Browsing by Author "Jafari, Hossein"

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Citation - WoS: 8
Citation - Scopus: 13
Analysis of Riccati Differential Equations Within a New Fractional Derivative Without Singular Kernel
(Ios Press, 2017) Lia, Atena; Tejadodi, Haleh; Baleanu, Dumitru; Jafari, Hossein
Recently Caputo and Fabrizio suggested new definition of fractional derivative that the new kernel has no singularity. In this paper, an analytical method for solving Riccati differential equation with a new fractional derivative is reported. We present numerical results of solving the fractional Riccati differential equations by using the variational iteration method and its modification. The obtained results of two methods demonstrate the efficiency and simplicity of the MVIM that gives good approximations for a larger interval.
Citation - WoS: 14
Citation - Scopus: 19
The Bernstein Operational Matrices for Solving the Fractional Quadratic Riccati Differential Equations With the Riemann-Liouville Derivative
(Hindawi Ltd, 2013) Alipour, Mohsen; Jafari, Hossein; Baleanu, Dumitru
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.
Citation - WoS: 52
Citation - Scopus: 58
Damped Wave Equation and Dissipative Wave Equation in Fractal Strings Within the Local Fractional Variational Iteration Method
(Springer international Publishing Ag, 2013) Baleanu, Dumitru; Yang, Xiao-Jun; Jafari, Hossein; Su, Wei-Hua
In this paper, the local fractional variational iteration method is given to handle the damped wave equation and dissipative wave equation in fractal strings. The approximation solutions show that the methodology of local fractional variational iteration method is an efficient and simple tool for solving mathematical problems arising in fractal wave motions. MSC: 74H10, 35L05, 28A80.
Citation - WoS: 33
Citation - Scopus: 34
Exact Solutions of Boussinesq and Kdv-Mkdv Equations by Fractional Sub-Equation Method
(Editura Acad Romane, 2013) Jafari, Hossein; Baleanu, Dumitru; Tajadodi, Haleh; Baleanu, Dumitru; Al-Zahrani, Abdulrahim A.; Alhamed, Yahia A.; Zahid, Adnan H.; Matematik
A fractional sub-equation method is introduced to solve fractional differential equations. By the aid of the solutions of the fractional Riccati equation, we construct solutions of the Boussinesq and KdV-mKdV equations of fractional order. The obtained results show that this method is very efficient and easy to apply for solving fractional partial differential equations.
Citation - WoS: 10
Exact Solutions of Two Nonlinear Partial Differential Equations by Using the First Integral Method
(Springer, 2013) Soltani, Rahmat; Khalique, Chaudry Masood; Baleanu, Dumitru; Jafari, Hossein
In recent years, many approaches have been utilized for finding the exact solutions of nonlinear partial differential equations. One such method is known as the first integral method and was proposed by Feng. In this paper, we utilize this method and obtain exact solutions of two nonlinear partial differential equations, namely double sine-Gordon and Burgers equations. It is found that the method by Feng is a very efficient method which can be used to obtain exact solutions of a large number of nonlinear partial differential equations.
Citation - WoS: 45
Citation - Scopus: 53
Existence Criterion for the Solutions of Fractional Order P-Laplacian Boundary Value Problems
(Springer, 2015) Baleanu, Dumitru; Khan, Hasib; Khan, Rahmat Ali; Khan, Aziz; Jafari, Hossein
The existence criterion has been extensively studied for different classes in fractional differential equations (FDEs) through different mathematical methods. The class of fractional order boundary value problems (FOBVPs) with p-Laplacian operator is one of the most popular class of the FDEs which have been recently considered by many scientists as regards the existence and uniqueness. In this scientific work our focus is on the existence and uniqueness of the FOBVP with p-Laplacian operator of the form: D-gamma(phi(p)(D-theta z(t))) + a(t)f(z(t)) = 0, 3 < theta, gamma <= 4, t is an element of [0, 1], z(0) = z'''(0), eta D(alpha)z(t)vertical bar(t=1) = z'(0), xi z ''(1) - z ''(0) = 0, 0 < alpha < 1, phi(p)(D-theta z(t))vertical bar(t=0) = 0 = (phi(p)(D-theta z(t)))'vertical bar(t=0), (phi(p)(D-theta z(t)))''vertical bar(t=1) = 1/2(phi(p)(D-theta z(t)))''vertical bar(t=0), (phi(p)(D-theta z(t)))'''vertical bar(t=0) = 0, where 0 < xi, eta < 1 and D-theta, D-gamma, D-alpha are Caputo's fractional derivatives of orders theta, gamma, alpha, respectively. For this purpose, we apply Schauder's fixed point theorem and the results are checked by illustrative examples.
Citation - WoS: 2
Citation - Scopus: 1
A Fixed Point Theorem on Multiplicative Metric Space With Integral-Type Inequality
(Journal Mathematics & Computer Science-jmcs, 2018) Khan, Hasib; Baleanu, Dumitru; Jafari, Hossein; Khan, Tahir Saeed; Alqurashi, Maysaa; Khan, Aziz
In this paper, we prove fixed point theorems (FPTs) on multiplicative metric space (MMS) (X, triangle) by the help of integral-type contractions of self-quadruple mappings (SQMs), i.e., for p(1), p(2), p(3), p(4) : X -> R. For this, we assume that the SQMs are weakly compatible mappings and the pairs (p(1), p(3)) and (p(2), p(4)) satisfy the property (CLRp3p4). Further, two corollaries are produced from our main theorem as special cases. The novelty of these results is that for the unique common fixed point (CFP) of the SQMs p(1), p(2), p(3), p(4), we do not need to the assumption of completeness of the MMS (X, triangle). These results generalize the work of Abdou, [A. A. N. Abdou, J. Nonlinear Sci. Appl., 9 (2016), 2244-2257], and many others in the available literature.
Citation - WoS: 67
Citation - Scopus: 83
Fractional Lie Group Method of the Time-Fractional Boussinesq Equation
(Springer, 2015) Kadkhoda, Nematollah; Baleanu, Dumitru; Jafari, Hossein
Finding the symmetries of the nonlinear fractional differential equations is a topic which has many applications in various fields of science and engineering. In this manuscript, firstly, we are interested in finding the Lie point symmetries of the time-fractional Boussinesq equation. After that, by using the infinitesimal generators, we determine their corresponding invariant solutions.
Citation - WoS: 44
Citation - Scopus: 50
Fractional Sub-Equation Method for the Fractional Generalized Reaction Duffing Model and Nonlinear Fractional Sharma-Tasso Equation
(de Gruyter Poland Sp Z O O, 2013) Tajadodi, Haleh; Baleanu, Dumitru; Al-Zahrani, Abdulrahim A.; Alhamed, Yahia A.; Zahid, Adnan H.; Jafari, Hossein
In this paper the fractional sub-equation method is used to construct exact solutions of the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation.The fractional derivative is described in the Jumarie's modified Riemann-Liouville sense. Two illustrative examples are given, showing the accuracy and convenience of the method.
Citation - WoS: 49
Citation - Scopus: 59
Fractional Subequation Method for Cahn-Hilliard and Klein-Gordon Equations
(Hindawi Ltd, 2013) Tajadodi, Haleh; Kadkhoda, Nematollah; Baleanu, Dumitru; Jafari, Hossein
The fractional subequation method is applied to solve Cahn-Hilliard and Klein-Gordon equations of fractional order. The accuracy and efficiency of the scheme are discussed for these illustrative examples.
Citation - WoS: 20
Citation - Scopus: 25
Homotopy Analysis Method for Solving Abel Differential Equation of Fractional Order
(de Gruyter Poland Sp Z O O, 2013) Sayevand, Khosro; Tajadodi, Haleh; Baleanu, Dumitru; Jafari, Hossein
In this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.
Citation - WoS: 29
Citation - Scopus: 33
A Mathematical Model for Simulation of a Water Table Profile Between Two Parallel Subsurface Drains Using Fractional Derivatives
(Pergamon-elsevier Science Ltd, 2013) Naseri, Abd Ali; Jafari, Hossein; Ghanbarzadeh, Afshin; Baleanu, Dumitru; Mehdinejadiani, Behrouz
By considering the initial and boundary conditions corresponding to parallel subsurface drains, the linear form of a one-dimensional fractional Boussinesq equation was solved and an analytical mathematical model was developed to predict the water table profile between two parallel subsurface drains. The developed model is a generalization of the Glover-Dumm's mathematical model. As a result, the new model is applicable for both homogeneous and heterogeneous soils. It considers the degree of heterogeneity of soil as a determinable parameter. This parameter was called the heterogeneity index. The laboratory and field tests were conducted to study the performance of the proposed mathematical model in a homogenous soil and in an agricultural soil. The optimal values of parameters of the fractional model developed in this study and Glover-Dumm's model were estimated using the inverse problem method. In the proposed inverse model, a bees algorithm (BA) was used. The results showed that in the homogenous soil, the heterogeneity index was nearly equal to 2 and therefore, the developed mathematical model reduced to the Glover-Dumm's mathematical model. The heterogeneity index of the experimental field soil considered was equal to 1.04; hence, this soil was classified as a very heterogeneous soil. In the experimental field soil, the proposed mathematical model better represented the water table profile between two parallel subsurface drains than the Glover-Dumm's mathematical model. Therefore, it appears that the proposed fractional model presented is a highly general and effective method to estimate the water table profile between two parallel subsurface drains, and the scale effects are robustly reflected by the introduced heterogeneity index. (C) 2013 Elsevier Ltd. All rights reserved.
Citation - WoS: 53
Citation - Scopus: 65
A Modified Variational Iteration Method for Solving Fractional Riccati Differential Equation by Adomian Polynomials
(Walter de Gruyter Gmbh, 2013) Tajadodi, Hale; Baleanu, Dumitru; Jafari, Hossein
In this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.
Citation - WoS: 9
Citation - Scopus: 10
More Efficient Estimates Via H-Discrete Fractional Calculus Theory and Applications
(Pergamon-elsevier Science Ltd, 2021) Sultana, Sobia; Jarad, Fahd; Jafari, Hossein; Hamed, Y. S.; Rashid, Saima
Discrete fractional calculus (DFC) is continuously spreading in the engineering practice, neural networks, chaotic maps, and image encryption, which is appropriately assumed for discrete-time modelling in continuum problems. First, we start with a novel discrete h-proportional fractional sum defined on the time scale hZ so as to give the premise to the more broad and complex structures, for example, the suitably accustomed transformations conjuring the property of observing the new chaotic behaviors of the logistic map. Here, we aim to present the novel discrete versions of Gruss and certain other associated variants by employing discrete h-proportional fractional sums are established. Moreover, several novel consequences are recaptured by the h-discrete fractional sums. The present study deals with the modification of Young, weighted-arithmetic and geometric mean formula by taking into account changes in the exponential function in the kernel represented by the parameters of the operator, varying delivery noted outcomes. In addition, two illustrative examples are apprehended to demonstrate the applicability and efficiency of the proposed technique. (C) 2021 Elsevier Ltd. All rights reserved.
Citation - WoS: 24
Citation - Scopus: 33
A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix
(Walter de Gruyter Gmbh, 2015) Tajadodi, Haleh; Baleanu, Dumitru; Jafari, Hossein
In this article, we develop an effective numerical method to achieve the numerical solutions of nonlinear fractional Riccati differential equations. We found the operational matrix within the linear B-spline functions. By this technique, the given problem converts to a system of algebraic equations. This technique is used to solve fractional Riccati differential equation. The obtained results are illustrated both applicability and validity of the suggested approach.
Citation - WoS: 41
Citation - Scopus: 43
A Numerical Approach for Solving Fractional Optimal Control Problems With Mittag-Leffler Kernel
(Sage Publications Ltd, 2022) Ganji, Roghayeh M.; Sayevand, Khosro; Baleanu, Dumitru; Jafari, Hossein
In this work, we present a numerical approach based on the shifted Legendre polynomials for solving a class of fractional optimal control problems. The derivative is described in the Atangana-Baleanu derivative sense. To solve the problem, operational matrices of AB-fractional integration and multiplication, together with the Lagrange multiplier method for the constrained extremum, are considered. The method reduces the main problem to a system of nonlinear algebraic equations. In this framework by solving the obtained system, the approximate solution is calculated. An error estimate of the numerical solution is also proved for the approximate solution obtained by the proposed method. Finally, some illustrative examples are presented to demonstrate the accuracy and validity of the proposed scheme.
Citation - WoS: 11
Citation - Scopus: 14
Numerical Investigation of Space Fractional Order Diffusion Equation by the Chebyshev Collocation Method of the Fourth Kind and Compact Finite Difference Scheme
(Amer inst Mathematical Sciences-aims, 2021) Safdari, Hamid; Azari, Yaqub; Jafari, Hossein; Baleanu, Dumitru; Aghdam, Yones Esmaeelzade
This paper develops a numerical scheme for finding the approximate solution of space fractional order of the diffusion equation (SFODE). Firstly, the compact finite difference (CFD) with convergence order O(delta tau 2) is used for discretizing time derivative. Afterwards, the spatial fractional derivative is approximated by the Chebyshev collocation method of the fourth kind. Furthermore, time-discrete stability and convergence analysis are presented. Finally, two examples are numerically investigated by the proposed method. The examples illustrate the performance and accuracy of our method compared to existing methods presented in the literature. 1. Introduction. One of the issues which have garnered researchers' attention these days is the fractional differential equations (FDEs) and have been numerically investigated by a huge number of authors [2, 3, 8, 9, 16, 21, 23, 25, 28, 29]. Fractional calculus is involved in many applications of science and engineering such as economics, physics, optimal control, and other applications, see [10, 11, 13, 19, 22, 26, 33, 34, 35]. A case in point is the diffusion and reaction-diffusion models in
On a Numerical Solution for Fractional Differential Equation Within B-Spline Operational Matrix
(Ieee, 2014) Tajadodi, Haleh; Baleanu, Dumitru; Jafari, Hossein
In our manuscript we suggest an approach to obtain the solutions of the fractional differential equations(FDEs). We found the operational matrix within the linear B-spline functions. In this way the investigated equations are turned into a set of algebraic equations. We provide examples to illustrate both accuracy and simplicity of the suggested approach.
Citation - WoS: 13
Citation - Scopus: 15
On Comparison Between Iterative Methods for Solving Nonlinear Optimal Control Problems
(Sage Publications Ltd, 2016) Ghasempour, Saber; Baleanu, Dumitru; Jafari, Hossein
Recently some semi-analytical methods have been introduced for solving a class of nonlinear optimal control problems such as the Adomian decomposition method, homotopy perturbation method and modified variational iteration method. In this manuscript we compare these methods for solving a type of nonlinear optimal control problem. We prove that these methods are equivalent, which means that they use the same iterative formula to obtain the approximate/analytical solution.
Citation - WoS: 60
Citation - Scopus: 78
On Existence Results for Solutions of a Coupled System of Hybrid Boundary Value Problems With Hybrid Conditions
(Springer, 2015) Khan, Hasib; Jafari, Hossein; Khan, Rahmat Ali; Alipour, Mohsen; Baleanu, Dumitru
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.