Wavelet Methods for the Solutions of Partial and Fractional Differential Equations Arising in Physical Problems

The subject of fractional calculus has gained considerable popularity and importance during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It deals with derivatives and integrals of arbitrary orders. The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PI D controller for the control of dynamical systems etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, control theory, neutron point kinetic model, anomalous diffusion, Brownian motion, signal and image processing, fluid dynamics and material science are well described by differential equations of fractional order. Generally, nonlinear partial differential equations of fractional order are difficult to solve. So for the last few decades, a great deal of attention has been directed towards the solution (both exact and numerical) of these problems. The aim of this dissertation is to present an extensive study of different wavelet methods for obtaining numerical solutions of mathematical problems occurring in disciplines of science and engineering. This present work also provides a comprehensive foundation of different wavelet methods comprising Haar wavelet method, Legendre wavelet method, Legendre multi-wavelet methods, Chebyshev wavelet method, Hermite wavelet method and Petrov-Galerkin method. The intension is to examine the accuracy of various wavelet methods and their efficiency for solving nonlinear fractional differential equations. With the widespread applications of wavelet methods for solving difficult problems in diverse fields of science and engineering such as wave propagation, data compression, image processing, pattern recognition, computer graphics and in medical technology, these methods have been implemented to develop accurate and fast algorithms for solving integral, differential and integro-differential equations, especially those whose solutions are highly localized in position and scale. The main feature of wavelets is its ability to convert the given differential and integral equations to a system of linear or nonlinear algebraic equations, which can be solved by numerical methods. Therefore, our main focus in the present work is to analyze the application of wavelet based transform methods for solving the problem of fractional order partial differential equations. The introductory concept of wavelet, wavelet transform and multi-resolution analysis (MRA) have been discussed in the preliminary chapter. The basic idea of various analytical and numerical methods viz. Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), First Integral Method (FIM), Optimal Homotopy Asymptotic Method (OHAM), Haar Wavelet Method, Legendre Wavelet Method, Chebyshev Wavelet Method and Hermite Wavelet Method have been presented in chapter 1. In chapter 2, we have considered both analytical and numerical approach for solving some particular nonlinear partial differential equations like Burgers’ equation, modified Burgers’ equation, Huxley equation, Burgers-Huxley equation and modified KdV equation, which have a wide variety of applications in physical models. Variational Iteration Method and Haar wavelet Method are applied to obtain the analytical and numerical approximate solution of Huxley and Burgers-Huxley equations. Comparisons between analytical solution and numerical solution have been cited in tables and also graphically. The Haar wavelet method has also been applied to solve Burgers’, modified Burgers’, and modified KdV equations numerically. The results thus obtained are compared with exact solutions as well as solutions available in open literature. Error of collocation method has been presented in this chapter. Methods like Homotopy Perturbation Method (HPM) and Optimal Homotopy Asymptotic Method (OHAM) are very powerful and efficient techniques for solving nonlinear PDEs. Using these methods, many functional equations such as ordinary, partial differential equations and integral equations have been solved. We have implemented HPM and OHAM in chapter 3, in order to obtain the analytical approximate solutions of system of nonlinear partial differential equation viz. the Boussinesq-Burgers’ equations. Also, the Haar wavelet method has been applied to obtain the numerical solution of BoussinesqBurgers’ equations. Also, the convergence of HPM and OHAM has been discussed in this chapter. The mathematical modeling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to differential equations of fractional order and the necessity to solve such equations. The mathematical preliminaries of fractional calculus, definitions and theorems have been presented in chapter 4. Next, in this chapter, the Haar wavelet method has been analyzed for solving fractional differential equations. The time-fractional Burgers-Fisher, generalized Fisher type equations, nonlinear time- and space-fractional Fokker-Planck equations have been solved by using two-dimensional Haar wavelet method. The obtained results are compared with the Optimal Homotopy Asymptotic Method (OHAM), the exact solutions and the results available in open literature. Comparison of obtained results with OHAM, Adomian Decomposition Method (ADM), VIM and Operational Tau Method (OTM) has been demonstrated in order to justify the accuracy and efficiency of the proposed schemes. The convergence of two-dimensional Haar wavelet technique has been provided at the end of this chapter. In chapter 5, the fractional differential equations such as KdV-Burger-Kuramoto (KBK) equation, seventh order KdV (sKdV) equation and Kaup-Kupershmidt (KK) equation have been solved by using two-dimensional Legendre wavelet and Legendre multi-wavelet methods. The main focus of this chapter is the application of two-dimensional Legendre wavelet technique for solving nonlinear fractional differential equations like timefractional KBK equation, time-fractional sKdV equation in order to demonstrate the efficiency and accuracy of the proposed wavelet method. Similarly in chapter 6, twodimensional Chebyshev wavelet method has been implemented to obtain the numerical solutions of the time-fractional Sawada-Kotera equation, fractional order Camassa-Holm equation and Riesz space-fractional sine-Gordon equations. The convergence analysis has been done for these wavelet methods. In chapter 7, the solitary wave solution of fractional modified Fornberg-Whitham equation has been attained by using first integral method and also the approximate solutions obtained by optimal homotopy asymptotic method (OHAM) are compared with the exact solutions acquired by first integral method. Also, the Hermite wavelet method has been implemented to obtain approximate solutions of fractional modified Fornberg-Whitham equation. The Hermite wavelet method is implemented to system of nonlinear fractional differential equations viz. the fractional Jaulent-Miodek equations. Convergence of this wavelet methods has been discussed in this chapter. Chapter 8 emphasizes on the application of Petrov-Galerkin method for solving the fractional differential equations such as the fractional KdV-Burgers’ (KdVB) equation and the fractional Sharma-TassoOlver equation with a view to exhibit the capabilities of this method in handling nonlinear equation. The main objective of this chapter is to establish the efficiency and accuracy of Petrov-Galerkin method in solving fractional differential equtaions numerically by implementing a linear hat function as the trial function and a quintic B-spline function as the test function. Various wavelet methods have been successfully employed to numerous partial and fractional differential equations in order to demonstrate the validity and accuracy of these procedures. Analyzing the numerical results, it can be concluded that the wavelet methods provide worthy numerical solutions for both classical and fractional order partial differential equations. Finally, it is worthwhile to mention that the proposed wavelet methods are promising and powerful methods for solving fractional differential equations in mathematical physics. This work also aimed at, to make this subject popular and acceptable to engineering and science community to appreciate the universe of wonderful mathematics, which is in between classical integer order differentiation and integration, which till now is not much acknowledged, and is hidden from scientists and engineers. Therefore, our goal is to encourage the reader to appreciate the beauty as well as the usefulness of these numerical wavelet based techniques in the study of nonlinear physical system

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

Fractional model of cancer immunotherapy and its optimal control

Cancer is one of the most serious illnesses in all of the world. Although most of the cancer patients are treated with chemotherapy, radiotherapy and surgery, wide research is conducted related to experimental and theoretical immunology. In recent years, the research on cancer immunotherapy has led to major medical advances. Cancer immunotherapy refers to the stimulation of immune system to deal with cancer cells. In medical practice, it is mainly achieved by using effector cells such as activated T-cells and Interleukin-2 (IL-2), which is the main cytokine responsible for lymphocyte activation, growth and differentiation. A well-known mathematical model, named as Kirschner-Panetta (KP) model, represents richly the dynamics of the interaction between cancer cells, IL-2 and the effector cells. The dynamics of the KP model is described and the solution to which is approximated by using polynomial approximation based methods such as Adomian decomposition method and differential transform method. The rich nonlinearity of the KP model causes these approaches to become so complicated in order to deal with the representation of polynomial approximations. It is illustrated that the approximated polynomials are in good agreement with the solution obtained by common numerical approaches. In the KP model, the growth of the tumour cells can be expressed by a linear function or any limited-growth function such as logistic equation, in which the cancer population possesses an upper bound mentioned as carrying capacity. Effector cells and IL-2 construct two external sources of medical treatment to stimulate immune system to eradicate cancer cells. Since the main goal in immunotherapy is to remove the tumour cells with the least probable medication side effects, an advanced version of the model may include a time dependent external sources of medical treatment, meaning that the external sources of medical treatment could be considered as control functions of time and therefore the optimum use of medical sources can be evaluated in order to achieve the optimal measure of an objective function. With this sense of direction, two distinct strategies are explored. The first one is to only consider the external source of effector cells as the control function to formulate an optimal control problem. It is shown under which circumstances, the tumour is eliminated. The approach in the formulation of the optimal control is the Pontryagin maximum principal. Furthermore the optimal control problem will be dealt with using particle swarm optimization (PSO). It is shown that the obtained results are significantly better than those obtained by previous researchers. The second strategy is to formulate an optimal control problem by considering both the two external sources as the controls. To our knowledge, it is the first time to present a multiple therapeutic protocol for the KP model. Some MATLAB routines are develop to solve the optimal control problems based on Pontryagin maximum principal and also the PSO. As known, fractional differential equations are more appropriate to describe the persistent memory of physical phenomena. Thus, the fractional KP model is defined in the sense of Caputo differentiation operator. An effective method for numerical treatment of the model is described, namely Predictor-Corrector method of Adams-Bashforth-Moulton type. A robust MATLAB routine is coded based on the mentioned approach and the solution obtained will be compared with those of the classical KP model. The code is prepared in such a way to be able to deal with systems of fractional differential equations, in which each equation has its own fractional order (i.e. multi-order systems of fractional differential equations). The theorems for existence of solutions and the stability analysis of the fractional KP model are represented. In this regard, a frequently used method of solving fractional differential equations (FDEs) is described in details, namely multi-step generalized differential transform method (MSGDTM), then it is illustrated that the method neglects the persistent memory property and takes the incorrect approach in dealing with numerical solutions of FDEs and therefore it is unfit to be used in differential equations governed by fractional differentiation operators. The sigmoidal behavior of the solution to the logistic equation caused it to be one of the most versatile models in natural sciences and therefore the fractional logistic equation would be a relevant problem to be dealt with. Thus, a power series of Mittag-Leffer functions is introduced, the behaviour of which is in good agreement with the solution to fractional logistic equation (FLE), and then a fractional integro-differential equation is represented and proved to be satisfied with the power series of Mittag-Leffler function. The obtained fractional integro-differential equation is named as modified fractional differential equation (MFDL) and possesses a nonlinear additive term related to the solution of the logistic equation (LE). The method utilized in the thesis, may be appropriately applied to the analysis of solutions to nonlinear fractional differential equations of mathematical physics. Inverse problems to FDEs occur in many branches of science. Such problems have been investigated, for instance, in fractional diffusion equation and inverse boundary value problem for semi- linear fractional telegraph equation. The determination of the order of fractional differential equations is an issue, which has been analyzed and discussed in, for instance, fractional diffusion equations. Thus, fractional order estimation has been conducted for some classes of linear fractional differential equations, by introducing the relationship between the fractional order and the asymptotic behaviour of the solutions to linear fractional differential equations. Fractional optimal control problems, in which the system and (or) the objective function are described based on fractional derivatives, are much more complicated to be solved by using a robust and reliable numerical approach. Thus, a MATLAB routine is provided to solve the optimal control for fractional KP model and the obtained solutions are compared with those of classical KP model. It is shown that the results for fractional optimal control problems are better than classical optimal control problem in the sense of the amount of drug administration

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable