Browsing by Author "Alipour, Mohsen"

Now showing 1 - 11 of 11

Citation - WoS: 13
Citation - Scopus: 16
Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations By Bps Operational Matrices
(Hindawi Ltd, 2013) Alipour, Mohsen; Baleanu, Dumitru; Baleanu, Dumitru; 56389; Matematik
We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1.
Citation - WoS: 7
Citation - Scopus: 14
Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis
(Hindawi Ltd, 2014) Alipour, Mohsen; Baleanu, Dumitru; Baleanu, Dumitru; Babaei, Fereshteh; 56389; Matematik
We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.
Numerical and Bifurcations Analysis for Multi Order Fractional Model of Hiv Infection of Cd4(+)T-Cells
(Univ Politehnica Bucharest, 2016) Baleanu, Dumitru; Arshad, Sadia; Baleanu, Dumitru; 56389; Matematik
In this paper, we solve the dynamical system of HIV infection of CD4(+) T cells within the multi-order fractional derivatives. The Bernstein operational matrices in arbitrary interval [a,b] are applied to obtain the approximate analytical solution of the model. In this way, the fractional differential equations are reduced to an algebraic easily solvable system. The obtained solutions are accurate and the method is very efficient and simple in implementation. With the help of bifurcation analysis, we acquired the critical value of viral death rate, that is, if viral death rate is greater than the critical value then level of virus particles starts to decline and thus free virus will eventually eliminate and patient is cured. Further, we found the threshold for viral infection rate analytically, which assures the stability of uninfected equilibrium and virus will ultimately eradicate.
Citation - WoS: 11
Citation - Scopus: 13
Numerical and Bifurcations Analysis for Multi Order Fractional Model of Hiv Infection of Cd4⁺t-cells
(Univ Politehnica Bucharest, Sci Bull, 2016) Baleanu, Dumitru; Arshad, Sadia; Baleanu, Dumitru; Matematik
In this paper, we solve the dynamical system of HIV infection of CD4(+) T cells within the multi-order fractional derivatives. The Bernstein operational matrices in arbitrary interval [a,b] are applied to obtain the approximate analytical solution of the model. In this way, the fractional differential equations are reduced to an algebraic easily solvable system. The obtained solutions are accurate and the method is very efficient and simple in implementation. With the help of bifurcation analysis, we acquired the critical value of viral death rate, that is, if viral death rate is greater than the critical value then level of virus particles starts to decline and thus free virus will eventually eliminate and patient is cured. Further, we found the threshold for viral infection rate analytically, which assures the stability of uninfected equilibrium and virus will ultimately eradicate.
Citation - WoS: 11
Numerical and bifurcations analysis for multi-order fractional model of HIV infection of CD4+T-cells
(Univ Politehnica Bucharest, Sci Bull, 2016) Alipour, Mohsen; Baleanu, Dumitru; Arshad, Sadia; Baleanu, Dumitru; 56389; Matematik
In this paper, we solve the dynamical system of HIV infection of CD4(+) T cells within the multi-order fractional derivatives. The Bernstein operational matrices in arbitrary interval [a,b] are applied to obtain the approximate analytical solution of the model. In this way, the fractional differential equations are reduced to an algebraic easily solvable system. The obtained solutions are accurate and the method is very efficient and simple in implementation. With the help of bifurcation analysis, we acquired the critical value of viral death rate, that is, if viral death rate is greater than the critical value then level of virus particles starts to decline and thus free virus will eventually eliminate and patient is cured. Further, we found the threshold for viral infection rate analytically, which assures the stability of uninfected equilibrium and virus will ultimately eradicate.
Citation - WoS: 55
Citation - Scopus: 73
On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions
(Springer, 2015) Baleanu, Dumitru; Baleanu, Dumitru; Khan, Hasib; Jafari, Hossein; Khan, Rahmat Ali; Alipour, Mohsen; Matematik
We investigate sufficient conditions for existence and uniqueness of solutions for a coupled system of fractional order hybrid differential equations (HDEs) with multi-point hybrid boundary conditions given by D-omega(x(t)/H(t, x(t), z(t))) = -K-1 (t, x(t), z(t)), omega epsilon (2, 3], D-epsilon(z(t)/G(t, x(t), z(t))) = -K-2 (t, x(t), z(t)), epsilon epsilon(2, 3] x(t)/H(t, x(t), z(t))vertical bar(t=1) = 0, D-mu(x(t)/H(t, x(t), z(t)))vertical bar(t=delta 1) =0, x((2))(0) = 0 z(t)/G(t, x(t), z(t))vertical bar(t=1) = 0, D-nu(z(t)/G(t, x(t), z(t)))vertical bar(t=delta 2) =0, z((2))(0) = 0 where t epsilon [0, 1], delta(1), delta(2), mu, upsilon epsilon (0, 1), and D-omega, D-epsilon, D-mu and D-upsilon are Caputo's fractional derivatives of order omega, is an element of, mu and nu, respectively, K-1, K-2 epsilon C([0, 1] x R x R, R) and G, H epsilon C([0, 1] x R x R, R - {0}). We use classical results due to Dhage and Banach's contraction principle (BCP) for the existence and uniqueness of solutions. For applications of our results, we include examples.
Citation - WoS: 1
Citation - Scopus: 1
On the Kolmogorov forward equations within Caputo and Riemann-Liouville fractions derivatives
(Ovidius Univ Press, 2016) Alipour, Mohsen; Baleanu, Dumitru; Baleanu, Dumitru; Matematik
In this work, we focus on the fractional versions of the well-known Kolmogorov forward equations. We consider the problem in two cases. In case 1, we apply the left Caputo fractional derivatives for alpha is an element of (0, 1] and in case 2, we use the right Riemann-Liouville fractional derivatives on R+, for alpha is an element of (1, + infinity). The exact solutions are obtained for the both cases by Laplace transforms and stable subordinators.
Citation - WoS: 7
Citation - Scopus: 7
On the motion of a heavy bead sliding on a rotating wire - Fractional treatment
(Elsevier, 2018) Baleanu, Dumitru; Baleanu, Dumitru; Asad, Jihad H.; Alipour, Mohsen; 56389; Matematik
In this work, we consider the motion of a heavy particle sliding on a rotating wire. The first step carried for this model is writing the classical and fractional Lagrangian. Secondly, the fractional Hamilton's equations (FHEs) of motion of the system is derived. The fractional equations are formulated in the sense of Caputo. Thirdly, numerical simulations of the FHEs within the fractional operators are presented and discussed for some fractional derivative orders. Numerical results are based on a discretization scheme using the Euler convolution quadrature rule for the discretization of the convolution integral. Finally, simulation results verify that, taking into account the fractional calculus provides more flexible models demonstrating new aspects of the real world phenomena.
Citation - WoS: 77
Citation - Scopus: 87
Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices
(Sage Publications Ltd, 2013) Alipour, Mohsen; Baleanu, Dumitru; Rostamy, Davood; Baleanu, Dumitru; 56389; Matematik
In this paper, we present a method for solving multi-dimensional fractional optimal control problems. Firstly, we derive the Bernstein polynomials operational matrix for the fractional derivative in the Caputo sense, which has not been done before. The main characteristic behind the approach using this technique is that it reduces the problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The results obtained are in good agreement with the existing ones in the open literature and it is shown that the solutions converge as the number of approximating terms increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach 1.
Citation - WoS: 4
Citation - Scopus: 7
SPECTRAL METHOD BASED ON BERNSTEIN POLYNOMIALS FOR COUPLED SYSTEM OF FREDHOLM INTEGRAL EQUATIONS
(Ministry Communications & High Technologies Republic Azerbaijan, 2016) Alipour, Mohsen; Baleanu, Dumitru; Baleanu, Dumitru; Karimi, Kobra; 56389; Matematik
In this paper, we apply Bernstein basis to solve the coupled system of Fredholm integral equations (CSFIE). This method transforms the problem to a system of linear algebraic equations that easily solvable. On the other hand, convergence analysis of this method is discussed. the examples show that the proposed method is implemented very simple and the results have high accuracy.
Citation - WoS: 12
Citation - Scopus: 19
The Bernstein Operational Matrices for Solving the Fractional Quadratic Riccati Differential Equations With The Riemann-Liouville Derivative
(Hindawi Ltd, 2013) Baleanu, Dumitru; Baleanu, Dumitru; Alipour, Mohsen; Jafari, Hossein; 56389; Matematik
We obtain the approximate analytical solution for the fractional quadratic Riccati differential equation with the Riemann-Liouville derivative by using the Bernstein polynomials (BPs) operational matrices. In this method, we use the operational matrix for fractional integration in the Riemann-Liouville sense. Then by using this matrix and operational matrix of product, we reduce the problem to a system of algebraic equations that can be solved easily. The efficiency and accuracy of the proposed method are illustrated by several examples.