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Granular Computing

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26-09-2022 | Original Paper

Solving Pythagorean fuzzy partial fractional diffusion model using the Laplace and Fourier transforms

Authors: Muhammad Akram, Tayyaba Ihsan

Published in: Granular Computing | Issue 4/2023

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Abstract

Many mathematical models describe the Corona virus disease 2019 (COVID-19) outbreak; however, they require advance mathematical skills. The need for this study is to determine the diffusion of the COVID-19 vaccine in humans. To this end, we first establish a Pythagorean fuzzy partial fractional differential equation using the Pythagorean fuzzy integral transforms to express the effects of COVID-19 vaccination on humans under the generalized Hukuhara partial differential conditions. We extract the analytical solution of the Pythagorean fuzzy partial fractional differential equation using the Pythagorean fuzzy Laplace transform under the generalized Hukuhara partial differential and the Pythagorean fuzzy Fourier transform using the Caputo generalized Hukuhara partial differential. Moreover, we present some essential postulates and results related to the Pythagorean fuzzy Laplace transform and the Pythagorean fuzzy Fourier transform. Furthermore, we develop the solution procedure to extract the solution of the Pythagorean fuzzy partial fractional differential equation. To grasp the considered approach, a mathematical model for the diffusion of the COVID-19 vaccination in the human body is provided and analyzed the behavior to visualize and support the proposed model. Our proposed method is efficient and has a great worth to discuss the bio-mathematical models in various fields of science and medicines.

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Agarwal RP, Lakshmikantham V, Nieto JJ (2010) On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal Theory Methods Appl 72(6):2859–2862MathSciNetMATH

Ahmad S, Ullah A, Abdeljawad T (2021) Computational analysis of fuzzy fractional order non-dimensional Fisher equation. Phys Scr 96(8):084004

Akram M, Ali G (2020) Hybrid models for decision making based on rough Pythagorean fuzzy bipolar soft information. Granul Comput 5(1):1–15MathSciNet

Akram M, Khan A (2021) Complex Pythagorean Dombi fuzzy graphs for decision making. Granul Comput 6(3):645–669MathSciNet

Akram M, Shahzadi G (2021) A hybrid decision making model under q-rung orthopair fuzzy Yager aggregation operators. Granul Compu 6(4):763–777

Akram M, Habib A, Davvaz B (2019) Direct sum of \(n\) Pythagorean fuzzy graphs with application to group decision-making. J Mult-Valued Log Soft Comput 33(1–2):75–115MathSciNetMATH

Akram M, Sattar A, Saeid AB (2022a) Competition graphs with complex intuitionistic fuzzy information. Granul Comput 7:25–47

Akram M, Shahzadi G, Alcantud JCR (2022b) Multi-attribute decision-making with q-rung picture fuzzy information. Granul Comput 7:197–215

Akram M, Ihsan T, Allahviranloo T (2022c) Solving Pythagorean fuzzy fractional differential equations using Laplace transform. Granul Comput. https://doi.org/10.1007/s41066-022-00344-zCrossRef

Akram M, Ihsan T, Allahviranloo T, Ali Al-Shamiri MM (2022d) Analysis on determining the solution of fourth-order fuzzy initial value problem with Laplace operator. Math Biosci Eng 19(12):11868–11902MathSciNetMATH

Akram M, Muhammad G, Allahviranloo T, Ali G (2022e) New analysis of fuzzy fractional Langevin differential equations in Caputo’s derivative sense. AIMS Math 7(10):18467–18496MathSciNet

Allahviranloo T, Ahmadi MB (2010) Fuzzy Laplace transforms. Soft Comput 14(3):235–243MATH

Allahviranloo T, Armand A, Gouyandeh Z (2014) Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. J Intell Fuzzy Syst 26:1481–1490MathSciNetMATH

Allahviranloo T, Ghaffari M, Abbasbandy S, Azhini M (2021) On the fuzzy solutions of time-fractional problems. Iran J Fuzzy Syst 18(3):51–66MathSciNetMATH

Arshad S, Lupulescu V (2011) On the fractional differential equations with uncertainty. Nonlinear Anal 75:3685–3693MathSciNetMATH

Asif M, Akram M, Ali G (2020) Pythagorean fuzzy matroids with application. Symmetry 12(3):423

Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96MathSciNetMATH

Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151(3):581–599MathSciNetMATH

Bede B, Stefanini L (2013) Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst 230:119–141MathSciNetMATH

Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2012) Fractional calculus: models and numerical methods. Sci World J 3:10–39MATH

Chalco-Canoa Y, Maqui-Huaman GG, Silva GN, Jimenez-Gamero MD (2019) Algebra of generalized Hukuhara differentiable interval-valued functions: review and new properties. Fuzzy Sets Syst 375:53–69MathSciNetMATH

Chang SSL, Zadeh LA (1972) On fuzzy mapping and control. IEEE Trans Syst Man Cyber net 2:30–34MathSciNetMATH

Dubios D, Prade H (1982) Towards fuzzy differential calculus. Fuzzy Sets Syst 8(3):225–233MathSciNet

Ezadi S, Allahviranloo T (2020) Artificial neural network approach for solving fuzzy fractional order initial value problems under \(gH\)-differentiability. Math Methods Appl Sci. https://doi.org/10.1002/mma.7287CrossRef

Gouyandeha Z, Allahviranloo T, Abbasbandy S, Armand A (2017) A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform. Fuzzy Sets Syst 309:81–97MathSciNetMATH

Hoa NV (2015) Fuzzy fractional functional integral and differential equations. Fuzzy Sets Syst 280:58–90MathSciNetMATH

Kaleva K (1987) Fuzzy differential equations. Fuzzy Sets Syst 24(3):301–317MathSciNetMATH

Khakrangin S, Allahviranloo T, Mikaeilvand N, Abbasbandy S (2021) Numerical solution of fuzzy fractional differential equation by haar wavelet. Appl Appl Math (AAM) 16(1):14MathSciNetMATH

Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier 204:1–523MathSciNetMATH

Keshavarz M, Allahviranloo T (2022) Fuzzy fractional diffusion proceses and drug release. Fuzzy Sets Syst 436:82–101

Keshavarz M, Qahremani E, Allahviranloo T (2022) Solving a fuzzy fractional diffusion model for cancer tumor by using fuzzy transforms. Fuzzy Sets Syst 443:198–220MathSciNet

Maxmen A (2021) Why did the world’s pandemic warning system fail when COVID hit? Nature 589:499–500

Melliani S, Elomari MH, Hilal K, Menchih M (2021) Fuzzy fractional differential wave equation. J Optim Theory Appl 1(2):42

Mondal SP, Roy TK (2015) System of differential equation with initial value as triangular intuitionistic fuzzy number and its application. Int J Appl Math 1(3):449–474MathSciNetMATH

Mondal SP, Goswami A, Kumar S (2019) Nonlinear triangular intuitionistic fuzzy number and its application in linear integral equation. Adv Fuzzy Syst. https://doi.org/10.1155/2019/4142382MathSciNetCrossRefMATH

Naz S, Ashraf S, Akram M (2018) A novel approach to decision-making with Pythagorean fuzzy information. Math 6(6):95MATH

Namazi H, Kulish VV (2015) Fractional diffusion based modelling and prediction of human brain response to external stimuli. Comput Math Methods Med. https://doi.org/10.1155/2015/148534MathSciNetCrossRefMATH

Peng X, Luo Z (2021) A review of \(q\)-rung orthopair fuzzy information: bibliometrics and future directions. Artif Intell Rev 54(5):3361–3430

Peng X, Selvachandran G (2019) Pythagorean fuzzy set: state of the art and future directions. Artif Intell Rev 52(3):1873–1927

Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier 198:62–86

Povstenko Y (2015) Linear fractional diffusion-wave equation for scientists and engineers, 1st edn. Birkhäuser Cham, p 460. https://doi.org/10.1007/978-3-319-17954-4

Rehman M (2011) Applications of Fourier transforms to generalized functions. WIT press

Rahman K (2022) Multiple attribute group decision-making based on generalized interval-valued Pythagorean fuzzy Einstein geometric aggregation operators. Granul Comput. https://doi.org/10.1007/s41066-022-00322-5CrossRef

Salahshour S, Allahviranloo T (2013) Applications of fuzzy Laplace transforms. Soft Comput 17(1):145–158MATH

Salahshour S, Allahviranloo T, Abbasbandy S (2012) Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun Nonlinear Sci Numer Simul 17(3):1372–1381MathSciNetMATH

Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 24(3):319–330MathSciNetMATH

Song S, Wu C (2000) Existence and uniqueness of solutions to Cauchy problem of fuzzy ordinary differential equations. Fuzzy Sets Syst 110(1):55–67MATH

Stefanini L, Bede B (2013) Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst 230:119–141MathSciNetMATH

Ullah K, Mahmood T, Ali Z, Jan N (2020) On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition. Complex Intell syst 6(1):15–27

Vu H, Hoa NV (2019) Uncertain fractional differential equations on a time scale under granular differentiability concept. Comput Appl Math 38(3):1–22MathSciNetMATH

Vu H, Rassias JM, Van Hoa N (2020) Ulam-Hyers-Rassias stability for fuzzy fractional integral equations. Iran J Fuzzy Syst 17(2):17–27MathSciNetMATH

Viet Long H, Thi Kim Son N, Thi Thanh Tam H (2017) The solve ability of fuzzy fractional partial differential equations under Caputo \(gH\)-differentiability. Fuzzy Sets Syst 309:35–63MATH

Van Hoa N (2015) Fuzzy fractional functional differential equations under Caputo \(gH\)-differentiability. Commun Nonlinear Sci Numer Simul 22(1–3):1134–1157MathSciNetMATH

Yager RR (2013a) Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 22(4):958–965

Yager RR (2013b) Pythagorean fuzzy subsets. In IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS):57-61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375

Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353MATH

Title: Solving Pythagorean fuzzy partial fractional diffusion model using the Laplace and Fourier transforms
Authors: Muhammad Akram
Tayyaba Ihsan
Publication date: 26-09-2022
Publisher: Springer International Publishing
Published in: Granular Computing / Issue 4/2023
Print ISSN: 2364-4966
Electronic ISSN: 2364-4974
DOI: https://doi.org/10.1007/s41066-022-00349-8

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