Browsing by Author "Golmankhaneh, Alireza K."

Now showing 1 - 20 of 34

Citation - WoS: 11
Citation - Scopus: 15
Heat and Maxwell's Equations on Cantor Cubes
(Editura Acad Romane, 2017) Golmankhaneh, Alireza K.; Baleanu, Dumitru; Baleanu, Dumitru; Matematik
The fractal physics is an important research domain due to its scaling properties that can be seen everywhere in the nature. In this work, the generalized Maxwell's equations are given using fractal differential equations on the Cantor cubes and the electric field for the fractal charge distribution is derived. Moreover, the fractal heat equation is defined, which can be an adequate mathematical model for describing the flowing of the heat energy in fractal media. The suggested models are solved and the plots of the corresponding solutions are presented. A few illustrative examples are given to demonstrate the application of the obtained results in solving diverse physical problems.
Citation - WoS: 23
Citation - Scopus: 29
Diffraction From Fractal Grating Cantor Sets
(Taylor & Francis Ltd, 2016) Baleanu, D.; Golmankhaneh, Alireza K.
In this paper, we have generalized the Fa-calculus by suggesting Fourier and Laplace transformations of the function with support of the fractals set which are the subset of the real line. Using this generalization, we have found the diffraction fringes from the fractal grating Cantor sets.
Citation - WoS: 87
Citation - Scopus: 96
On Electromagnetic Field in Fractional Space
(Pergamon-elsevier Science Ltd, 2010) Golmankhaneh, Alireza K.; Golmankhaneh, Ali K.; Baleanu, Dumitru
Laplacian equation in fractional space describes complex phenomena of physics. With this view, potential of charge distribution in fractional space is derived using Gegenbauer polynomials. Multipoles and magnetic field of charges in fractional space have been obtained. (C) 2008 Elsevier Ltd. All rights reserved.
Calculus on Fractals
(De Gruyter, 2015) Golmankhaneh, Alireza K.; Baleanu, Dumitru
In this chapter we present a framework and a calculus on fractals. The sug-gested equation has been solved and applied in physics and dynamics.
Citation - WoS: 15
Citation - Scopus: 10
Einstein Field Equations Within Local Fractional Calculus
(Editura Acad Romane, 2015) Golmankhaneh, Alireza K.; Baleanu, Dumitru; Yang, Xiao-Jun; Baleanu, D.; Matematik
In this paper, we introduce the local fractional Christoffel index symbols of the first and second kind. The divergence of a local fractional contravariant vector and the curl of local fractional covariant vector are defined. The fractional intrinsic derivative is given. The local fractional Riemann-Christoffel and Ricci tensors are obtained. Finally, the Einstein tensor and Einstein field are generalized by involving the fractional derivatives. Illustrative examples are presented.
Calculus on fractals
(2015) Golmankhaneh, Alireza K.; Baleanu, Dumitru
In this chapter we present a framework and a calculus on fractals. The suggested equation has been solved and applied in physics and dynamics.
Citation - WoS: 16
Citation - Scopus: 21
Using Anns Approach for Solving Fractional Order Volterra Integro-Differential Equations
(Springernature, 2017) Rostami, Fariba; Golmankhaneh, Alireza K.; Baleanu, Dumitru; Jafarian, Ahmad
Indeed, interesting properties of artificial neural networks approach made this non-parametric model a powerful tool in solving various complicated mathematical problems. The current research attempts to produce an approximate polynomial solution for special type of fractional order Volterra integrodifferential equations. The present technique combines the neural networks approach with the power series method to introduce an efficient iterative technique. To do this, a multi-layer feed-forward neural architecture is depicted for constructing a power series of arbitrary degree. Combining the initial conditions with the resulted series gives us a suitable trial solution. Substituting this solution instead of the unknown function and employing the least mean square rule, converts the origin problem to an approximated unconstrained optimization problem. Subsequently, the resulting nonlinear minimization problem is solved iteratively using the neural networks approach. For this aim, a suitable three-layer feed-forward neural architecture is formed and trained using a back-propagation supervised learning algorithm which is based on the gradient descent rule. In other words, discretizing the differential domain with a classical rule produces some training rules. By importing these to designed architecture as input signals, the indicated learning algorithm can minimize the defined criterion function to achieve the solution series coefficients. Ultimately, the analysis is accompanied by two numerical examples to illustrate the ability of the method. Also, some comparisons are made between the present iterative approach and another traditional technique. The obtained results reveal that our method is very effective, and in these examples leads to the better approximations.
Citation - WoS: 124
Citation - Scopus: 136
On Nonlinear Fractional Klein-Gordon Equation
(Elsevier, 2011) Golmankhaneh, Ali K.; Baleanu, Dumitru; Golmankhaneh, Alireza K.
Numerical methods are used to find exact solution for the nonlinear differential equations. In the last decades Iterative methods have been used for solving fractional differential equations. In this paper, the Homotopy perturbation method has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein-Cordon equation can be used as numerical algorithm. The behavior of solutions and the effects of different values of fractional order a are shown graphically. Some examples are given to show ability of the method for solving the fractional nonlinear equation. Crown Copyright (C) 2010 Published by Elsevier B.V. All rights reserved.
Citation - WoS: 52
Citation - Scopus: 54
Non-Local Integrals and Derivatives on Fractal Sets With Applications
(de Gruyter Open Ltd, 2016) Baleanu, D.; Golmankhaneh, Alireza K.
In this paper, we discuss non-local derivatives on fractal Cantor sets. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared. Related physical models are also suggested.
Citation - WoS: 20
Citation - Scopus: 24
The Proposed Modified Liu System With Fractional Order
(Hindawi Ltd, 2013) Arefi, Roohiyeh; Baleanu, Dumitru; Golmankhaneh, Alireza K.
The chaos in a new system with order 3 is studied. We have shown that this chaotic system again will be chaotic when the order of system is less than 3. Generalized Adams-Bashforth algorithm has been used for investigating in stability of fixed points and existence of chaos.
Citation - WoS: 20
Citation - Scopus: 22
Analytical Treatment of System of Abel Integral Equations by Homotopy Analysis Method
(Editura Acad Romane, 2014) Jafarian, A.; Baleanu, Dumitru; Ghaderi, P.; Golmankhaneh, Alireza K.; Baleanu, D.; Matematik
Abel equation has important applications in describing the least time for an object which is sliding on surface without friction in uniform gravity, and the classical theory of elasticity of materials is modeled by a system of Abel integral equations. In this manuscript, the homotopy analysis method is presented for obtaining analytical solutions of a system of Abel integral equations as fractional equations. The applied method has lessened the size of calculation and improved the accuracy of solution in the case of the singular Abel integral equation. The illustrated examples and numerical results have proved the assertion.
Citation - WoS: 5
Citation - Scopus: 6
Analytic Solution for a Nonlinear Problem of Magneto-Thermoelasticity
(Pergamon-elsevier Science Ltd, 2013) Ghaderi, P.; Golmankhaneh, Alireza K.; Baleanu, D.; Jafarian, A.
In this paper, we present a comparative study of the homotopy analysis method (HAM), the variational iteration method (VIM) and the iterative method (He's polynomials). The approximate solution of the coupled harmonic waves nonlinear magneto-thermoelasticity equations under influence of rotation is obtained. In order to control and adjust the convergence region and the rate of solution series, we show that it is possible to choose a valid auxiliary parameter h of HAM. Using the boundary and the initial conditions we select a suitable initial approximation. The results show that these methods are very efficient, convenient and applicable to a large class of problems.
Citation - WoS: 13
Citation - Scopus: 24
On a New Measure on Fractals
(Springer, 2013) Baleanu, Dumitru; Golmankhaneh, Alireza K.
Fractals are sets whose Hausdorff dimension strictly exceeds their topological dimension. The algorithmic Riemannian-like method, F-alpha-calculus, has been suggested very recently. Henstock-Kurzweil integral is the generalized Riemann integral method by using the gauge function. In this paper we generalize the F-alpha-calculus as a fractional local calculus that is more suitable to describe some physical process. We introduce the new measure using the gauge function on fractal sets that gives a finer dimension in comparison with the Hausdorff and box dimension. Hilbert F-alpha-spaces are defined. We suggest the self-adjoint F-alpha-differential operator so that it can be applied in the fractal quantum mechanics and on the fractal curves.
Citation - WoS: 48
Citation - Scopus: 52
Mean Square Solutions of Second-Order Random Differential Equations by Using Homotopy Analysis Method
(Editura Acad Romane, 2013) Golmankhaneh, Alireza K.; Baleanu, Dumitru; Porghoveh, Neda A.; Baleanu, D.; Matematik
In this paper, the Homotopy Analysis Method (HAM) is successfully applied for solving second-order random differential equations, homogeneous or inhomogeneous. Expectation and variance of the approximate solutions are computed. Several numerical examples are presented to show the ability and efficiency of this method.
Citation - WoS: 46
Synchronization in a Nonidentical Fractional Order of a Proposed Modified System
(Sage Publications Ltd, 2015) Arefi, Rouhiyeh; Baleanu, Dumitru; Golmankhaneh, Alireza K.
In this paper, the fractional order of a new chaotic system was studied and the minimum effective dimension for which the system remains chaotic was calculated. We have presented the chaos synchronization of two identical and nonidentical fractional orders of the new system by using active control. Furthermore, the Laplace transform and the Niemann-Trouvaille fractional integral operator were used for synchronizing two system.
Citation - WoS: 65
Citation - Scopus: 72
Newtonian Law With Memory
(Springer, 2010) Golmankhaneh, Alireza K.; Golmankhaneh, Ali K.; Nigmatullin, Raoul R.; Baleanu, Dumitru
In this study we analyzed the Newtonian equation with memory. One physical model possessing memory effect is analyzed in detail. The fractional generalization of this model is investigated and the exact solutions within Caputo and Riemann-Liouville fractional derivatives are reported.
Citation - WoS: 9
Citation - Scopus: 10
On Bernstein Polynomials Method To the System of Abel Integral Equations
(Hindawi Ltd, 2014) Nia, S. Measoomy; Golmankhaneh, Alireza K.; Baleanu, D.; Jafarian, A.
This paper deals with a new implementation of the Bernstein polynomials method to the numerical solution of a special kind of singular system. For this aim, first the truncated Bernstein series polynomials of the solution functions are substituted in the given problem. Using some properties of these polynomials, the solution of the problem is reduced to solve a linear system of algebraic equations. In order to confirm the reliability and accuracy of the proposed method, some weakly Abel integral equations systems with comparisons are solved in detail as numerical examples.
Citation - WoS: 20
Homotopy Analysis Method for Solving Coupled Ramani Equations
(Editura Acad Romane, 2014) Jafarian, A.; Baleanu, Dumitru; Ghaderi, P.; Golmankhaneh, Alireza K.; Baleanu, D.; Matematik
In this manuscript, coupled Ramani equations are solved by means of an analytic technique, namely the homotopy analysis method (HAM). The HAM is a capable and a straightforward analytic tool for solving nonlinear problems and does not need small parameters in the governing equations and boundary/initial conditions. The result of this study presents the utility and sufficiency of HAM method. Comparisons demonstrate that there is a good agreement between the HAM solutions and the exact solutions.
Citation - WoS: 23
Citation - Scopus: 26
Numerical Solution of Linear Integral Equations System Using the Bernstein Collocation Method
(Springer international Publishing Ag, 2013) Nia, Safa A. Measoomy; Golmankhaneh, Alireza K.; Baleanu, Dumitru; Jafarian, Ahmad
Since in some application mathematical problems finding the analytical solution is too complicated, in recent years a lot of attention has been devoted by researchers to find the numerical solution of this equations. In this paper, an application of the Bernstein polynomials expansion method is applied to solve linear second kind Fredholm and Volterra integral equations systems. This work reduces the integral equations system to a linear system in generalized case such that the solution of the resulting system yields the unknown Bernstein coefficients of the solutions. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.
Citation - WoS: 3
Citation - Scopus: 7
A Numerical Solution of the Urysohn-Type Fredholm Integral Equations
(Editura Acad Romane, 2014) Jafarian, A.; Baleanu, Dumitru; Measoomy, S. A.; Golmankhaneh, Alireza K.; Baleanu, D.; Matematik
In the present paper, a combination of the Bernstein polynomials and artificial neural networks (ANNs) is presented for solving the non-linear Urysohn equation. These polynomials are utilized to reduce the solution of the given problem to the solution of a system of non-linear algebraic equations. The remaining set of non-linear equations is solved numerically by using the ANNs approach to yield truncated Bernstein series coefficients a the solution function. Several illustrative examples with numerical simulations are provided to support the theoretical claims.