Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5, 1393-1405 2025 Publisher: Learning Gate DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate Β© 2025 by the author; licensee Learning Gate History: Received: 12 February 2025; Revised: 22 April 2025; Accepted: 25 April 2025; Published: 14 May 2025 * Correspondence: m.fadlallah@qu.edu.sa Solution of neutrosophic fractional differential equations by neutrosophic extension principal method Mohammed Nour A. Rabih1* 1Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia; m.fadlallah@qu.edu.sa, mohammednoor81@gmail.co (M.N.A.R.). Abstract: This paper aims at a new approach for finding the solution of neutrosophic fuzzy fractional differential equations (NFFDEs) based on the Zadeh’s Extension Principle method. NFFDEs combine fractional-order systems with uncertainty, which deals with truth, indeterminacy, and falsity information. This approach competently addresses the challenges modeled by both the fractional derivatives and the indeterminate constructions characteristic of neutrosophic systems. The paper frames the theoretical framework, advances the solution process, and validates the usefulness of the method. Theoretical and numerical results validate that the Extension Principle method conserves vital properties of the fundamental systems while providing flexible and inclusive representations of uncertainty. Keywords: Fractional derivative, Fractional differential equation, Fuzzy logic, Neutrosphic set theory. 1. Introduction Differential equation is very important techniques for theoretical Cooke [1] and Ross [2] as well as modelling Braun and Golubitsky [3] and Sobczyk [4] based study. If formed in continuous system. Several modifications and variation are already done in the wide field of research involving differential equations. In that context the order of any differential equation need not always be integer, it may be fractional order [5-8]. A fractional differential equation includes derivatives of non-integer (fractional) order which encompassing the concept of classical calculus. It models systems with memory and hereditary belongings, apprehending complex behaviours better than traditional differential equations ideology. These types of equations are extensively used in fields like physical sciences [9] biology inspired model [10] and financial analysis [11]. The solutions methodology is quite different and it needs more specialized techniques [12-15]. Theory based on uncertainty play important role for real world modelling now a days. There is several well know ideology to capture the uncertainty when modelling. Few concepts such as interval quantification [16] fuzzy set theory [17] intutionistic fuzzy set [18] theory etc. Fuzzy set consider the degree of belonging ness where as intuitionistic fuzzy set capture both the belongingness and non- belongingness [19, 20]. Apart from the previously mention settings neutrosophic set [21, 22] capture uncertainty than others. The idea of a neutrosophic set is significant because it spreads classical fuzzy set philosophies by letting the depiction of truth, indeterminacy and falsity by making it ideal for manage uncertain, incomplete, and inconsistent info. Contrasting traditional models that might strict boundaries or membership where as neutrosophic sets offer better flexibility and pragmatism in complex decision-making problem like site selection problem [23, 24] mathematical biology [25] Inventory control problem [26]. Transportation problem [27]. Graph theory [28] etc . This makes it mostly powerful in settings in uncertainty modelling. Differential equation with uncertainty is not new. The most popular differential equation with uncertainty is fuzzy differential equation, which have importance both in theoretical [29, 30] and 1394 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate modelling [31, 32] purposes. Although fuzzy differential equation have different variation like fuzzy fractional differential equation [33, 34] fuzzy delay differential equation [35, 36]. In other hand the neutrosophic differential equation [25, 37-44]. Is taken and solved by few researchers whereas neutrosophic fractional differential equation work is very rare [45]. The details comparative analysis is of published paper based on neutrosophic differential equation show in table 1. In that context we propose the neutrosphic fractional differential equation by neutrosophic extension principle. The structure of the paper is are follows: Section 1 describes preliminary introduction of related keyworks, Section 2 describes the preliminary ideas. Neutrosophic extension principle is defined in Section 3. The formulation of neutrosophic differential equation is addressed in Section 4. Section 5 stands for the solution strategy of neutrosophic fractional differential equation. Numerical example is illustrated in Section 6. Section 7 stand for conclusion and future research scope. Table 1. Comparative study of published neutrosophic differential equation Sl. No. Paper details Approaches used Type of differential equation Applications/ Theory 1 Sumathi and Antony Crispin Sweety [37] Generalized neutrosohic hukuhara differentiability Second order linear differential equation Theory 2 Mondal, et al. [38] (𝛼, 𝛽, 𝛾)-cut of neutrosophic function method First order system of differential equation Application 3 Sumathi and Priya [39] [39] Sumathi et al. (𝛼, 𝛽, 𝛾)-cut of neutrosophic function method First order linear homogeneous differential equation Theory and application both 4 Parikh and Sahni [40] [40] Parikh et al. Generalized Hukuhara neutrosophic differentiability First order linear differential equation Application 5 Rahaman, et al. [41] [41] Rahaman et al. generalized Neutrosophic derivative System of linear differential equation Theory and applications both 6 Acharya, et al. [46] [42] Acharya et al. Generalized Hukuhara neutrosophic differentiability First order linear non homogeneous differential equation Applications 7 Acharya, et al. [25] Generalized neutrosophic derivative System of linear non homogenous differential equation Applications 8 Kamal, et al. [42] Generalized Hukuhara Differentiability Second order linear homogeneous Theory 9 Mera, et al. [43] [45] Neutrosophic mathematical transform First order linear homogeneous Theory 10 Momena, et al. [44] generalized neutrosophic derivative; generalized neutrosophic derivative Application 2. Preliminaries and Basic Concepts Fuzzy Set: Zadeh [17] A fuzzy set οΏ½ΜƒοΏ½ is well-defined as a set of ordered pair, notationaly as (π‘Ÿ, πœ‡οΏ½ΜƒοΏ½(π‘Ÿ)), where π‘Ÿ ∈ 𝑋, where 𝑋 is nonempty universal set. The function πœ‡οΏ½ΜƒοΏ½(π‘Ÿ): 𝑋 β†’ [0,1], is called membership function. Zadeh’s extension principle: Zadeh [47] Let 𝐽 be a crisp set and οΏ½ΜƒοΏ½ be a fuzzy set in 𝐽. The function 𝑔: 𝐽 β†’ 𝐾 is defined by π‘˜ = 𝑔(𝑗), then the extension principle introduces a fuzzy set οΏ½ΜƒοΏ½ in 𝐾 as οΏ½ΜƒοΏ½ = {(π‘˜, πœ‡οΏ½ΜƒοΏ½(π‘˜)|π‘˜ = 𝑔(𝑗), 𝑗 ∈ 𝐽)} where, πœ‡οΏ½ΜƒοΏ½(π‘˜) = { (πœ‡οΏ½ΜƒοΏ½(𝑗)), 𝑖𝑓𝑓 𝑔 βˆ’1(π‘˜) β‰  πœ™ π‘—βˆˆπ‘”βˆ’1(π‘˜) 𝑠𝑒𝑝 0 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ . Note: Zadeh's Extension Principle permits crisp functions to work on fuzzy sets uncertainty. It encompasses functions by mapping fuzzy inputs functions to fuzzy outputs functions whereas conserving membership grades. The output's membership functions are resolute using the supremum of input memberships function that map to respective output value. This ideology is introductory in fuzzy 1395 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate arithmetic operations and fuzzy based decision-making. It permits applying in models to capture imprecise data. Example: Let, �̃�𝑓 be a fuzzy set given by the membership function as follows: πœ‡οΏ½ΜƒοΏ½π‘“ (𝑗) = { 0 𝑖𝑓 𝑗 ≀ 2 𝑗 βˆ’ 2 4 𝑖𝑓 2 ≀ 𝑗 < 6 1 𝑖𝑓 𝑗 = 6 10 βˆ’ 𝑗 4 𝑖𝑓 6 < 𝑗 ≀ 10 0 𝑖𝑓 𝑗 β‰₯ 10 Let us choose a function 𝐹(𝑗) = 3𝑗 + 2. Using the concept of Zadeh’s extension principle, another fuzzy set 𝐹(�̃�𝑓) can be determined. The membership function of 𝐹(�̃�𝑓) is obtained as follows: πœ‡πΉ(�̃�𝑓) (π‘˜) = { 0 𝑖𝑓 π‘˜ ≀ 8 π‘˜ βˆ’ 8 12 𝑖𝑓 8 ≀ π‘˜ < 20 1 𝑖𝑓 π‘˜ = 20 32 βˆ’ π‘˜ 12 𝑖𝑓 20 < π‘˜ ≀ 32 0 𝑖𝑓 π‘˜ β‰₯ 32 Note: The above concepts define that how we construct a fuzzy function by considering a parameter or variable as fuzzy in nature. Since the resulting function also obey the fuzzy rules. Neutrosophic Set: The extension of fuzzy sets is neutrosophic fuzzy sets. Here in the neutrosophic set Smarandache [48] one step forward of the Intuitionistic fuzzy set theory ideology. There exists several form of the said set. One of them is single-valued neutrosophic set. Consider an neutrosophic set �̃�𝑁𝑒 on universal set π‘ˆ is defined as �̃�𝑁𝑒 = {(𝑇𝑁𝑒(π‘˜), 𝐼𝑁𝑒(π‘˜), 𝐹𝑁𝑒(π‘˜)) ∢ π‘˜ ∈ U}, where 𝑇𝑁𝑒(π‘˜), 𝐼𝑁𝑒(π‘˜), 𝐹𝑁𝑒(π‘˜) ∢ U β†’ [0,1] are considered as the degree of truthness, degree of indeterministic and degree of falsity function respectively for π‘˜ ∈ U, such that 0 ≀ 𝑇𝑁𝑒(π‘˜), 𝐼𝑁𝑒(π‘˜), 𝐹𝑁𝑒(π‘˜) ≀ 3. Table 2. Comparison between Fuzzy, Intutitionistic fuzzy and Neutrosophic fuzzy set idea Sl. No. Set Associated functions Conditions Advantage Disadvantage 1 Fuzzy set Membership function (πœ‡π΄(π‘₯)) 0 ≀ πœ‡π΄(π‘₯) ≀ 1 Simple and for vague concepts. No way to represent contradiction in data. 2 Intutitionistic fuzzy set Membership and non- membership (πœ‡π΄(π‘₯), πœ—π΄(π‘₯)) 0 ≀ πœ‡π΄(π‘₯) + πœ—π΄(π‘₯) ≀ 1 Captures hesitation Conditions restricts handling inconsistent or contradictory data. 3 Neutrosophic fuzzy set Truth, Indeterminacy and Falsity (𝑇𝐴(π‘₯), 𝐼𝐴(π‘₯), 𝐼𝐴(π‘₯)) 0 ≀ 𝑇𝐴(π‘₯) + 𝐼𝐴(π‘₯) + 𝐼𝐴(π‘₯) ≀ 1 π‘œπ‘Ÿ, 2 π‘œπ‘Ÿ, 3 deal with incomplete, indeterminate, and inconsistent information, More complex computation and interpretation. Note: Here in table 2 the comparison between fuzzy set, intuitionistic fuzzy and neutrosophic fuzzy set. 1396 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate Triangular neutrosophic number: Wang, et al. [22] A Triangular neutrosophic number is taken as οΏ½ΜƒοΏ½ = (π‘š01,π‘š02,π‘š03; π‘š11,π‘š12,π‘š13; π‘š21,π‘š22, π‘š23), where the truth membership, indeterminacy and falsity function is fixed as follows: 𝑇𝑁𝑒(π‘˜) = { π‘˜ βˆ’π‘š01 π‘š02 βˆ’π‘š01 π‘€β„Žπ‘’π‘› π‘š01 ≀ π‘˜ < π‘š02 1 π‘€β„Žπ‘’π‘› π‘˜ = π‘š02 π‘š03 βˆ’ π‘˜ π‘š03 βˆ’π‘š02 π‘€β„Žπ‘’π‘› π‘š02 < π‘˜ ≀ π‘š03 0 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ 𝐼𝑁𝑒(π‘˜) = { π‘š12 βˆ’ π‘˜ π‘š12 βˆ’π‘š11 π‘€β„Žπ‘’π‘› π‘š11 ≀ π‘˜ < π‘š12 0 π‘€β„Žπ‘’π‘› π‘˜ = π‘š12 π‘˜ βˆ’π‘š12 π‘š13 βˆ’π‘š12 π‘€β„Žπ‘’π‘› π‘š12 < π‘˜ ≀ π‘š13 1 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ 𝐹𝑁𝑒(π‘˜) = { π‘š22 βˆ’ π‘˜ π‘š22 βˆ’π‘š21 π‘€β„Žπ‘’π‘› π‘š21 ≀ π‘˜ < π‘š22 0 π‘€β„Žπ‘’π‘› π‘˜ = π‘š22 π‘˜ βˆ’π‘š22 π‘š23 βˆ’π‘š22 π‘€β„Žπ‘’π‘› π‘š22 < π‘˜ ≀ π‘š23 1 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ Where, 0 ≀ 𝑇𝑁𝑒(π‘˜) + 𝐼𝑁𝑒(π‘˜) + 𝐹𝑁𝑒(π‘˜) ≀ 3 and 𝑦 ∈ οΏ½ΜƒοΏ½. Parametric form of triangular neutrosophic number or, (𝛼, 𝛽, 𝛾)-cut: The parametric setting of the above number or the (𝛼, 𝛽, 𝛾)-cut is [οΏ½ΜƒοΏ½]𝛼,𝛽,𝛾 = {[𝑀𝛼 𝐿 ,𝑀𝛼 𝑅]; [𝑀𝛽 𝐿 ,𝑀𝛽 𝑅]; [𝑀𝛾 𝐿 ,𝑀𝛾 𝑅]} Where { 𝑀𝛼 𝐿 = π‘š01 + Ξ±(π‘š02 βˆ’ π‘š01) 𝑀𝛼 𝑅 = π‘š03 βˆ’ Ξ±(π‘š03 βˆ’ π‘š02) 𝑀𝛽 𝐿 = π‘š12 βˆ’ Ξ²(π‘š12 βˆ’ π‘š11) 𝑀𝛽 𝑅 = π‘š12 + Ξ²(π‘š13 βˆ’ π‘š12) 𝑀𝛾 𝐿 = π‘š22 βˆ’ Ξ²(π‘š22 βˆ’ π‘š21) 𝑀𝛾 𝑅 = π‘š22 + Ξ²(π‘š23 βˆ’ π‘š22) with 0 < 𝛼, 𝛽, 𝛾 ≀ 1 and 0 < 𝛼 + 𝛽 + 𝛾 ≀ 3. Note: Its is need not necessary that we have to take triangular neutrosophic number. There exist different variation of neutrosophic number such as trapezoidal neutrosophic number, pentagonal neutrosophic number etc. Mittag-Leffler function: If 𝐷𝑏 𝛿𝑐 𝑦(𝑑) = 𝑦(𝑑), with 𝑦(0) = 𝑦0 then the solution is 𝑦(𝑑) = 𝑦0𝐸𝛿(𝑑 𝛿), where 𝐸𝛿(𝑑 𝛿) is called Mittag-Leffler function and it expressed as, 𝐸𝛿(𝑑 𝛿) = βˆ‘ 𝑒𝑝 Ξ“(𝑝𝛿 + 1) ∞ 𝑝=0 , 𝛿 > 0 Note: The Mittag-Leffler function have a vital role in fractional calculus. It generalizes the exponential function and arises in the solutions of fractional differential equations. In particularly those problems involving Caputo or Riemann–Liouville derivatives. Dissimilar the exponential, which defines 1397 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate memoryless progressions, the Mittag-Leffler function imprisonments power-law based decay and memory-based effects for making it perfect for modelling real-world spectacles such as anomalous diffusion, fluid dynamics, biological systems with hereditary properties. It appears when solving the fractional differential equation by using Laplace transform also. Caputo Derivative: For a function βˆ‡(π‘Ÿ) the Caputo derivative of order 𝜌 ∈ (𝑛 βˆ’ 1, 𝑛) where 𝑛 ∈ 𝑁, is defined as, 𝐷𝑐 π‘Ÿ 𝜌 βˆ‡(π‘Ÿ) = 1 Ξ“(𝑛 βˆ’ 𝜌) ∫ βˆ‡(𝑛)(𝑦) (π‘Ÿ βˆ’ 𝑦)πœŒβˆ’π‘›+1 𝑑𝑦 π‘Ÿ 0 Where Ξ“(. ) is the Gamma function and βˆ‡(𝑛) denoted the nth derivative of βˆ‡. So, for 0 < 𝜌 < 1, the above definitions become 𝐷𝑐 π‘Ÿ 𝜌 βˆ‡(π‘Ÿ) = 1 Ξ“(1 βˆ’ 𝜌) ∫ βˆ‡(1)(𝑦) (π‘Ÿ βˆ’ 𝑦)𝜌 𝑑𝑦 π‘Ÿ 0 Note: The Caputo derivative is very important in fractional calculus theory because it allows fractional differential equations with initial conditions which may expressed in terms of classical integer-order derivatives. The Caputo derivative particularly suitable for modelling dynamical systems in science, engineering, physics and biological science where initial states are known in classical terms. Moreover, it conserves key properties like linearity and convulsions naturally into Laplace transform methods. It simplifying the analytical solution of fractional differential equations and attractive its real- world applicability in initial value problems. Caputo derivative for initial value problem: Consider the fractional differential equation of type 𝐷𝑦 πœŒπ‘ 𝑦(π‘Ÿ) = π‘šπ‘¦(π‘Ÿ) with initial value 𝑦(0) = 𝑦0, then the solution is written as 𝑦(π‘Ÿ) = 𝐸𝜌(βˆ’π‘šπ‘Ÿ 𝜌) 3. Neutrosophic Extension Principle Extension Zadeh’s extension principle with neutrosophic uncertainty: Let 𝑉 be a crisp set and οΏ½ΜƒοΏ½ be a neutrosophic set in 𝑉. The function 𝑔: 𝑉 β†’ 𝑄 is defined by π‘ž = 𝑔(𝑣), then the extension principle introduces a neutrosophic set fuzzy set οΏ½ΜƒοΏ½ in 𝑄 as οΏ½ΜƒοΏ½ = {(π‘ž, 𝑇𝑐̃(π‘ž), 𝐼𝑐̃(π‘ž), 𝐹𝑐̃(π‘ž)|π‘ž = 𝑔(𝑣), 𝑒 ∈ π‘ˆ)} where, 𝑇𝑐̃(π‘ž) = { (𝑇�̃�(𝑣)), 𝑖𝑓𝑓 𝑔 βˆ’1(π‘ž) β‰  πœ™ π‘£βˆˆπ‘”βˆ’1(π‘ž) 𝑠𝑒𝑝 0 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ 𝐼𝑐̃(π‘ž) = { (𝐼�̃�(𝑣)), 𝑖𝑓𝑓 𝑔 βˆ’1(π‘ž) β‰  πœ™ π‘£βˆˆπ‘”βˆ’1(π‘ž) 𝑖𝑛𝑓 1 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ and 𝐹𝑐̃(π‘ž) = { (𝐹�̃�(𝑣)), 𝑖𝑓𝑓 𝑔 βˆ’1(π‘ž) β‰  πœ™ π‘£βˆˆπ‘”βˆ’1(π‘ž) 𝑖𝑛𝑓 1 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ Note: Neutrosophic extension principle ultimately is the extension of Zadey’s extension principal. Our main aim is to use the theory for finding the solution of neutrosophic fractional differential equation. Example: Let, �̃�𝑔 be a neutrosophic set given by the truth, indeterminacy and falsity function as follows: 1398 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate 𝑇𝐡𝑔(π‘₯) = { 0 𝑖𝑓 π‘₯ ≀ 80 ( π‘₯ βˆ’ 80 20 ) 𝑖𝑓 80 ≀ π‘₯ < 100 1 𝑖𝑓 π‘₯ = 100 ( 120 βˆ’ π‘₯ 20 ) 𝑖𝑓 100 < π‘₯ ≀ 120 0 𝑖𝑓 π‘₯ β‰₯ 120 𝐼𝐡𝑔(π‘₯) = { 0 𝑖𝑓 π‘₯ ≀ 90 ( 100 βˆ’ π‘₯ 10 ) 𝑖𝑓 90 ≀ π‘₯ < 100 0 𝑖𝑓 π‘₯ = 100 ( π‘₯ βˆ’ 100 10 ) 𝑖𝑓 100 < π‘₯ ≀ 110 0 𝑖𝑓 π‘₯ β‰₯ 110 and 𝐹𝐡𝑔(π‘₯) = { 0 𝑖𝑓 π‘₯ ≀ 95 ( 100 βˆ’ π‘₯ 5 ) 𝑖𝑓 95 ≀ π‘₯ < 100 0 𝑖𝑓 π‘₯ = 100 ( π‘₯ βˆ’ 100 5 ) 𝑖𝑓 100 < π‘₯ ≀ 105 0 𝑖𝑓 π‘₯ β‰₯ 115 Consider a function 𝐺(π‘₯) = π‘₯ + 50. Using the concept of Zadeh’s extension principle, the neutrosophic set 𝐺(𝐡𝑔) can be determined. The function of 𝐺(𝐡𝑔) is obtained as follows: 𝑇𝐡𝑔(π‘₯) = { 0 𝑖𝑓 π‘₯ ≀ 130 ( π‘₯ βˆ’ 130 20 ) 𝑖𝑓 130 ≀ π‘₯ < 150 1 𝑖𝑓 π‘₯ = 150 ( 170 βˆ’ π‘₯ 20 ) 𝑖𝑓 150 < π‘₯ ≀ 170 0 𝑖𝑓 π‘₯ β‰₯ 170 𝐼𝐡𝑔(π‘₯) = { 0 𝑖𝑓 π‘₯ ≀ 90 ( 150 βˆ’ π‘₯ 10 ) 𝑖𝑓 140 ≀ π‘₯ < 140 0 𝑖𝑓 π‘₯ = 150 ( π‘₯ βˆ’ 150 10 ) 𝑖𝑓 150 < π‘₯ ≀ 160 0 𝑖𝑓 π‘₯ β‰₯ 160 and 1399 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate 𝐹𝐡𝑔(π‘₯) = { 0 𝑖𝑓 π‘₯ ≀ 245 ( 150 βˆ’ π‘₯ 5 ) 𝑖𝑓 145 ≀ π‘₯ < 150 0 𝑖𝑓 π‘₯ = 150 ( π‘₯ βˆ’ 150 5 ) 𝑖𝑓 150 < π‘₯ ≀ 165 0 𝑖𝑓 π‘₯ β‰₯ 165 Theorem: If οΏ½ΜƒοΏ½(𝑑): [𝑑0, 𝑇] β†’ 𝐹(𝑅) is a neutrosophic fuzzy function whose (𝛼, 𝛽, 𝛾)- cut are denoted by (οΏ½ΜƒοΏ½(𝑑))𝛼,𝛽,𝛾 = {[𝑦11(𝑑, 𝛼), 𝑦12(𝑑, 𝛼)]; [𝑦21(𝑑, 𝛽), 𝑦22(𝑑, 𝛽)]; [𝑦31(𝑑, 𝛾), 𝑦32(𝑑, 𝛾)]} for 𝛼, 𝛽, 𝛾 ∈ [0,1], then (i) (οΏ½ΜƒοΏ½(𝑑))𝛼,𝛽,𝛾 is nonempty compact subset of 𝑅. (ii) (οΏ½ΜƒοΏ½(𝑑))𝛼1 βŠ† (οΏ½ΜƒοΏ½(𝑑))𝛼2, (οΏ½ΜƒοΏ½(𝑑))𝛽1 βŠ† (οΏ½ΜƒοΏ½(𝑑))𝛽2 and (οΏ½ΜƒοΏ½(𝑑))𝛾1 βŠ† (οΏ½ΜƒοΏ½(𝑑))𝛾2 for 0 ≀ 𝛼1 ≀ 𝛼2 ≀ 1, 0 ≀ 𝛽1 ≀ 𝛽2 ≀ 1, 0 ≀ 𝛾1 ≀ 𝛾2 ≀ 1. 4. Neutrosophic Fractional Differential Equation (NFDE) In this section we introduce neutrosophic fractional differential equation. It is obvious that the crisp fractional differential equation and NFDE is different in nature. The idea of fuzzy differential equation is extended here. Let us consider the crisp fractional differential equation of the form { 𝐷𝑏 𝛿𝑐 𝑦(𝑑) = 𝐹(𝑑, π‘˜, 𝑦(𝑑)) 𝑦(𝑑0) = 𝑦0 (1) Where 𝐹: [𝑑0, 𝑇] Γ— 𝑅 β†’ 𝑅 is a real valued function, 𝑦0 ∈ 𝑅, π‘˜ ∈ 𝑅 is constant and 𝛿 ∈ (0,1]. The above fractional differential equation (1) is said to be neutrosohic fractional differential equation if one of the following conditions holds: I. The initial conditions 𝑦0 is neutrosophic fuzzy valued number II. The coefficient or constant π‘˜ is neutrosophic fuzzy valued number III. Both the initial conditions 𝑦0 and coefficient or constant π‘˜ is neutrosophic fuzzy valued number Here a question arises that when we consider the above cases ? Basically, for theoretical study any one or all three may considered. But for real life model the one has to take which is best fit for the model considered. In this study we only consider the first cases i.e., the initial condition is neutrosophic fuzzy valued number. In future study all cases are considered still the idea for solution strategy is quite similar. Since the initial condition is neutrosophic fuzzy valued so the solutions also neutrosophic fuzzy valued, so we take the whole equations form as follows and treated as neutrosophic fuzzy fractional differential equations: { 𝐷𝑏 𝛿𝑐 οΏ½ΜƒοΏ½(𝑑) = οΏ½ΜƒοΏ½(𝑑, π‘˜, 𝑦(𝑑)) 𝑦(𝑑0) = οΏ½ΜƒοΏ½0 (2) Where οΏ½ΜƒοΏ½: [𝑑0, 𝑇] Γ— 𝑅𝑓 β†’ 𝑅𝑓 is a real valued neutrosophic function, οΏ½ΜƒοΏ½0 ∈ 𝑅𝑓, π‘˜ ∈ 𝑅 is constant and 𝛿 ∈ (0,1]. Note 1: For 𝛿 = 1, the equation (1) converted to simple crisp ordinary differential equation and (2) converted to neutrosophic fuzzy differential equation. Also, it should be noted that in solution if we put the integer value of 𝛿 then the solution is quite similar by not fully because we have to use some numerical approximation restrictions. 1400 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate 5. Solution of Neutrosophic Fractional Differential Equation Using Extension Principal Method There are several articles where extension principal is used to solve fuzzy differential equation such as [1-4]. The idea is very popular since the derivative of fuzzy functions concepts or interval arithmetic property is not necessary. The solution using the extension principle is very much straight forward. Strating from crisp solution, then maximum and minimum consideration with respect to uncertain parameters, then submission the corresponding interval ends the solution procedure is completed. It should be noted that different method may give different solution. Consider the crisp solution of (2), that is the solution of (1) is as follows: 𝑦(𝑑) = 𝑔(π‘˜, 𝑦0, 𝑑) (3) In equation (3) the function 𝑔 is crisp. The idea is that fuzzification have to done by extension principal. In the function π‘˜, 𝑦0 may be neutrosophic in nature. In this particular paper we take only the initial value as neutrosophic number, so only we have to focus the parameter 𝑦0 for applying Zadeh’s extension principle via neutrosophic approach. and let the (𝛼, 𝛽, 𝛾)-cut of the neutrosophic initial value of (2) is (οΏ½ΜƒοΏ½0)𝛼,𝛽,𝛾 = {[𝑝1(𝛼), 𝑝2(𝛼)]; [π‘ž1(𝛽), π‘ž2(𝛽)]; [π‘Ÿ1(𝛾), π‘Ÿ2(𝛾)]} (4) This above is parametric representation or in interval form. Theorem: If (οΏ½ΜƒοΏ½(𝑑))𝛼,𝛽,𝛾 = {𝑔(π‘˜, 𝑦0, 𝑑)|𝛼; 𝑔(π‘˜, 𝑦0, 𝑑)|𝛽; 𝑔(π‘˜, 𝑦0, 𝑑)|𝛾} is the solution of (2) then 𝑔(π‘˜, 𝑦0, 𝑑)|𝛼 = [𝑦11(𝑑, 𝛼) = π‘šπ‘–π‘› {οΏ½Μ‚οΏ½(π‘˜, 𝑦0, 𝑑)}, 𝑦12(𝑑, 𝛼) = π‘šπ‘Žπ‘₯ {𝑔(π‘˜, 𝑦0, 𝑑)}| 𝑦0 ∈ [𝑝1(𝛼), 𝑝2(𝛼)]] 𝑔(π‘˜, 𝑦0, 𝑑)|𝛽 = [𝑦21(𝑑, 𝛽) = π‘šπ‘–π‘› {οΏ½Μ‚οΏ½(π‘˜, 𝑦0, 𝑑)}, 𝑦22(𝑑, 𝛽) = π‘šπ‘Žπ‘₯ {𝑔(π‘˜, 𝑦0, 𝑑)}| 𝑦0 ∈ [π‘ž1(𝛽), π‘ž2(𝛽)]] 𝑔(π‘˜, 𝑦0, 𝑑)|𝛾 = [𝑦31(𝑑, 𝛾) = π‘šπ‘–π‘› {𝑔(π‘˜, 𝑦0, 𝑑)}, 𝑦32(𝑑, 𝛾) = π‘šπ‘Žπ‘₯ {𝑔(π‘˜, 𝑦0, 𝑑)}| 𝑦0 ∈ [π‘Ÿ1(𝛾), π‘Ÿ2(𝛾)]] For 𝛼, 𝛽, 𝛾 ∈ [0,1] it is obvious that 𝑦11(𝑑, 𝛼) ≀ 𝑦12(𝑑, 𝛼), 𝑦21(𝑑, 𝛽) ≀ 𝑦22(𝑑, 𝛽) and 𝑦31(𝑑, 𝛾) ≀ 𝑦32(𝑑, 𝛾). The above theory shows that how we may find the parametric neutrosophic function with respect to a neutrosophic parameter. Two cases happen for the 𝑔(π‘˜, 𝑦0, 𝑑). Case 1: 𝑔(π‘˜, 𝑑) is increasing with respect to 𝑦0 Then by Zade’s extension principle the solutions are written as follows (οΏ½ΜƒοΏ½(𝑑))𝛼,𝛽,𝛾 = {𝑔(π‘˜, 𝑦0, 𝑑)|𝛼; 𝑔(π‘˜, 𝑦0, 𝑑)|𝛽; 𝑔(π‘˜, 𝑦0, 𝑑)|𝛾} (5) Where { 𝑔(π‘˜, 𝑦0, 𝑑)|𝛼 = [𝑔(π‘˜, 𝑝1(𝛼), 𝑑), 𝑔(π‘˜, 𝑝2(𝛼), 𝑑)] 𝑔(π‘˜, 𝑦0, 𝑑)|𝛽 = [𝑔(π‘˜, π‘ž1(𝛽), 𝑑), 𝑔(π‘˜, π‘ž2(𝛽), 𝑑)] 𝑔(π‘˜, 𝑦0, 𝑑)|𝛾 = [𝑔(π‘˜, π‘Ÿ1(𝛾), 𝑑), 𝑔(π‘˜, π‘Ÿ2(𝛾), 𝑑)] (6) and another one is Case 2: 𝑔(π‘˜, 𝑑) is decreasing with respect to 𝑦0 Then by Zade’s extension principle the solutions are written as follows (οΏ½ΜƒοΏ½(𝑑))𝛼,𝛽,𝛾 = {𝑔(π‘˜, 𝑦0, 𝑑)|𝛼; 𝑔(π‘˜, 𝑦0, 𝑑)|𝛽; 𝑔(π‘˜, 𝑦0, 𝑑)|𝛾} (7) Where { 𝑔(π‘˜, 𝑦0, 𝑑)|𝛼 = [𝑔(π‘˜, 𝑝2(𝛼), 𝑑), 𝑔(π‘˜, 𝑝1(𝛼), 𝑑)] 𝑔(π‘˜, 𝑦0, 𝑑)|𝛽 = [𝑔(π‘˜, π‘ž2(𝛽), 𝑑), 𝑔(π‘˜, π‘ž1(𝛽), 𝑑)] 𝑔(π‘˜, 𝑦0, 𝑑)|𝛾 = [𝑔(π‘˜, π‘Ÿ2(𝛾), 𝑑), 𝑔(π‘˜, π‘Ÿ1(𝛾), 𝑑)] (8) Note 2: Same concept is applicable if only coefficient π‘˜ is neutrosophic number. Note 3: Four cases happen if k and 𝑦0 both are neutrosophic valued, and the cases are as follows: Case 1: 𝑔(π‘˜, 𝑑) is increasing with respect to k and 𝑦0 both Case 2: 𝑔(π‘˜, 𝑑) is increasing with respect to k but decreasing with respect to 𝑦0 Case 3: 𝑔(π‘˜, 𝑑) is decreasing with respect to k and increasing with respect to 𝑦0 1401 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate Case 4: 𝑔(π‘˜, 𝑑) is decreasing with respect to k and 𝑦0 both It should be noted that every time the result should be checked whether it is obeying the neutrosophic rules or not. 6. Numerical Illustrations Example 1: Consider the fractional differential equation with neutrosophic initial value { 𝐷𝑏 πœŒπ‘ Ξ©Μƒ(𝑑) = οΏ½ΜƒοΏ½(𝑑) Ξ©Μƒ0 = (8,12,16; 10, 12,14; 11, 12, 15) Solution: The solution associated with crisp fractional differential equation is Ξ©(𝑑) = Ξ©0𝐸𝜌(𝑑 𝜌) = Ξ©0 (1 + π‘‘πœŒ Ξ“(ρ+1) + 𝑑2𝜌 Ξ“(2ρ+1) ) (we take the approximated form of Mitag-Lafler function up to third term). Now 𝑑Ω(𝑑) 𝑑𝑑 = Ξ©0 (1 + π‘‘πœŒ Ξ“(ρ+1) + 𝑑2𝜌 Ξ“(2ρ+1) ) > 0, It shows that the function Ξ©(𝑑) is an increasing function with respect to 𝑑. Now the (𝛼, 𝛽, 𝛾)-cut of the initial value of οΏ½ΜƒοΏ½0 is (Ξ©Μƒ0)𝛼,𝛽,𝛾 = {[8 + 4𝛼, 16 βˆ’ 4𝛼]; [12 βˆ’ 2𝛽, 12 + 2𝛽]; [12 βˆ’ 𝛽, 12 + 𝛽]} Using equation (5) in section 5, we get the neutrosophic solution (Ξ©Μƒ(𝑑)) 𝛼,𝛽,𝛾 = {[Ω𝛼 𝐿 (𝑑), Ω𝛼 𝑅(𝑑)]; [Ω𝛽 𝐿(𝑑), Ω𝛽 𝑅(𝑑)]; [Ω𝛾 𝐿(𝑑), Ω𝛾 𝑅(𝑑)]} = {[(8 + 4𝛼)(1 + π‘‘πœŒ Ξ“(ρ + 1) + 𝑑2𝜌 Ξ“(2ρ + 1) ) , (16 βˆ’ 4𝛼)(1 + π‘‘πœŒ Ξ“(ρ + 1) + 𝑑2𝜌 Ξ“(2ρ + 1) )] ; [(12 βˆ’ 2𝛽)(1 + π‘‘πœŒ Ξ“(ρ + 1) + 𝑑2𝜌 Ξ“(2ρ + 1) ) , (12 + 2𝛽)(1 + π‘‘πœŒ Ξ“(ρ + 1) + 𝑑2𝜌 Ξ“(2ρ + 1) )] ; [(12 βˆ’ 𝛾) (1 + π‘‘πœŒ Ξ“(ρ + 1) + 𝑑2𝜌 Ξ“(2ρ + 1) ) , (12 + 𝛾) (1 + π‘‘πœŒ Ξ“(ρ + 1) + 𝑑2𝜌 Ξ“(2ρ + 1) )]} Now the pictorial representation of the solution for ρ = 0.75 and 𝑑 ∈ [0,100] is as follows: Figure 1. Solution for ρ = 0.75 and 𝑑 ∈ [0,100] Note: Clearly, we that the solution obeys the conditions 1402 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate Ω𝛼 𝐿 (𝑑) ≀ Ω𝛼 𝑅(𝑑), Ω𝛽 𝐿(𝑑) ≀ Ω𝛽 𝑅(𝑑), Ω𝛾 𝐿(𝑑) ≀ Ω𝛾 𝑅(𝑑) for particular ρ = 0.75 and 𝑑 ∈ [0,100], 𝛼, 𝛽, 𝛾 ∈ [0,1], therefore it also is a neutrosophic solution. Example 2: Consider the fractional differential equation with neutrosophic initial value { 𝐷𝑏 πœŒπ‘ Ξ”Μƒ(𝑑) = βˆ’Ξ”Μƒ(𝑑) Ξ”Μƒ0 = (30,50,70; 40, 50,60; 45,50,55) Solution: The crisp solution is Ξ”(𝑑) = Ξ”0𝐸𝜌(βˆ’π‘‘ 𝜌) = Ξ”0 (βˆ’1 βˆ’ π‘‘πœŒ Ξ“(ρ+1) βˆ’ 𝑑2𝜌 Ξ“(2ρ+1) ) (approximated up to third term). Now 𝑑Δ(𝑑) 𝑑𝑑 = Ξ”0 (βˆ’1 βˆ’ π‘‘πœŒ Ξ“(ρ+1) βˆ’ 𝑑2𝜌 Ξ“(2ρ+1) ) < 0, which is an increasing function with respect to 𝑑. The (𝛼, 𝛽, 𝛾)-cut of the initial value of Ξ”Μƒ0 is (Ξ”Μƒ0)𝛼,𝛽,𝛾 = {30 + 20𝛼, 70 βˆ’ 20𝛼]; [50 βˆ’ 10𝛽, 50 + 10𝛽]; [50 βˆ’ 5𝛽, 50 + 5𝛽]} Using equation (6) from section 5 we get (Ξ”Μƒ(𝑑)) 𝛼,𝛽,𝛾 = {[Δ𝛼 𝐿 (𝑑), Δ𝛼 𝑅(𝑑)]; [Δ𝛽 𝐿 (𝑑), Δ𝛽 𝑅(𝑑)]; [Δ𝛾 𝐿(𝑑), Δ𝛾 𝑅(𝑑)]} = {[(70 βˆ’ 20𝛼)(βˆ’1 βˆ’ π‘‘πœŒ Ξ“(ρ + 1) βˆ’ 𝑑2𝜌 Ξ“(2ρ + 1) ) , (30 + 20𝛼)(βˆ’1 βˆ’ π‘‘πœŒ Ξ“(ρ + 1) βˆ’ 𝑑2𝜌 Ξ“(2ρ + 1) )] ; [(50 + 10𝛽)(βˆ’1 βˆ’ π‘‘πœŒ Ξ“(ρ + 1) βˆ’ 𝑑2𝜌 Ξ“(2ρ + 1) ) , (50 βˆ’ 10𝛽)(βˆ’1 βˆ’ π‘‘πœŒ Ξ“(ρ + 1) βˆ’ 𝑑2𝜌 Ξ“(2ρ + 1) )] ; [(50 + 5𝛾) (βˆ’1 βˆ’ π‘‘πœŒ Ξ“(ρ + 1) βˆ’ 𝑑2𝜌 Ξ“(2ρ + 1) ) , (50 βˆ’ 5𝛾) (βˆ’1 βˆ’ π‘‘πœŒ Ξ“(ρ + 1) βˆ’ 𝑑2𝜌 Ξ“(2ρ + 1) )]} Now the pictorial representation of the solution for ρ = 0.25 and 𝑑 ∈ [0,100] is as follows: Figure 2. Solution for ρ = 0.25 and 𝑑 ∈ [0,100] Note: Clearly, we that the solution obeys the conditions 1403 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 1393-1405, 2025 DOI: 10.55214/25768484.v9i5.7172 Β© 2025 by the author; licensee Learning Gate Δ𝛼 𝐿 (𝑑) ≀ Δ𝛼 𝑅(𝑑), Δ𝛽 𝐿 (𝑑) ≀ Δ𝛽 𝑅(𝑑), Δ𝛾 𝐿 (𝑑) ≀ Δ𝛾 𝑅(𝑑) for particular ρ = 0.25 and 𝑑 ∈ [0,100], 𝛼, 𝛽, 𝛾 ∈ [0,1], therefore it also is a neutrosophic solution. 7. Conclusion and Future Research Scope In this paper, we have illustrated neutrosophic extension principal method for efficiently solving the neutrosophic fuzzy fractional differential equations (NFFDEs). By participating the extension principal method into the framework of fractional calculus ideology in neutrosophic fuzzy environments a complex system formed. The advanced method extends classical solution strategy to handle neutrosophic fuzzy initial conditions. Several illustrative examples demonstrated the viability, reliability, and flexibility of the proposed method. Overall, the proposed methods provide a systematic, reliable, and adaptable tool for dealing with fractional calculus systems influenced by multifaceted uncertainties like neutrosophic sets. There are various aspects for future research extension based on the present work. One of the proposals is to extend the methodology into system of fractional differential equations with neutrosophic uncertainty. Another development done for nonlinear systems numerical algorithm findings rather than the analytical solutions which may difficult sometimes to obtained. By considering several fractional derivatives like the Caputo–Fabrizio or Atangana–Baleanu derivatives with the neutrosophic fuzzy settings which could also improve the model's capability to capture different types of memory effects. The uncertainty parameters also change with respect to decision makers need, the extension of neutrosophic sets such as cylindrical neutrosophic sets may considered. The core applications may be found from the field like engineering science, mathematical biology and economics for adopting the fractional calculus theory with uncertainty. Future research also focusses on by integrating optimization methods in the solution methdology, which allowing for parameter identification and system optimization in various complex neutrosophic fractional modelling. Transparency: The author confirms that the manuscript is an honest, accurate, and transparent account of the study; that no vital features of the study have been omitted; and that any discrepancies from the study as planned have been explained. This study followed all ethical practices during writing. Copyright: Β© 2025 by the author. 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