[Paper Review] Solving nth order fuzzy differential equation by fuzzy Laplace transform

Latif Ahmad, Muhammad Shoaib Farooq+1 more|arXiv (Cornell University)|Mar 2, 2014

Fuzzy Systems and Optimization9 references3 citations

TL;DR

This paper generalizes the fuzzy Laplace transform (FLT) for solving nth-order fuzzy initial value problems (FIVPs) using strongly generalized differentiability. It introduces an nth-derivative theorem for FLT, enabling analytical solutions of FIVPs via Laplace inversion, with examples solved using MATLAB, demonstrating the method's effectiveness for arbitrary-order fuzzy differential equations under generalized differentiability.

ABSTRACT

In this paper, we generalize the fuzzy Laplace transformation (FLT) for the nth derivative of a fuzzy-valued function named as nth derivative theorem and under the strongly generalized differentiability concept, we use it in an analytical solution method for the solution of an nth order fuzzy initial value problem (FIVP). This is a simple approach toward the solution of nth order fuzzy initial value problem (FIVP) by the nth generalized (FLT) form, and then we can use it to solve any order of FIVP. The related theorems and properties are proved. The method is illustrated with the help of some examples. We use MATLAB to evaluate the inverse Laplace transform.

Motivation & Objective

• To extend the fuzzy Laplace transform (FLT) to the nth derivative of fuzzy-valued functions for solving higher-order fuzzy differential equations.
• To establish a systematic analytical method for solving nth-order fuzzy initial value problems (FIVPs) using the generalized differentiability concept.
• To prove the nth-derivative theorem for FLT and related properties, ensuring theoretical consistency and applicability.
• To demonstrate the method's effectiveness through numerical examples solved using MATLAB's inverse Laplace transform functionality.
• To provide a unified framework applicable to any order of FIVP under the strongly generalized differentiability condition.

Proposed method

• The paper introduces an nth-derivative theorem for the fuzzy Laplace transform, extending FLT to handle the nth-order derivative of fuzzy-valued functions.
• It applies the generalized Hukuhara differentiability concept to define derivatives of fuzzy functions, allowing for multiple solution branches in FIVPs.
• The method transforms the nth-order FIVP into an algebraic equation in the Laplace domain using the nth-derivative theorem.
• Solutions are obtained by solving the transformed algebraic equations for the lower and upper α-cuts of the fuzzy function.
• Inverse Laplace transforms are computed using MATLAB to recover the time-domain fuzzy solution in parametric form.
• The approach distinguishes between (1)- and (2)-differentiability cases, leading to different algebraic forms for the transformed equations.

Experimental results

Research questions

• RQ1 Can the fuzzy Laplace transform be generalized to handle the nth derivative of a fuzzy-valued function?
• RQ2 How can the nth-derivative theorem of FLT be formulated and proven under the concept of strongly generalized differentiability?
• RQ3 What is the analytical solution method for an nth-order fuzzy initial value problem using the generalized FLT?
• RQ4 How does the choice of differentiability type (1- or 2-differentiability) affect the solution structure of the FIVP?
• RQ5 Can the proposed method be applied to solve FIVPs of arbitrary order using a consistent and systematic framework?

Key findings

• The paper successfully generalizes the fuzzy Laplace transform to the nth derivative, establishing a new nth-derivative theorem for fuzzy-valued functions.
• The method enables analytical solutions of nth-order FIVPs by transforming the problem into algebraic equations in the Laplace domain.
• For a fourth-order FIVP, the solution is derived in parametric form as $\underline{y}(t,r)$ and $\overline{y}(t,r)$, involving trigonometric, exponential, and hyperbolic functions.
• The inverse Laplace transform is computed using MATLAB, confirming the feasibility of numerical implementation for complex fuzzy solutions.
• The solution structure differs under (1)- and (2)-differentiability, with distinct algebraic forms derived for each case.
• The method is validated through multiple examples, showing consistency and applicability across different orders and differentiability types.