Abstract
We present an efficient numerical method to approximate the solution of a system of fractional-order linear semi-explicit differential-algebraic equations with variable coefficients. The method is based on the use of the direct and inverse fuzzy transforms (\({\mathcal {F}}\)-transforms). By employing this method, we obtain an analytical approximate solution to the main problem in terms of flexible basic functions. The non-local property of fuzzy transforms helps us to have an efficient method for problems involving non-singular kernels. The error analysis and convergence evaluation of the method are demonstrated in detail. We give some examples to illustrate the significant features of the method.
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References
Alikhani R, Zeinali M, Bahrami F, Shahmorad S, Perfilieva I (2017) Trigonometric \(F^{m}\)-transform and its approximation properties. Soft Comput 21(13):3567–3577
Atkinson K (2009) The numerical solution of integral equations of the second kind, Cambridge
Bede B (2013) Mathematics of Fuzzy sets and Fuzzy logic. Springer
Campbell S, Ilchmann A, Mehrmann V, Reis T (2019) Applications of differential-algebraic equations. Springer, Examples and Benchmarks
Damarla SK, Kundu M (2015) Numerical solution of fractional order differential-algebraic equations using generalized triangular function operational matrices. Fract Calculus Appl 6:31–55
Di Martino F, Loia V, Perfilieva I, Sessa S (2008) An image coding/decoding method based on direct and inverse fuzzy transforms. Int J Approx Reason 48(1):110–131
Diethelm K (2010) The analysis of fractional differential equations: an application-oriented exposition using differential operators of caputo typ. Springer
Ghanbari F, Ghanbari K, Mokhtary P (2018) High-order Legendre collocation method for fractional-order linear semi-explicit differential algebraic equations. Electronic Trans Numer Anal 48:387–406
Hurtik P, Tomasiello S (2019) A review on the application of fuzzy transform in data and image compression. Soft Comput 23(23):12641–12653
Ibis B, Bayram M (2011) Numerical comparison of methods for solving fractional differential-algebraic equations(FDAEs). Comput Math Appl 62:3270–3278
Khastan A, Alijani Z, Perfilieva I (2016) Fuzzy transform to approximate solution of two-point boundary value problems. Math Methods Appl Sci 40(17):6147–6154
Khastan A, Perfilieva I, Alijani Z (2016) A new fuzzy approximation method to Cauchy problems by fuzzy transform. Fuzzy Sets Syst 288:75–95
Mainardi F (1997) Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and fractional calculus in continuum mechanics. Springer-Verlag, New York, pp 291–348
Novák V, Perfilieva I, Dvořák A, Chen G, Wei Q, Yan P (2008) Mining pure linguistic associations from numerical data. Int J Approx Reason 48(1):4–22
Perfilieva I (2004) Fuzzy transform: Application to the reef growth problem. Fuzzy Logic Geol, pp 275–300
Perfilieva I (2006) Fuzzy transforms: theory and applications. Fuzzy Sets Syst 157(8):993–1023
Perfilieva I (2007) Fuzzy transforms: a challenge to conventional transforms. Adv Imag Electron Phys 147:137–196
Perfilieva I, Daňková M, Bede B (2011) Towards a higher degree \({\cal{F} }\)-transform. Fuzzy Sets Syst 180(1):3–19
Perfilieva I, Daňková M (2008) Image fusion on the basis of fuzzy transforms. Computational Intelligence in Decision and Control, pp 471–476
Perfilieva I, De Meyer, De Baets B, Piskova D (2008) Cauchy problem with fuzzy initial condition and its approximate solution with the help of fuzzy transform. In: IEEE international conference on fuzzy systems (IEEE World Congress on Computational Intelligence), pp 2285–2290
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press, San Diego
Tomasiello S (2021) Least-squares fuzzy Transforms and autoencoders: some remarks and application. IEEE Trans Fuzzy Syst 29(1):129–136
Tomasiello S (2017) An alternative use of fuzzy transform with application to a class of delay differential equations. Int J Computer Math 94(9):1719–1726
Wang HS, Jhu WL, Yung CF, Wang PF (2011) Numerical solutions of differential-algebraic equations and its applications in solving TPPC problems. Marine Sci Technol 19(1):76–88
Wang L, Chen N (2014) The predictor-corrector solution for fractional order differential-algebraic equation. In: ICFDA’14 international conference on fractional differentiation and its applications, pp 1–6
Westerlund S, Ekstam L (1994) Capacitor theory. IEEE Trans Dielectrics Electrical Insulat 1(5):826–839
Yang XJ (2019) General fractional derivatives: theory, methods and applications. Chapman and Hall/CRC
Zeinali M, Alikhani R, Bahrami F, Shahmorad S, Perfilieva I (2018) On the structural properties of \(F^{m}\)-transform with applications. Fuzzy Sets Syst 342:32–52
Zurigat M, Momani S, Alawneh A (2010) Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method. Comput Math Appl 59(3):1227–1235
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All authors contributed to the study conception and design. Analysis of the method and results were performed by [FB], [SM] and [SS]. The first draft of the manuscript was written by [FB], and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. The authors declare that contributed equally to this work. All authors agree that authors list is correct in its content and order.
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Bahrami, F., Mirzajani, S. & Shahmorad, S. Fuzzy transform-based approximation method for solving fractional semi-explicit differential-algebraic equations. Soft Comput 27, 1389–1400 (2023). https://doi.org/10.1007/s00500-022-07638-z
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DOI: https://doi.org/10.1007/s00500-022-07638-z