Browsing by Author "Machado, J. A. Tenreiro"
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- Absolutely stable difference scheme for a general class of singular perturbation problemsPublication . El-Zahar, Essam R.; Alotaibi, A. M.; Ebaid, Abdelhalim; Baleanu, Dumitru; Machado, J. A. Tenreiro; Hamed, Y. S.This paper presents an absolutely stable noniterative difference scheme for solving a general class of singular perturbation problems having left, right, internal, or twin boundary layers. The original two-point second-order singular perturbation problem is approximated by a first-order delay differential equation with a variable deviating argument. This delay differential equation is transformed into a three-term difference equation that can be solved using the Thomas algorithm. The uniqueness and stability analysis are discussed, showing that the method is absolutely stable. An optimal estimate for the deviating argument is obtained to take advantage of the second-order accuracy of the central finite difference method in addition to the absolute stability property. Several problems having left, right, interior, or twin boundary layers are considered to validate and illustrate the method. The numerical results confirm that the deviating argument can stabilize the unstable discretized differential equation and that the new approach is effective in solving the considered class of singular perturbation problems.
- Accessing complexity from genome informationPublication . Machado, J. A. TenreiroThis paper studies the information content of the chromosomes of 24 species. In a first phase, a scheme inspired in dynamical system state space representation is developed. For each chromosome the state space dynamical evolution is shed into a two dimensional chart. The plots are then analyzed and characterized in the perspective of fractal dimension. This information is integrated in two measures of the species’ complexity addressing its average and variability. The results are in close accordance with phylogenetics pointing quantitative aspects of the species’ genomic complexity.
- Adaptive tackling of the swinging problem for a 2 DOF crane – payload systemPublication . Tar, József K.; Rudas, Imre J.; Bitó, János F.; Machado, J. A. Tenreiro; Kozłowski, Krzysztof R.The control of a crane carrying its payload by an elastic string corresponds to a task in which precise, indirect control of a subsystem dynamically coupled to a directly controllable subsystem is needed. This task is interesting since the coupled degree of freedom has little damping and it is apt to keep swinging accordingly. The traditional approaches apply the input shaping technology to assist the human operator responsible for the manipulation task. In the present paper a novel adaptive approach applying fixed point transformations based iterations having local basin of attraction is proposed to simultaneously tackle the problems originating from the imprecise dynamic model available for the system to be controlled and the swinging problem, too. The most important phenomenological properties of this approach are also discussed. The control considers the 4th time-derivative of the trajectory of the payload. The operation of the proposed control is illustrated via simulation results.
- Adomian Decomposition and Fractional Power Series Solution of a Class of Nonlinear Fractional Differential EquationsPublication . Mohammed, Pshtiwan Othman; Machado, J. A. Tenreiro; Guirao, Juan L. G.; Agarwal, Ravi P.Nonlinear fractional differential equations reflect the true nature of physical and biological models with non-locality and memory effects. This paper considers nonlinear fractional differential equations with unknown analytical solutions. The Adomian decomposition and the fractional power series methods are adopted to approximate the solutions. The two approaches are illustrated and compared by means of four numerical examples.
- Advances in fractional differential equations (IV): Time-fractional PDEsPublication . Zhou, Yong; Feckan, Michal; Liu, Fawang; Machado, J. A. TenreiroThe fractional calculus (FC) started more than three centuries ago. In the last years, FC is playing a very important role in various scientific fields. In fact, FC has been recognized as one of the best tools to describe long-memory processes. Fractional-order models are interesting not only for engineers and physicists, but also for mathematicians. Among such models those described by partial differential equations (PDEs) containing fractional derivatives are of utmost importance. Their evolution was more complex than for the classical integer-order counterpart. Nonetheless, classical PDEs’ methods are hardly applicable directly to fractional PDEs. Therefore, new theories and methods are required, with concepts and algorithms specifically developed for fractional PDEs. This is the fourth special issue on Advances in Fractional Differential Equations of the journal Computers and Mathematics with Applications. This selection of 38 papers focuses on innovative theoretical and numerical methods, and in applications of FC to important problems that encompass the most relevant areas of current research on fractional PDEs.
- An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion modelPublication . Nikan, O.; Avazzadeh, Z.; Machado, J. A. TenreiroThis paper adopts an efficient meshless approach for approximating the nonlinear fractional fourth-order diffusion model described in the Riemann–Liouville sense. A second-order difference technique is applied to discretize temporal derivatives, while the radial basis function meshless generated the finite difference scheme approximates the spatial derivatives. One key advantage of the local collocation method is the approximation of the derivatives via the finite difference formulation, for each local-support domain, by deriving the basis functions expansion. Another advantage of this method is that it can be applied in problems with non-regular geometrical domains. For the proposed time discretization, the unconditional stability is examined and an error bound is obtained. Numerical results illustrate the applicability and validity of the scheme and confirm the theoretical formulation.
- An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance indexPublication . Bhrawy, A. H.; Doha, E.H.; Machado, J. A. Tenreiro; Ezz-Eldien, S. S.The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.
- An Efficient Operational Matrix Technique for Multidimensional Variable-Order Time Fractional Diffusion EquationsPublication . Zaky, M. A.; Ezz-Eldien, S. S.; Doha, E. H.; Machado, J. A. Tenreiro; Bhrawy, A. H.This paper derives a new operational matrix of the variable-order (VO) time fractional partial derivative involved in anomalous diffusion for shifted Chebyshev polynomials. We then develop an accurate numerical algorithm to solve the 1þ1 and 2þ1 VO and constant-order fractional diffusion equation with Dirichlet conditions. The contraction of the present method is based on shifted Chebyshev collocation procedure in combination with the derived shifted Chebyshev operational matrix. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations, and it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, we analyze the convergence of the present method graphically. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.
- An evolutionary approach for the motion planning of redundant and hyper-redundant manipulatorsPublication . Marcos, Maria da Graça; Machado, J. A. Tenreiro; Azevedo-Perdicoúlis, T. P.The trajectory planning of redundant robots is an important area of research and efficient optimization algorithms are needed. The pseudoinverse control is not repeatable, causing drift in joint space which is undesirable for physical control. This paper presents a new technique that combines the closed-loop pseudoinverse method with genetic algorithms, leading to an optimization criterion for repeatable control of redundant manipulators, and avoiding the joint angle drift problem. Computer simulations performed based on redundant and hyper-redundant planar manipulators show that, when the end-effector traces a closed path in the workspace, the robot returns to its initial configuration. The solution is repeatable for a workspace with and without obstacles in the sense that, after executing several cycles, the initial and final states of the manipulator are very close.
- An Evolutionary Perspective of Virus PropagationPublication . Machado, J. A. TenreiroThis paper presents an evolutionary algorithm that simulates simplified scenarios of the diffusion of an infectious disease within a given population. The proposed evolutionary epidemic diffusion (EED) computational model has a limited number of variables and parameters, but is still able to simulate a variety of configurations that have a good adherence to real-world cases. The use of two space distances and the calculation of spatial 2-dimensional entropy are also examined. Several simulations demonstrate the feasibility of the EED for testing distinct social, logistic and economy risks. The performance of the system dynamics is assessed by several variables and indices. The global information is efficiently condensed and visualized by means of multidimensional scaling.