Approximate Analytic–Numeric Fuzzy Solutions of Fuzzy Fractional Equations Using a Residual Power Series Approach

• Published in: Symmetry (Basel) Vol. 14; no. 4; p. 804
• Main Authors: Al-qudah, Yousef, Alaroud, Mohammed, Qoqazeh, Hamza, Jaradat, Ali, Alhazmi, Sharifah E., Al-Omari, Shrideh
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.04.2022
• Subjects: Applied mathematics; Approximation; Calculus; Differential equations; Error functions; Exact solutions; fractional series expansion; fuzzy fractional initial value problems; Fuzzy sets; Mathematicians; Partial differential equations; Power; Power series; residual error function; strongly generalized differentiability; Taylor series
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym14040804

In this article, we consider a reliable analytical and numerical approach to create fuzzy approximated solutions for differential equations of fractional order with appropriate uncertain initial data by the means of a residual error function. The concept of strongly generalized differentiability is utilized to introduce the fuzzy fractional derivatives. The proposed method provides a systematic scheme based on generalized Taylor expansion and minimization of the residual error function, so as to obtain the coefficients values of a fractional series based on the given initial data of triangular fuzzy numbers in the parametric form. The obtained approximated solutions are provided within an appropriate radius to the requisite domain in the form of rapidly convergent fractional series according to their parametric form. The method’s performance and applicability are verified by applying it on some numerical examples. The impact of r-levels and fractional order γ is presented quantitatively and graphically, showing the coincidence between the exact and the fuzzy approximated solutions. Moreover, for reliability and accuracy, our obtained results are numerically compared with the exact solutions and with results obtained using other methods described in the literature. This indicates that the proposed approach overcomes the difficulties that appear in other approaches to create fractional series solutions for varied uncertain natural problems arising within the fields of applied physics and engineering.