A Modified Differential Transform Method for Solving Nonlinear Systems

Many natural phenomena are now being modelled by fractional calculus, however, there is still need for improvements of the present numerical approaches due to the non-local property of the fractional derivative and difficulty in solving problems related to physical units. In this paper, we present an improved numerical method based on Newton-Raphson fractional method (NFM) for solution of some nonlinear systems in complex space. Unlike in the Newton-Raphson fractional method, the commands of fractional derivatives are replaced with functions which is valid for one and several variables. The conformable fractional derivative of fractional order was employed to replace the first order derivative in the Newton method. Numerical results have been presented which show that the proposed numerical approach is efficient and promising

Adaptive Pinning Synchronization Control of the Fractional-Order Chaos Nodes in Complex Networks

Adaptive pinning synchronization control is studied for a class of fractional-order complex network systems which are constructed depending on small-world network algorithm. Based on the fractional-order stability theory, the suitable adaptive control scheme is designed to guarantee global asymptotic stability of all the nodes in complex network systems and the node selected algorithm is given. In numerical implementation, it is shown that the numerical solution of the fractional-order complex network systems can be obtained by applying an improved version of Adams-Bashforth-Moulton algorithm. Furthermore, simulation results are provided to confirm the validity and synchronization performance of the advocated design methodology

On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain. Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense. Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Method

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