Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

Abstract

This paper aims to solve the nonlinear two-point fuzzy boundary value problem (TPFBVP) using approximate analytical methods. Most fuzzy boundary value problems cannot be solved exactly or analytically. Even if the analytical solutions exist, they may be challenging to evaluate. Therefore, approximate analytical methods may be necessary to consider the solution. Hence, there is a need to formulate new, efficient, more accurate techniques. This is the focus of this study: two approximate analytical methods-homotopy perturbation method (HPM) and the variational iteration method (VIM) is proposed. Fuzzy set theory properties are presented to formulate these methods from crisp domain to fuzzy domain to find approximate solutions of nonlinear TPFBVP. The presented algorithms can express the solution as a convergent series form. A numerical comparison of the mean errors is made between the HPM and VIM. The results show that these methods are reliable and robust. However, the comparison reveals that VIM convergence is quicker and offers a swifter approach over HPM. Hence, VIM is considered a more efficient approach for nonlinear TPFBVPs.

Higher Derivative Block Method with Generalised Steplength for Solving First-Order Fuzzy Initial Value Problems

“…According to [4], the crisp equation can be modelled in a fuzzy form using the definition of fuzzy theory, which is given in Section 2. The uncertain behaviour of a capacitor using the voltage, capacitance, or resistance of the circuit current is defined as triangular fuzzy numbers.…”

“…Numerous researchers have developed different numerical methods [1][2][3][4][5][6] to solve problems in the form of Equation (1), however, the major problem encountered is that these existing numerical methods give a low level of accuracy in terms of absolute error due to order of the method used. Specifically, researchers considered the use of linear multistep methods implemented in predictor-corrector mode (a non-self-starting approach with low accuracy) as seen in studies [7,8].…”