• The method incorporates tools for verifying candidate solutions, including fractional derivatives of Mittag-Leffler functions and differentiation rules for parameter-dependent integrals.
• Explicit formulas for the fractional Green’s function are derived for one- to four-term equations, culminating in a general n-term expression.

Experimental results

Research questions

• RQ1 How can the Laplace transform be systematically applied to solve linear fractional differential equations with arbitrary real-order derivatives?
• RQ2 What is the structure of the fractional Green’s function for general n-term linear fractional differential equations with constant coefficients?
• RQ3 How do solutions differ between standard and sequential fractional derivatives, and what common structure underlies both?
• RQ4 Can the Mittag-Leffler function and its Laplace transform be used to derive closed-form solutions for initial-value problems?
• RQ5 What is the role of the Wright function and multinomial expansions in the inversion process of fractional Laplace transforms?

Key findings

• The Laplace transform of the two-parameter Mittag-Leffler function $ E_{\alpha,\beta}(z) $ is derived and used as a foundational tool for solving fractional differential equations.
• Explicit expressions for the fractional Green’s function are obtained for one- to four-term equations, with the general n-term case expressed via a series involving multinomial coefficients.
• For the n-term equation $ \sum_{k=0}^n a_k D^{\beta_k} y(t) = f(t) $, the solution is given by $ y(t) = \int_0^t G_n(t-\tau) f(\tau) d\tau $, where $ G_n(t) $ is the fractional Green’s function.
• The fractional Green’s function for the general n-term equation is expressed as $ G_n(t) = \frac{1}{a_n} \sum_{m=0}^\infty \frac{(-1)^m}{m!} \sum_{\substack{k_0+\cdots+k_{n-2}=m \\ k_i \geq 0}} (m; k_0,\dots,k_{n-2}) \prod_{i=0}^{n-2} \left(\frac{a_i}{a_n}\right)^{k_i} t^{\gamma m + \delta} E_{\gamma, \epsilon}^{(m)}(\cdot) $, with $ \gamma = \beta_n - \beta_{n-1} $, and $ E_{\gamma, \epsilon}^{(m)} $ denoting the m-th derivative of the Mittag-Leffler function.
• The method successfully handles both standard and sequential fractional derivatives, yielding consistent Green’s functions across both formulations.
• The approach enables analytical solutions for a broad class of initial-value problems in fractional calculus, including those arising in physics, engineering, and finance.