Discrete Nonlinear Planar Systems and Applications to Biological Population Models

We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential. We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system. We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd. The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates

Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations

Solutions of fractional gas dynamics equation by a new technique

[EN] In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section.This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.Akgül, A.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2020). Solutions of fractional gas dynamics equation by a new technique. Mathematical Methods in the Applied Sciences. 43(3):1349-1358. https://doi.org/10.1002/mma.5950S13491358433Singh, J., Kumar, D., & Kılıçman, A. (2013). Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform. Abstract and Applied Analysis, 2013, 1-8. doi:10.1155/2013/934060Momani, S. (2005). Analytic and approximate solutions of the space- and time-fractional telegraph equations. Applied Mathematics and Computation, 170(2), 1126-1134. doi:10.1016/j.amc.2005.01.009Hajipour, M., Jajarmi, A., Baleanu, D., & Sun, H. (2019). On an accurate discretization of a variable-order fractional reaction-diffusion equation. Communications in Nonlinear Science and Numerical Simulation, 69, 119-133. doi:10.1016/j.cnsns.2018.09.004Meng, R., Yin, D., & Drapaca, C. S. (2019). Variable-order fractional description of compression deformation of amorphous glassy polymers. Computational Mechanics, 64(1), 163-171. doi:10.1007/s00466-018-1663-9Baleanu, D., Jajarmi, A., & Hajipour, M. (2018). On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel. Nonlinear Dynamics, 94(1), 397-414. doi:10.1007/s11071-018-4367-yJajarmi, A., & Baleanu, D. (2018). A new fractional analysis on the interaction of HIV withCD4+T-cells. Chaos, Solitons & Fractals, 113, 221-229. doi:10.1016/j.chaos.2018.06.009Baleanu, D., Jajarmi, A., Bonyah, E., & Hajipour, M. (2018). New aspects of poor nutrition in the life cycle within the fractional calculus. Advances in Difference Equations, 2018(1). doi:10.1186/s13662-018-1684-xJajarmi, A., & Baleanu, D. (2017). Suboptimal control of fractional-order dynamic systems with delay argument. Journal of Vibration and Control, 24(12), 2430-2446. doi:10.1177/1077546316687936Singh, J., Kumar, D., & Baleanu, D. (2018). On the analysis of fractional diabetes model with exponential law. Advances in Difference Equations, 2018(1). doi:10.1186/s13662-018-1680-1Kumar, D., Singh, J., Tanwar, K., & Baleanu, D. (2019). A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws. International Journal of Heat and Mass Transfer, 138, 1222-1227. doi:10.1016/j.ijheatmasstransfer.2019.04.094Kumar, D., Singh, J., Al Qurashi, M., & Baleanu, D. (2019). A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying. Advances in Difference Equations, 2019(1). doi:10.1186/s13662-019-2199-9Kumar, D., Singh, J., Purohit, S. D., & Swroop, R. (2019). A hybrid analytical algorithm for nonlinear fractional wave-like equations. Mathematical Modelling of Natural Phenomena, 14(3), 304. doi:10.1051/mmnp/2018063Kumar, D., Tchier, F., Singh, J., & Baleanu, D. (2018). An Efficient Computational Technique for Fractal Vehicular Traffic Flow. Entropy, 20(4), 259. doi:10.3390/e20040259Goswami, A., Singh, J., Kumar, D., & Sushila. (2019). An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Physica A: Statistical Mechanics and its Applications, 524, 563-575. doi:10.1016/j.physa.2019.04.058Mohyud-Din, S. T., Bibi, S., Ahmed, N., & Khan, U. (2018). Some exact solutions of the nonlinear space–time fractional differential equations. Waves in Random and Complex Media, 29(4), 645-664. doi:10.1080/17455030.2018.1462541Momani, S., & Shawagfeh, N. (2006). Decomposition method for solving fractional Riccati differential equations. Applied Mathematics and Computation, 182(2), 1083-1092. doi:10.1016/j.amc.2006.05.008Hashim, I., Abdulaziz, O., & Momani, S. (2009). Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation, 14(3), 674-684. doi:10.1016/j.cnsns.2007.09.014Yıldırım, A. (2010). He’s homotopy perturbation method for solving the space- and time-fractional telegraph equations. International Journal of Computer Mathematics, 87(13), 2998-3006. doi:10.1080/00207160902874653Momani, S., & Odibat, Z. (2007). Numerical comparison of methods for solving linear differential equations of fractional order. Chaos, Solitons & Fractals, 31(5), 1248-1255. doi:10.1016/j.chaos.2005.10.068Rida, S. Z., El-Sayed, A. M. A., & Arafa, A. A. M. (2010). On the solutions of time-fractional reaction–diffusion equations. Communications in Nonlinear Science and Numerical Simulation, 15(12), 3847-3854. doi:10.1016/j.cnsns.2010.02.007Machado, J. A. T., & Mata, M. E. (2014). A fractional perspective to the bond graph modelling of world economies. Nonlinear Dynamics, 80(4), 1839-1852. doi:10.1007/s11071-014-1334-0Raja Balachandar, S., Krishnaveni, K., Kannan, K., & Venkatesh, S. G. (2018). Analytical Solution for Fractional Gas Dynamics Equation. National Academy Science Letters, 42(1), 51-57. doi:10.1007/s40009-018-0662-xWang, Y.-L., Liu, Y., Li, Z., & zhang, H. (2018). Numerical solution of integro-differential equations of high-order Fredholm by the simplified reproducing kernel method. International Journal of Computer Mathematics, 96(3), 585-593. doi:10.1080/00207160.2018.1455091Gumah, G. N., Naser, M. F. M., Al-Smadi, M., & Al-Omari, S. K. (2018). Application of reproducing kernel Hilbert space method for solving second-order fuzzy Volterra integro-differential equations. Advances in Difference Equations, 2018(1). doi:10.1186/s13662-018-1937-8Al-Smadi, M. (2018). Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation. Ain Shams Engineering Journal, 9(4), 2517-2525. doi:10.1016/j.asej.2017.04.006Kashkari, B. S. H., & Syam, M. I. (2018). Reproducing Kernel Method for Solving Nonlinear Fractional Fredholm Integrodifferential Equation. Complexity, 2018, 1-7. doi:10.1155/2018/2304858Akgül, A., & Grow, D. (2019). Existence of Unique Solutions to the Telegraph Equation in Binary Reproducing Kernel Hilbert Spaces. Differential Equations and Dynamical Systems, 28(3), 715-744. doi:10.1007/s12591-019-00453-3Akgül, A., Khan, Y., Akgül, E. K., Baleanu, D., & Al Qurashi, M. M. (2017). Solutions of nonlinear systems by reproducing kernel method. The Journal of Nonlinear Sciences and Applications, 10(08), 4408-4417. doi:10.22436/jnsa.010.08.33Karatas Akgül, E. (2018). Reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(2), 145-151. doi:10.11121/ijocta.01.2018.00568Akgül, A., Inc, M., & Karatas, E. (2015). Reproducing kernel functions for difference equations. Discrete & Continuous Dynamical Systems - S, 8(6), 1055-1064. doi:10.3934/dcdss.2015.8.1055Akgül, A., Inc, M., Karatas, E., & Baleanu, D. (2015). Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique. Advances in Difference Equations, 2015(1). doi:10.1186/s13662-015-0558-8Aronszajn, N. (1950). Theory of reproducing kernels. Transactions of the American Mathematical Society, 68(3), 337-337. doi:10.1090/s0002-9947-1950-0051437-7Bergman, S. (1950). The Kernel Function and Conformal Mapping. Mathematical Surveys and Monographs. doi:10.1090/surv/005Inc, M., & Akgül, A. (2014). Approximate solutions for MHD squeezing fluid flow by a novel method. Boundary Value Problems, 2014(1). doi:10.1186/1687-2770-2014-18Inc, M., Akgül, A., & Geng, F. (2014). Reproducing Kernel Hilbert Space Method for Solving Bratu’s Problem. Bulletin of the Malaysian Mathematical Sciences Society, 38(1), 271-287. doi:10.1007/s40840-014-0018-8Wang, Y., & Chao, L. (2008). Using reproducing kernel for solving a class of partial differential equation with variable-coefficients. Applied Mathematics and Mechanics, 29(1), 129-137. doi:10.1007/s10483-008-0115-yWu, B. Y., & Li, X. Y. (2011). A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method. Applied Mathematics Letters, 24(2), 156-159. doi:10.1016/j.aml.2010.08.036Yao, H., & Lin, Y. (2009). Solving singular boundary-value problems of higher even-order. Journal of Computational and Applied Mathematics, 223(2), 703-713. doi:10.1016/j.cam.2008.02.01

Asymptotic properties of discrete linear fractional equations

In this paper we study the dynamical behavior of linear discrete-time fractional systems. The first main result is that the norm of the difference of two different solutions of a time-varying discrete-time Caputo equation tends to zero not faster than polynomially. The second main result is a complete description of the decay to zero of the trajectories of one-dimensional time-invariant stable Caputo and Riemann-Liouville equations. Moreover, we present Volterra convolution equations, that are equivalent to Caputo equations

On a moment generalization of some classical second-order differential equations generating classical orthogonal polynomials

The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of Jacobi, Laguerre, Hermite and Bessel. These functional equations can be chosen to be of different type: fractional differential equations, q-difference equations, etc, which converge to their respective differential equations of the aforesaid classical orthogonal polynomials. In addition to this, there exists a confluence of both the families of polynomials constructed and the functional equations who approach to the classical families of polynomials and second-order differential equations, respectivel

An Efficient Computational Method for Solving a System of FDEs via Fractional Finite Difference Method

This paper aims to provide a numerical method for solving systems of fractional (Caputo sense) differential equations (FDEs). This method is based on the fractional finite difference method (FDM), where we implemented the Grünwald-Letnikov’s approach. This method is computationally very efficient and gives very accurate solutions. In this study, the stability of the obtained numerical scheme is given. The numerical results show that the proposed approach is easy to be implemented and are accurate when applied to system of FDEs. The method introduces promising tool for solving many systems of FDEs. Two examples are given to demonstrate the applicability and the effectiveness of our method

Nonlocal operators are chaotic

[EN] We characterize for the first time the chaotic behavior of nonlocal operators that come from a broad class of time-stepping schemes of approximation for fractional differential operators. For that purpose, we use criteria for chaos of Toeplitz operators in Lebesgue spaces of sequences. Surprisingly, this characterization is proved to be-in some cases-dependent of the fractional order of the operator and the step size of the scheme.C. Lizama is partially supported by FONDECYT (Grant No. 1180041) and DICYT, Universidad de Santiago de Chile, USACH. M. Murillo-Arcila is supported by MICINN and FEDER, Projects MTM2016-75963-P and PID2019-105011GB-I00, and by Generalitat Valenciana, Project GVA/2018/110. A. Peris is supported by MICINN and FEDER, Projects MTM2016-75963-P and PID2019-105011GB-I00, and by Generalitat Valenciana, Project PROMETEO/2017/102.Lizama, C.; Murillo Arcila, M.; Peris Manguillot, A. (2020). Nonlocal operators are chaotic. Chaos An Interdisciplinary Journal of Nonlinear Science. 30(10):1-8. https://doi.org/10.1063/5.0018408183010Abadias, L., & Miana, P. J. (2018). Generalized Cesàro operators, fractional finite differences and Gamma functions. Journal of Functional Analysis, 274(5), 1424-1465. doi:10.1016/j.jfa.2017.10.010Atici, F. M., & Eloe, P. (2009). Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory of Differential Equations, (3), 1-12. doi:10.14232/ejqtde.2009.4.3Atıcı, F. M., & Eloe, P. W. (2011). Two-point boundary value problems for finite fractional difference equations. Journal of Difference Equations and Applications, 17(4), 445-456. doi:10.1080/10236190903029241Atici, F. M., & Eloe, P. W. (2008). Initial value problems in discrete fractional calculus. Proceedings of the American Mathematical Society, 137(03), 981-989. doi:10.1090/s0002-9939-08-09626-3Atıcı, F. M., & Şengül, S. (2010). Modeling with fractional difference equations. Journal of Mathematical Analysis and Applications, 369(1), 1-9. doi:10.1016/j.jmaa.2010.02.009Banks, J., Brooks, J., Cairns, G., Davis, G., & Stacey, P. (1992). On Devaney’s Definition of Chaos. The American Mathematical Monthly, 99(4), 332-334. doi:10.1080/00029890.1992.11995856Baranov, A., & Lishanskii, A. (2016). Hypercyclic Toeplitz Operators. Results in Mathematics, 70(3-4), 337-347. doi:10.1007/s00025-016-0527-xBayart, F., & Matheron, E. (2009). Dynamics of Linear Operators. doi:10.1017/cbo9780511581113DELAUBENFELS, R., & EMAMIRAD, H. (2001). Chaos for functions of discrete and continuous weighted shift operators. Ergodic Theory and Dynamical Systems, 21(05). doi:10.1017/s0143385701001675Edelman, M. (2014). Caputo standard α-family of maps: Fractional difference vs. fractional. Chaos: An Interdisciplinary Journal of Nonlinear Science, 24(2), 023137. doi:10.1063/1.4885536Edelman, M. (2015). On the fractional Eulerian numbers and equivalence of maps with long term power-law memory (integral Volterra equations of the second kind) to Grünvald-Letnikov fractional difference (differential) equations. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(7), 073103. doi:10.1063/1.4922834Edelman, M. (2015). Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α- Families of Maps. The interdisciplinary journal of Discontinuity, Nonlinearity, and Complexity, 4(4), 391-402. doi:10.5890/dnc.2015.11.003Erbe, L., Goodrich, C. S., Jia, B., & Peterson, A. (2016). Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions. Advances in Difference Equations, 2016(1). doi:10.1186/s13662-016-0760-3Ferreira, R. A. C. (2012). A discrete fractional Gronwall inequality. Proceedings of the American Mathematical Society, 140(5), 1605-1612. doi:10.1090/s0002-9939-2012-11533-3Ferreira, R. A. C. (2013). Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. Journal of Difference Equations and Applications, 19(5), 712-718. doi:10.1080/10236198.2012.682577Goodrich, C., & Peterson, A. C. (2015). Discrete Fractional Calculus. doi:10.1007/978-3-319-25562-0Goodrich, C. S. (2012). On discrete sequential fractional boundary value problems. Journal of Mathematical Analysis and Applications, 385(1), 111-124. doi:10.1016/j.jmaa.2011.06.022Goodrich, C. S. (2014). A convexity result for fractional differences. Applied Mathematics Letters, 35, 58-62. doi:10.1016/j.aml.2014.04.013Goodrich, C., & Lizama, C. (2020). A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity. Israel Journal of Mathematics, 236(2), 533-589. doi:10.1007/s11856-020-1991-2Gray, H. L., & Zhang, N. F. (1988). On a new definition of the fractional difference. Mathematics of Computation, 50(182), 513-529. doi:10.1090/s0025-5718-1988-0929549-2Li, K., Peng, J., & Jia, J. (2012). Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives. Journal of Functional Analysis, 263(2), 476-510. doi:10.1016/j.jfa.2012.04.011Lizama, C. (2017). The Poisson distribution, abstract fractional difference equations, and stability. Proceedings of the American Mathematical Society, 145(9), 3809-3827. doi:10.1090/proc/12895Lizama, C. (2015). lp-maximal regularity for fractional difference equations on UMD spaces. Mathematische Nachrichten, 288(17-18), 2079-2092. doi:10.1002/mana.201400326Lizama, C., & Murillo-Arcila, M. (2020). Discrete maximal regularity for volterra equations and nonlocal time-stepping schemes. Discrete & Continuous Dynamical Systems - A, 40(1), 509-528. doi:10.3934/dcds.2020020Martínez-Giménez, F. (2007). Chaos for power series of backward shift operators. Proceedings of the American Mathematical Society, 135(6), 1741-1752. doi:10.1090/s0002-9939-07-08658-3Radwan, A. G., AbdElHaleem, S. H., & Abd-El-Hafiz, S. K. (2016). Symmetric encryption algorithms using chaotic and non-chaotic generators: A review. Journal of Advanced Research, 7(2), 193-208. doi:10.1016/j.jare.2015.07.002Radwan, A. G., Moaddy, K., Salama, K. N., Momani, S., & Hashim, I. (2014). Control and switching synchronization of fractional order chaotic systems using active control technique. Journal of Advanced Research, 5(1), 125-132. doi:10.1016/j.jare.2013.01.003Wu, G.-C., & Baleanu, D. (2013). Discrete fractional logistic map and its chaos. Nonlinear Dynamics, 75(1-2), 283-287. doi:10.1007/s11071-013-1065-7Wu, G.-C., Baleanu, D., & Zeng, S.-D. (2014). Discrete chaos in fractional sine and standard maps. Physics Letters A, 378(5-6), 484-487. doi:10.1016/j.physleta.2013.12.01