Novel Approaches for Solving Fuzzy Fractional Partial Differential Equations
Abstract
:1. Introduction
- •
- We present the comparison for a fuzzy -dimensional RDTM, ADM, VIM [1], and fuzzy HPM [2] demonstrates that even though the results of these approaches when implemented to the fuzzy wave-like and heat-like equations are the same. But, the fuzzy -dimensional RDTM, like fuzzy HPM, does not require specific algorithms and complex calculations such as fuzzy ADM or construction of correction functionals using general Lagranges multipliers in the fuzzy variational iteration method. In particular, the fuzzy RDTM and HPM are simple to apply and represent two successful techniques to obtain the solution of fuzzy PDEs.
- •
- We investigated the comparison of fuzzy -dimensional RDTM, ADM, HPM, and fuzzy HAM to obtain the solutions of fuzzy wave-like, heat-like and Zakharov-Kuznetsov equations. Although the results of these methods are the same when applied to problems. Moreover, the fuzzy -dimensional RDTM, HPM, and HAM don’t require complex techniques and computations as fuzzy ADM. The results recall that the fuzzy RDTM, HPM, and HAM are easy to use for solving fuzzy partial differential equations.
- •
- We propose the solutions of fuzzy fractional wave-like, heat-like, and Zakharov-Kuznetsov equations using -dimensional fuzzy fractional RDTM. The method is flexible and can solve problems without calculating complicated Adomian polynomials or making unrealistic assumptions about nonlinear behavior. The provided technique is thus an influential way of solving fuzzy fractional PDEs and fractional order problems in physics, engineering, and other areas.
2. Preliminaries
- ,
- with ,
- ,
- If the -difference exists, it is unique.
- or whenever the statement on the right exists, especially, .
- If exists in sense (i), then exists in sense (ii) and vice versa.
- .
- .
- if and only if ; moreover, if and only if .
- f is -differentiable at if
- f is -differentiable at if
- is -differentiable whenever the type of -differentiability and is the same:
- is -differentiable if the type of -differentiability and is different:
3. Fuzzy Partial Differential Equations
3.1. Fuzzy -Dimensional Reduced Differential Transform
3.1.1. The Properties of Fuzzy -Dimensional Reduced Differential Transform
- If then
- If then
- If then where c is a constant.
- If or then ,
- If , then ,
3.1.2. Applications
3.2. Fuzzy Zakharov-Kuznetsov Equations
3.3. Fuzzy Adomian Decomposition Method
3.4. The Fuzzy Homotopy Perturbation Method
3.5. The Fuzzy Homotopy Analysis Method
3.6. Applications
4. Fuzzy Fractional Partial Differential Equations
4.1. Fuzzy Fractional Calculus
- (i)
- if is (i)-differentiable fuzzy-valued function, then
- (ii)
- if is (ii)-differentiable fuzzy-valued function, then
4.2. Fuzzy -Dimensional Fractional Reduced Differential Transform
The Properties of Fuzzy -Dimensional Fractional Reduced Differential Transform
- If then
- If then
- If then where c is a constant,
- If or , then .
- If , then
4.3. Examples
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Osman, M.; Gong, Z.T.; Mustafa, A.M. Comparison of fuzzy Adomian decomposition method with fuzzy VIM for solving fuzzy heat-like and wave-like equations with variable coefficients. Adv. Diff. Equ. 2020, 2020, 327. [Google Scholar] [CrossRef]
- Osman, M.; Xia, Y.; Omer, O.A.; Hamoud, A. On the fuzzy solution of linear-nonlinear partial differential equations. Mathematics 2022, 10, 2295. [Google Scholar] [CrossRef]
- Stefanini, L.; Bede, B. Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal. 2009, 71, 1311–1328. [Google Scholar] [CrossRef] [Green Version]
- Bede, B.; Stefanini, L. Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 2013, 230, 119–141. [Google Scholar] [CrossRef] [Green Version]
- Gomes, L.T.; Barros, L.C. A note on the generalized difference and the generalized differentiability. Fuzzy Sets Syst. 2015, 280, 142–145. [Google Scholar] [CrossRef]
- Suna, H.G.; Zhang, Y.; Baleanua, D. A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 2018, 64, 213–231. [Google Scholar] [CrossRef]
- Agarwal, R.P.; Lakshmikantham, V.; Nieto, J.J. On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 2010, 72, 2859–2862. [Google Scholar] [CrossRef]
- Agarwal, R.P.; Arshad, S.; O’Regan, D.; Lupulescu, V. Fuzzy fractional integral equations under compactness type condition. Fract. Calc. Appl. Anal. 2012, 15, 572–590. [Google Scholar] [CrossRef]
- Alikhani, R.; Bahrami, F. Global solutions for nonlinear fuzzy fractional integral and integro differential equations. Commun. Nonlinear Sci. Numer. Simul. 2013, 18, 2007–2017. [Google Scholar] [CrossRef]
- Allahviranloo, T.; Gouyandeh, Z.; Armand, A. Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. J. Intell. Fuzzy Syst. 2014, 26, 1481–1490. [Google Scholar] [CrossRef]
- Hoa, N.V.; Lupulescu, V.; O’Regan, D. Solving interval-valued fractional initial value problems under Caputo gH-fractional differentiability. Fuzzy Sets Syst. 2017, 309, 1–34. [Google Scholar] [CrossRef]
- Long, H.V.; Son, N.T.K.; Hoa, N.V. Fuzzy fractional partial differential equations in partially ordered metric spaces. Iran. J. Fuzzy Syst. 2017, 14, 107–126. [Google Scholar]
- Lupulescu, V. Fractional calculus for interval-valued functions. Fuzzy Sets Syst. 2015, 265, 63–85. [Google Scholar] [CrossRef]
- Lupulescu, V.; Hoa, N.V. Interval Abel integral equation. Soft Comput. 2017, 21, 2777–2784. [Google Scholar] [CrossRef]
- Mazandarani, M.; Kamyad, A.V. Modified fractional Euler method for solving fuzzy fractional initial value problem. Commun. Nonlinear Sci. Numer. Simul. 2013, 18, 12–21. [Google Scholar] [CrossRef]
- Prakash, P.; Nieto, J.J.; Senthilvelavan, S.; Priya, G.S. Fuzzy fractional initial value problem. J. Intell. Fuzzy Syst. 2015, 28, 2691–2704. [Google Scholar] [CrossRef]
- Salahshour, S.; Allahviranloo, T.; Abbasbandy, S.; Baleanu, D. Existence and uniqueness results for fractional differential equations with uncertainty. Adv. Differ. Equ. 2012, 2012, 1311–1328. [Google Scholar] [CrossRef] [Green Version]
- Siryk, S.V.; Salnikov, N.N. Numerical solution of Burgers’ equation by Petrov-Galerkin method with adaptive weighting functions. J. Autom. Inf. Sci. 2012, 44, 50–67. [Google Scholar] [CrossRef]
- Keshavarz, M.; Qahremani, E.; Allahviranloo, T. Solving a fuzzy fractional diffusion model for cancer tumor by using fuzzy transforms. Fuzzy Sets Syst. 2022, 443, 198–220. [Google Scholar] [CrossRef]
- Keshavarz, M.; Allahviranloo, T. Fuzzy fractional diffusion processes and drug release. Fuzzy Sets Syst. 2022, 436, 82–101. [Google Scholar] [CrossRef]
- Allahviranloo, T. Difference methods for fuzzy partial differential equations. Comput. Methods Appl. Math. 2022, 2, 233–242. [Google Scholar] [CrossRef]
- Alihani, R.; Bahram, F. Fuzzy partial differential equations under the cross product of fuzzy numbers. Inf. Sci. 2019, 494, 80–99. [Google Scholar] [CrossRef]
- Buckley, J.J.; Feuring, T. Introduction to fuzzy partial differential equations. Fuzzy Sets Syst. 1999, 105, 241–248. [Google Scholar] [CrossRef]
- Osman, M.; Gong, Z.T.; Mustafa, A.M.; Yang, H. Solving fuzzy (1+n)-dimensional Burgers equation. Adv. Diff. Equ. 2021, 219, 1–51. [Google Scholar] [CrossRef]
- Stynes, M.; Stynes, D. Convection Diffusion Problems: An Introduction to Their Analysis and Numerical Solution. Am. Math. Soc. 2018, 196, 156. [Google Scholar]
- John, V.; Knobloch, P.; Novo, J. Finite elements for scalar convection-dominated equations and incompressible flow problems: A neverending story? Comput. Vis. Sci. 2018, 19, 47–63. [Google Scholar] [CrossRef] [Green Version]
- Zhou, J.K. Differential Transformation and Its Applications for Electrical Circuits (in Chinese); Huazhong University Press: Wuhan, China, 1986. [Google Scholar]
- Rivaz, A.; Fard, O.S.; Bidgoli, T.A. Solving fuzzy fractional differential equations by generalized differential transform method. SeMA J. 2016, 73, 149–170. [Google Scholar] [CrossRef]
- Salahshour, S.; Allahviranloo, T. Application of fuzzy differential transform method for solving fuzzy Volterra integral equations. Appl. Math. Model. 2013, 37, 1016–1027. [Google Scholar] [CrossRef]
- Allahviranloo, T.; Kiani, N.A.; Motamedi, N. Solving fuzzy differential equations by differential transform method. Inf. Sci. 2009, 170, 956–966. [Google Scholar] [CrossRef]
- Abazari, R.; Ganji, M. Extended two-dimensional DTM and its application on nonlinear PDEs with proportional delay. Int. J. Comput. Math. 2011, 88, 1749–1762. [Google Scholar] [CrossRef]
- Keskin, Y.; Oturanc, G. Reduced differential transform method for partial diferential equations. Int. J. Nonlinear Sci. Numer. Simul. 2009, 10, 741–749. [Google Scholar] [CrossRef]
- Keskin, Y.; Oturanc, G. Reduced differential transform method for fractional partial diferential equations. Nonlinear Sci. Lett. A 2010, 1, 61–72. [Google Scholar]
- Abazari, R.; Abazari, M. Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM. Commun. Nonli. Sci. Numer. Simul. 2012, 17, 619–629. [Google Scholar] [CrossRef]
- Saadatmandi, A.; Dehghan, M. Numerical solution of hyperbolic telegraph equation using the ChebyshevTau Method. Meth. Part. Diff. Equ. 2010, 26, 239–252. [Google Scholar] [CrossRef]
- Aloy, R.; Casaban, M.C.; Caudillo-Mata, L.A.; Jodar, L. Computing the variable coefficient telegraph equation using a discrete eigenfunctions method. Comput. Math. Appl. 2007, 54, 448–458. [Google Scholar] [CrossRef] [Green Version]
- Owyed, S.; Abdou, M.A.; Abdel-Aty, A.H.; Alharbi, W.; Nekhili, R. Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method. Chaos Solitons Fractals 2020, 131, 109474. [Google Scholar] [CrossRef]
- Osman, M.; Gong, Z.; Mustafa, A.M. A fuzzy solution of nonlinear partial differential equations. Open J. Math. Anal. 2021, 5, 51–63. [Google Scholar] [CrossRef]
- Srivastava, V.K.; Awasthi, M.K.; Chaurasia, R.K. Reduced differential transform method to solve two and three dimensional second order hyperbolic telegraph equations. J. King Saud Univ. Engin. Sci. 2017, 29, 166–171. [Google Scholar] [CrossRef] [Green Version]
- Tamboli, V.K.; Tandel, P.V. Solution of the time-fractional generalized Burger-Fisher equation using the fractional reduced differential transform method. J. Ocean. Eng. Sci. 2022, 7, 399–407. [Google Scholar] [CrossRef]
- Siryk, S.V.; Salnikov, N.N. Accuracy and stability of the Petrov-Galerkin method for solving the stationary convection-diffusion equation. Cybern. Syst. Anal. 2014, 50, 278–287. [Google Scholar] [CrossRef]
- Saelao, J.; Yokchoo, N. The solution of Klein-Gordon equation by using modified Adomian decomposition method. Math. Compu. Simul. 2020, 171, 94–102. [Google Scholar] [CrossRef]
- Lu, T.T.; Zheng, W.Q. Adomian decomposition method for first order PDEs with unprescribed data. Alex. Eng. J. 2021, 60, 2563–2572. [Google Scholar] [CrossRef]
- Siryk, S.V.; Salnikov, N.N. Analysis of lumped approximations in the finite-element method for convection-diffusion problems Cybern. Syst. Anal. 2013, 49, 774–784. [Google Scholar] [CrossRef]
- He, J.H. Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 1999, 178, 257–262. [Google Scholar] [CrossRef]
- He, J.H. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int. J. Non-Linear Mech. 2000, 35, 37–43. [Google Scholar] [CrossRef]
- He, J.H. Homotopy perturbation method: A new nonlinear analytical technique. Appl. Math. Comput. 2003, 135, 73–79. [Google Scholar] [CrossRef]
- Naik, P.A.; Zu, J.; Ghoreishi, M. Estimating the approximate analytical solution of HIV viral dynamic model by using homotopy analysis method. Chaos Solitons Fractal 2020, 131, 109500. [Google Scholar] [CrossRef]
- Fadugba, S.E. Homotopy analysis method and its applications in the valuation of European call options with time-fractional Black-Scholes equation. Chaos Solitons Fractals 2020, 141, 110351. [Google Scholar] [CrossRef]
- Deniz, S. Optimal perturbation iteration method for solving fractional FitzHugh-Nagumo equation. Chaos Solitons Fractals 2021, 142, 110417. [Google Scholar] [CrossRef]
- Kashkari, B.S.; El-Tantawy, S.A.; Salas, A.H.; El-Sherif, L.S. Homotopy perturbation method for studying dissipative nonplanar solitons in an electronegative complex plasma. Chaos Solitons Fractals 2020, 130, 109457. [Google Scholar] [CrossRef]
- Kanth, A.S.V.R.; Aruna, K. He’s homotopy-perturbation method for solving higher-order boundary value problems. Chaos Solitons Fractals 2009, 41, 1905–1909. [Google Scholar] [CrossRef]
- Biazar, J.; Ghazvini, H.; Eslami, M. He’s homotopy perturbation method for systems of integro-differential equations. Chaos Solitons Fractals 2020, 39, 1253–1258. [Google Scholar] [CrossRef]
- Xu, Y. Similarity solution and heat transfer characteristics for a class of nonlinear convection-diffusion equation with initial value conditions. Math. Probl. Eng. 2019, 2019, 3467276. [Google Scholar] [CrossRef]
- Ahmad, S.; Ullah, A.; Akgul, A.; De la Sen, M. A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel. J. Funct. Spaces 2021, 2021, 8770488. [Google Scholar] [CrossRef]
- Liao, S.J. The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, China, 1992. [Google Scholar]
- Liao, J.S. Beyond Perturbation: Introduction to the Homotopy Analysis Method; Chapman and Hall/CRC Press: Boca Raton, FL, USA, 2003. [Google Scholar]
- Liao, J.S. Notes on the homotopy analysis method: Some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 983–997. [Google Scholar] [CrossRef]
- Xu, H.; Liao, S.J.; You, X.C. Analysis of nonlinear fractional partial differential equations with the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 1152–1156. [Google Scholar] [CrossRef]
- Cang, J.; Tan, Y.; Xu, H.; Liao, S.J. Series solutions of non-linear Riccati differential equations with fractional order. Chaos Solitons Fractals 2009, 40, 1–9. [Google Scholar] [CrossRef]
- Saratha, S.R.; Krishnan, G.S.S.; Bagyalakshmi, M. Analysis of a fractional epidemic model by fractional generalised homotopy analysis method using modified Riemann-Liouville derivative. Appl. Math. Modell. 2021, 92, 525–545. [Google Scholar] [CrossRef]
- Li, J.X.; Yan, Y.; Wang, W.Q. Time-delay feedback control of a cantilever beam with concentrated mass based on the homotopy analysis method. Appl. Math. Model. 2022, 108, 629–645. [Google Scholar] [CrossRef]
- Allahviranloo, T.; Gouyandeh, Z.; Armand, A.; Hasanoglu, A. On fuzzy solutions for heat equation based on generalized Hukuhara differentiability. Fuzzy Sets Syst. 2015, 265, 1–23. [Google Scholar] [CrossRef]
- Negoita, C.V.; Ralescu, D. Applications of Fuzzy Sets to Systems Analysis; Wiley: New York, NY, USA, 1975. [Google Scholar]
- Lakshmikantham, V.; Bhaskar, T.; Devi, J. Theory of Set Diffenerntial Equations in Metric Spaces; Cambridge Scientific Publishers: Cottenham, UK, 2006. [Google Scholar]
- Gong, Z.T.; Yang, H. lll-Posed fuzzy initial-boundary value problems based on generalized differentiability and regularization. Fuzzy Sets Syst. 2016, 295, 99–113. [Google Scholar] [CrossRef]
- Stefanini, L. A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst. 2010, 161, 1564–1584. [Google Scholar] [CrossRef]
- Congxin, W.; Ming, M. Embedding problem of fuzzy number space: Part III. Fuzzy Sets Syst. 1992, 46, 281–286. [Google Scholar] [CrossRef]
- Yang, H.; Gong, Z. I11-Posedness for fuzzy Fredholm integral equations of the first kind and regularization methods. Fuzzy Sets Syst. 2019, 358, 132–149. [Google Scholar] [CrossRef]
- Anastassiou, G.A. Fuzzy Mathematics: Approximation Theory; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Salahshour, S.; Allahviranloo, T.; Abbasbandy, S. Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear Sci. Numer. Simul. 2012, 17, 1372–1381. [Google Scholar] [CrossRef]
Lower ES | Lower AS | Lower Error | Upper ES | Upper AS | Upper Error | |
---|---|---|---|---|---|---|
0 | −3.1552 × | 0.00011003 | −0.00014158 | 0.99997 | 1.0001 | −0.00014158 |
0.1 | −3.124 × | 0.00011034 | −0.00014158 | 0.77375 | 0.77389 | −0.00014158 |
0.2 | −2.1552 × | 0.00012003 | −0.00014158 | 0.59046 | 0.5906 | −0.00014158 |
0.3 | 4.4385 × | 0.00018597 | −0.00014158 | 0.44367 | 0.44382 | −0.00014158 |
0.4 | 0.00028845 | 0.00043003 | −0.00014158 | 0.32765 | 0.32779 | −0.00014158 |
0.5 | 0.00094501 | 0.0010866 | −0.00014158 | 0.23727 | 0.23741 | −0.00014158 |
0.6 | 0.0023984 | 0.00254 | −0.00014158 | 0.16804 | 0.16818 | −0.00014158 |
0.7 | 0.0052206 | 0.0053622 | −0.00014158 | 0.116 | 0.11614 | −0.00014158 |
0.8 | 0.010208 | 0.01035 | −0.00014158 | 0.077728 | 0.07787 | −0.00014158 |
0.9 | 0.018421 | 0.018563 | −0.00014158 | 0.050297 | 0.050438 | −0.00014158 |
1 | 0.031218 | 0.03136 | −0.00014158 | 0.031218 | 0.03136 | −0.00014158 |
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Osman, M.; Xia, Y.; Marwan, M.; Omer, O.A. Novel Approaches for Solving Fuzzy Fractional Partial Differential Equations. Fractal Fract. 2022, 6, 656. https://doi.org/10.3390/fractalfract6110656
Osman M, Xia Y, Marwan M, Omer OA. Novel Approaches for Solving Fuzzy Fractional Partial Differential Equations. Fractal and Fractional. 2022; 6(11):656. https://doi.org/10.3390/fractalfract6110656
Chicago/Turabian StyleOsman, Mawia, Yonghui Xia, Muhammad Marwan, and Omer Abdalrhman Omer. 2022. "Novel Approaches for Solving Fuzzy Fractional Partial Differential Equations" Fractal and Fractional 6, no. 11: 656. https://doi.org/10.3390/fractalfract6110656
APA StyleOsman, M., Xia, Y., Marwan, M., & Omer, O. A. (2022). Novel Approaches for Solving Fuzzy Fractional Partial Differential Equations. Fractal and Fractional, 6(11), 656. https://doi.org/10.3390/fractalfract6110656