[Paper Review] Solving fuzzy two-point boundary value problem using fuzzy Laplace transform

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

Fuzzy Systems and Optimization14 references3 citations

TL;DR

This paper proposes a novel analytical approach to solving fuzzy two-point boundary value problems (FBVPs) using the fuzzy Laplace transform (FLT) under generalized Hukuhara differentiability. By applying FLT to transform the fuzzy differential equation into an algebraic system, then applying boundary conditions to eliminate unknown constants, the method derives exact solutions for both the Schrödinger equation and homogeneous FBVPs, demonstrating FLT's effectiveness as a new tool for fuzzy boundary value problems.

ABSTRACT

A natural way to model dynamic systems under uncertainty is to use fuzzy boundary value problems (FBVPs) and related uncertain systems. In this paper we use fuzzy Laplace transform to find the solution of two-point boundary value under generalized Hukuhara differentiability. We illustrate the method for the solution of the well known two-point boundary value problem Schrodinger equation, and homogeneous boundary value problem. Consequently, we investigate the solutions of FBVPs under as a new application of fuzzy Laplace transform.

Motivation & Objective

• To develop a new analytical method for solving fuzzy two-point boundary value problems (FBVPs) under uncertainty.
• To extend the application of fuzzy Laplace transform (FLT) beyond initial value problems to boundary value problems.
• To demonstrate the feasibility and accuracy of FLT in solving second-order fuzzy differential equations with fuzzy boundary conditions.
• To provide exact solutions for the Schrödinger equation and homogeneous FBVPs using FLT under generalized Hukuhara differentiability.
• To establish a framework that can be extended to higher-order FBVPs.

Proposed method

• The fuzzy Laplace transform (FLT) is applied to the fuzzy differential equation, transforming it into an algebraic equation in the Laplace domain.
• The solution is derived in parametric form using the lower and upper functions of the fuzzy number, represented as $\underline{u}(r)$ and $\overline{u}(r)$.
• Boundary conditions are applied after transformation to eliminate unknown constants, using the fuzzy boundary values to solve for the constants in the transformed solution.
• The inverse fuzzy Laplace transform is applied to obtain the analytical solution in the time domain, expressed as fuzzy-valued functions.
• The method is validated using two examples: the Schrödinger equation and a homogeneous second-order FBVP with fuzzy boundary conditions.
• The solution is constructed by solving the transformed equations for the lower and upper functions separately, ensuring consistency with generalized H-differentiability.

Experimental results

Research questions

• RQ1 Can the fuzzy Laplace transform be effectively applied to solve second-order fuzzy two-point boundary value problems?
• RQ2 How does the solution obtained via FLT compare to classical solutions in terms of validity and structure under generalized H-differentiability?
• RQ3 What are the conditions under which the fuzzy solution remains a valid level set, particularly when the upper and lower solutions are not monotonic?
• RQ4 Can FLT be used to derive analytical solutions for the fuzzy Schrödinger equation with fuzzy boundary conditions?
• RQ5 Is the FLT-based method extendable to higher-order fuzzy boundary value problems?

Key findings

• The fuzzy Laplace transform successfully solves the fuzzy Schrödinger equation with fuzzy boundary conditions, yielding a complete analytical solution in parametric form.
• For the homogeneous FBVP $x''(t) - 3x'(t) + 2x(t) = 0$, the method produces exact solutions for both the lower and upper functions of the fuzzy-valued solution.
• The solution constants are determined by applying boundary conditions after transformation, ensuring consistency with the given fuzzy boundary values.
• The method ensures that the fuzzy solution remains valid as a level set, provided the upper and lower functions satisfy the required monotonicity conditions.
• The derived solution for the homogeneous problem shows that the crisp solution lies between the upper and lower solutions, confirming the method's reliability.
• The approach is extendable to $n$th-order FBVPs, as suggested by the authors in the conclusion.