An RBF-FD Method for Numerical Solutions of 2D Diffusion-Wave and Diffusion Equations of Distributed Fractional Order
The subject of this paper is to propose a numerical algorithm for solving 2D diffusion and diffusion-wave equations of distributed order fractional derivatives. Such equations arise in modelling complex systems and have many important applications. Existence of integral term over the order of fracti...
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| Published in | Journal of nonlinear mathematical physics Vol. 30; no. 4; pp. 1357 - 1374 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.12.2023
Springer Nature B.V |
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| ISSN | 1776-0852 1402-9251 1776-0852 |
| DOI | 10.1007/s44198-023-00153-1 |
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| Abstract | The subject of this paper is to propose a numerical algorithm for solving 2D diffusion and diffusion-wave equations of distributed order fractional derivatives. Such equations arise in modelling complex systems and have many important applications. Existence of integral term over the order of fractional derivative causes the high complexity of these equations and so their numerical solutions needs special cares. Using Gauss quadrature approach for discretizing the integral term of fractional derivative converts the distributed equation into a multi-term fractional differential equation. Then, the time variable is discretized with a suitable finite difference approach. The resultant semi-discretized equations are fully discretized by a radial basis function-generated finite difference based method. Convergence of the method are studied numerically. Various kind of test problems are considered for a comprehensive numerical study and the results confirm the efficiency of the method. |
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| AbstractList | The subject of this paper is to propose a numerical algorithm for solving 2D diffusion and diffusion-wave equations of distributed order fractional derivatives. Such equations arise in modelling complex systems and have many important applications. Existence of integral term over the order of fractional derivative causes the high complexity of these equations and so their numerical solutions needs special cares. Using Gauss quadrature approach for discretizing the integral term of fractional derivative converts the distributed equation into a multi-term fractional differential equation. Then, the time variable is discretized with a suitable finite difference approach. The resultant semi-discretized equations are fully discretized by a radial basis function-generated finite difference based method. Convergence of the method are studied numerically. Various kind of test problems are considered for a comprehensive numerical study and the results confirm the efficiency of the method. |
| Author | Safarpoor, Mansour Taghipour, Fatemeh Shirzadi, Ahmad |
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Syst. 40(6), 3509–3527 (2020). https://doi.org/10.3934/dcds.2020027. https://www.aimsciences.org/article/id/2ded8560-83bf-43a5-800e-db6a0330137e PodlubnyISkovranekTJaraBMVPetrasIVerbitskyVChenYMatrix approach to discrete fractional calculus iii: non-equidistant grids, variable step length and distributed orders, Philosophical Transactions of the Royal Society A: MathematicalPhys. Eng. Sci.20133711990201201533044228 WeiLLiuLSunHStability and convergence of a local discontinuous galerkin method for the fractional diffusion equation with distributed orderJ. Appl. Math. Comput.201959323341393522810.1007/s12190-018-1182-z GhehsarehHRZaghianAMajlesiAThe method of approximate particular solutions to simulate an anomalous mobile-immobile transport processMath. Methods Appl. Sci.202043636373649408550110.1002/mma.6144 ShirzadiMRostamiMDehghanMLiXAmerican options pricing under regime-switching jump-diffusion models with meshfree finite point methodChaos Solitons Fract.2023166451737610.1016/j.chaos.2022.112919 AbbaszadehMDehghanMAn improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimateNumer. Algorithms2017751173211363600910.1007/s11075-016-0201-0 KilbasAASrivastavaHMTrujilloJJTheory and Applications of Fractional Differential Equations2006North-HollandElsevier10.1016/S0304-0208(06)80001-0 Ding, W., Patnaik, S., Sidhardh, S., Semperlotti, F.: Applications of distributed-order fractional operators: a review. Entropy. https://doi.org/10.3390/e23010110. https://www.mdpi.com/1099-4300/23/1/110 Hua GaoGZhong SunZTwo alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equationsComput. Math. 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Lett.201996159165395062210.1016/j.aml.2019.04.030 Liu, P., Shi, J., Wang, Z.-A.: Pattern formation of the attraction-repulsion Keller-Segel system. Discrete Contin. Dyn. Syst. B 18(10), 2597–2625 (2013). https://doi.org/10.3934/dcdsb.2013.18.2597. https://www.aimsciences.org/article/id/07a4507d-9ffb-4e2c-9c13-ef57dbd913c4 SandevTChechkinAVKorabelNKantzHSokolovIMMetzlerRDistributed-order diffusion equations and multifractality: models and solutionsPhys. Rev. E201592355586310.1103/PhysRevE.92.042117 WangFWangHZhouXFuRA driving fatigue feature detection method based on multifractal theoryIEEE Sens. J.20222219190461905910.1109/JSEN.2022.3201015 FasshauerGEMeshfree Approximation Methods with MATLAB2007SingaporeWorld Scientific ShirzadiMDehghanMGeneralized regularized least-squares approximation of noisy data with application to stochastic PDEsAppl. Math. Lett.2021111414209510.1016/j.aml.2020.106598 KochubeiANDistributed order calculus and equations of ultraslow diffusionJ. Math. Anal. Appl.20083401252281237615210.1016/j.jmaa.2007.08.024 AslefallahMAbbasbandySShivanianEMeshless singular boundary method for two-dimensional pseudo-parabolic equation: analysis of stability and convergenceJ. Appl. Math. Comput.202063585606410099310.1007/s12190-020-01330-x Luo, R., Peng, Z., Hu, J.: On model identification based optimal control and its applications to multi-agent learning and control. Mathematics. https://doi.org/10.3390/math11040906. https://www.mdpi.com/2227-7390/11/4/906 HosseinzadehHShirzadiAA new meshless local integral equation methodAppl. Numer. Math.20231944458463550610.1016/j.apnum.2023.08.007 FornbergBFlyerNA Primer on Radial Basis Functions with Applications to the Geosciences2015PhiladelphiaSociety for Industrial and Applied Mathematics10.1137/1.9781611974041 Nava-Antonio, G., Fernandez-Anaya, G., Hernandez-Martinez, E.G., Jamous-Galante, J., Ferreira-Vazquez, E., Flores-Godoy, J.: Consensus of multi-agent systems with distributed fractional order dynamics. In: International workshop on complex systems and networks (IWCSN) vol. 2017, pp. 190–197 (2017). https://doi.org/10.1109/IWCSN.2017.8276526 GE Fasshauer (153_CR32) 2007 M Dehghan (153_CR29) 2016; 39 153_CR7 M Shirzadi (153_CR25) 2021; 111 AN Kochubei (153_CR8) 2008; 340 153_CR4 G Hua Gao (153_CR17) 2015; 69 B Fornberg (153_CR28) 2015 153_CR3 153_CR6 H Hosseinzadeh (153_CR23) 2023; 213 Q Yu (153_CR9) 2022; 427 A Mohebbi (153_CR30) 2013; 63 M Safarpoor (153_CR24) 2019; 98 H Hosseinzadeh (153_CR27) 2023; 194 T Sandev (153_CR16) 2015; 92 H Wendland (153_CR31) 2004 I Ali (153_CR21) 2023; 30 153_CR11 153_CR10 F Wang (153_CR15) 2022; 22 F Mainardi (153_CR2) 2010 153_CR12 M Abbaszadeh (153_CR33) 2017; 75 HR Ghehsareh (153_CR20) 2020; 43 E Shivanian (153_CR22) 2020; 17 J Ren (153_CR34) 2019; 96 AA Kilbas (153_CR1) 2006 I Podlubny (153_CR13) 2013; 371 L Wei (153_CR18) 2019; 59 M Aslefallah (153_CR19) 2020; 63 JT Katsikadelis (153_CR14) 2014; 259 C Escudero (153_CR5) 2006; 19 M Shirzadi (153_CR26) 2023; 166 |
| References_xml | – reference: Caputo, M.: Diffusion with space memory modelled with distributed order space fractional differential equations. Ann. Geophys. 46(2) – reference: ShivanianEPseudospectral meshless radial point hermit interpolation versus pseudospectral meshless radial point interpolationInt. J. Comput. Methods202017071950023413573010.1142/S0219876219500233 – reference: EscuderoCThe fractional Keller–Segel modelNonlinearity200619122909227376510.1088/0951-7715/19/12/010 – reference: Nava-Antonio, G., Fernandez-Anaya, G., Hernandez-Martinez, E.G., Jamous-Galante, J., Ferreira-Vazquez, E., Flores-Godoy, J.: Consensus of multi-agent systems with distributed fractional order dynamics. In: International workshop on complex systems and networks (IWCSN) vol. 2017, pp. 190–197 (2017). https://doi.org/10.1109/IWCSN.2017.8276526 – reference: ShirzadiMRostamiMDehghanMLiXAmerican options pricing under regime-switching jump-diffusion models with meshfree finite point methodChaos Solitons Fract.2023166451737610.1016/j.chaos.2022.112919 – reference: Ding, W., Patnaik, S., Sidhardh, S., Semperlotti, F.: Applications of distributed-order fractional operators: a review. Entropy. https://doi.org/10.3390/e23010110. https://www.mdpi.com/1099-4300/23/1/110 – reference: FasshauerGEMeshfree Approximation Methods with MATLAB2007SingaporeWorld Scientific – reference: SandevTChechkinAVKorabelNKantzHSokolovIMMetzlerRDistributed-order diffusion equations and multifractality: models and solutionsPhys. Rev. E201592355586310.1103/PhysRevE.92.042117 – reference: MainardiFFractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models2010SingaporeWorld Scientific10.1142/p614 – reference: KochubeiANDistributed order calculus and equations of ultraslow diffusionJ. Math. Anal. Appl.20083401252281237615210.1016/j.jmaa.2007.08.024 – reference: PodlubnyISkovranekTJaraBMVPetrasIVerbitskyVChenYMatrix approach to discrete fractional calculus iii: non-equidistant grids, variable step length and distributed orders, Philosophical Transactions of the Royal Society A: MathematicalPhys. Eng. Sci.20133711990201201533044228 – reference: RenJChenHA numerical method for distributed order time fractional diffusion equation with weakly singular solutionsAppl. Math. 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Appl.2015699926948333180810.1016/j.camwa.2015.02.023 – reference: MohebbiAAbbaszadehMCompact finite difference scheme for the solution of time fractional advection-dispersion equationNumer. Algorithms2013633431452307118510.1007/s11075-012-9631-5 – reference: YuQTurnerILiuFVeghVThe application of the distributed-order time fractional Bloch model to magnetic resonance imagingAppl. Math. Comput.2022427441268210.1016/j.amc.2022.127188 – reference: Luo, R., Peng, Z., Hu, J.: On model identification based optimal control and its applications to multi-agent learning and control. Mathematics. https://doi.org/10.3390/math11040906. https://www.mdpi.com/2227-7390/11/4/906 – reference: AslefallahMAbbasbandySShivanianEMeshless singular boundary method for two-dimensional pseudo-parabolic equation: analysis of stability and convergenceJ. Appl. Math. Comput.202063585606410099310.1007/s12190-020-01330-x – reference: AbbaszadehMDehghanMAn improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimateNumer. Algorithms2017751173211363600910.1007/s11075-016-0201-0 – reference: SafarpoorMShirzadiAA localized RBF-MLPG method for numerical study of heat and mass transfer equations in elliptic finsEng. Anal. Bound. Elem.2019983545387264910.1016/j.enganabound.2018.09.016 – reference: WangFWangHZhouXFuRA driving fatigue feature detection method based on multifractal theoryIEEE Sens. J.20222219190461905910.1109/JSEN.2022.3201015 – reference: AliIHaqSUllahRArifeenSUApproximate solution of second order singular perturbed and obstacle boundary value problems using meshless method based on radial basis functionsJ. Nonlinear Math. Phys.202330215234455741710.1007/s44198-022-00080-7 – reference: Liu, P., Shi, J., Wang, Z.-A.: Pattern formation of the attraction-repulsion Keller-Segel system. Discrete Contin. Dyn. Syst. B 18(10), 2597–2625 (2013). https://doi.org/10.3934/dcdsb.2013.18.2597. https://www.aimsciences.org/article/id/07a4507d-9ffb-4e2c-9c13-ef57dbd913c4 – reference: GhehsarehHRZaghianAMajlesiAThe method of approximate particular solutions to simulate an anomalous mobile-immobile transport processMath. Methods Appl. Sci.202043636373649408550110.1002/mma.6144 – reference: FornbergBFlyerNA Primer on Radial Basis Functions with Applications to the Geosciences2015PhiladelphiaSociety for Industrial and Applied Mathematics10.1137/1.9781611974041 – reference: KilbasAASrivastavaHMTrujilloJJTheory and Applications of Fractional Differential Equations2006North-HollandElsevier10.1016/S0304-0208(06)80001-0 – reference: HosseinzadehHShirzadiAA new meshless local integral equation methodAppl. Numer. Math.20231944458463550610.1016/j.apnum.2023.08.007 – reference: Jin, H.-Y., Wang, Z.-A.: Global stabilization of the full attraction-repulsion Keller–Segel system. Discrete Contin. Dyn. Syst. 40(6), 3509–3527 (2020). https://doi.org/10.3934/dcds.2020027. https://www.aimsciences.org/article/id/2ded8560-83bf-43a5-800e-db6a0330137e – reference: KatsikadelisJTNumerical solution of distributed order fractional differential equationsJ. Comput. Phys.20142591122314855610.1016/j.jcp.2013.11.013 – reference: HosseinzadehHShirzadiAOn optimal radius of sub-domains in meshless LBIE methodMath. Comput. Simul.2023213145160460301610.1016/j.matcom.2023.06.006 – reference: DehghanMSafarpoorMThe dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equationsMath. Methods Appl. 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| SubjectTerms | Accuracy Algorithms Approximation Boundary conditions Complex systems Complexity Derivatives Differential equations Discretization Finite difference method Fractional calculus Integrals Mathematical Physics Mathematics Mathematics and Statistics Methods Numerical analysis Quadratures Radial basis function Research Article Wave equations |
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| Title | An RBF-FD Method for Numerical Solutions of 2D Diffusion-Wave and Diffusion Equations of Distributed Fractional Order |
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