Browsing by Author "Jafari, H."

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Citation - Scopus: 15
A Decomposition Method for Solving Q-Difference Equations
(Natural Sciences Publishing Co., 2015) Jafari, H.; Baleanu, Dumitru; Johnston, S.J.; Sani, S.M.; Baleanu, D.; 56389; Matematik
The q-difference equations are important in q-calculus. In this paper, we apply the iterative method which is suggested by Daftardar and Jafari, hereafter called the Daftardar-Jafari method, for solving a type of q-partial differential equations. We discuss the convergency of this method. In the implementation of this technique according to other iterative methods such as Adomian decomposition and homotopy perturbation methods, one does not need the calculation of the Adomian's polynomials for nonlinear terms. It is proven that under a special constraint, the given result by this method converges to exact solution of nonlinear q-ordinary or q-partial differential equations. © 2015 NSP Natural Sciences Publishing Cor.
A new algorithm for solving dynamic equations on a time scale
(2017) Baleanu, Dumitru; Haghbin, A.; Johnston, S. J.; Baleanu, Dumitru; 56389; Matematik
In this paper, we propose a numerical algorithm to solve a class of dynamic time scale equation which is called the q-difference equation. First, we apply the method for solving initial value problems (IVPs) which contain the first and second order delta derivatives. Illustrative examples show the usefulness of the method. Then we present applications of the method for solving the strongly non-linear damped q-difference equation. The results show that our method is more accurate than the other existing method. (C) 2016 Elsevier B.V. All rights reserved.
Citation - WoS: 123
Citation - Scopus: 147
A New Approach For Solving A System of Fractional Partial Differential Equations
(Pergamon-elsevier Science Ltd, 2013) Jafari, H.; Baleanu, Dumitru; Nazari, M.; Baleanu, D.; Khalique, C. M.; 56389; Matematik
In this paper we propose a new method for solving systems of linear and nonlinear fractional partial differential equations. This method is a combination of the Laplace transform method and the Iterative method and here after called the Iterative Laplace transform method. This method gives solutions without any discretization and restrictive assumptions. The method is free from round-off errors and as a result the numerical computations are reduced. The fractional derivative is described in the Caputo sense. Finally, numerical examples are presented to illustrate the preciseness and effectiveness of the new technique. (C) 2012 Elsevier Ltd. All rights reserved.
Citation - WoS: 123
Citation - Scopus: 143
A New Approach for Solving Multi Variable Orders Differential Equations With Mittag–Leffler Kernel
(Pergamon-elsevier Science Ltd, 2020) Ganji, R. M.; Baleanu, Dumitru; Jafari, H.; Baleanu, D.; 56389; Matematik
In this paper we consider multi variable orders differential equations (MVODEs) with non-local and no-singular kernel. The derivative is described in Atangana and Baleanu sense of variable order. We use the fifth-kind Chebyshev polynomials as basic functions to obtain operational matrices. We transfer the original equations to a system of algebraic equations using operational matrices and collocation method. The convergence analysis of the presented method is discussed. Few examples are presented to show the efficiency of the presented method. (C) 2019 Elsevier Ltd. All rights reserved.
Citation - WoS: 77
Citation - Scopus: 86
Application of A Homogeneous Balance Method To Exact Solutions of Nonlinear Fractional Evolution Equations
(Asme, 2014) Jafari, H.; Baleanu, Dumitru; Tajadodi, H.; Baleanu, D.; 56389; Matematik
The fractional Fan subequation method of the fractional Riccati equation is applied to construct the exact solutions of some nonlinear fractional evolution equations. In this paper, a powerful algorithm is developed for the exact solutions of the modified equal width equation, the Fisher equation, the nonlinear Telegraph equation, and the Cahn-Allen equation of fractional order. Fractional derivatives are described in the sense of the modified Riemann-Liouville derivative. Some relevant examples are investigated.
Citation - WoS: 5
Citation - Scopus: 9
Complex B-Spline Collocation Method for Solving Weakly Singular Volterra Integral Equations of the Second Kind
(Univ Miskolc inst Math, 2015) Ramezani, M.; Jafari, H.; Johnston, S. J.; Baleanu, D.
In this paper we propose a new collocation type method for solving Volterra integral equations of the second kind with weakly singular kernels. In this method we use the complex B-spline basics in collocation method for solving Volterra integral. We compare the results obtained by this method with exact solution. A few numerical examples are presented to demonstrate the effectiveness of the proposed method.
Complex b-spline collocation method for solving weaklysingular volterra integral equations of the second kind
(Univ Miskolc Inst Math, 2015) Baleanu, Dumitru; Ramezani, M.; Johnston, S. J.; Baleanu, Dumitru; 56389; Matematik
In this paper we propose a new collocation type method for solving Volterra integral equations of the second kind with weakly singular kernels. In this method we use the complex B-spline basics in collocation method for solving Volterra integral. We compare the results obtained by this method with exact solution. A few numerical examples are presented to demonstrate the effectiveness of the proposed method.
Citation - WoS: 35
Citation - Scopus: 40
Derivation of a fractional Boussinesq equation for modelling unconfined groundwater
(Springer Heidelberg, 2013) Mehdinejadiani, B.; Baleanu, Dumitru; Jafari, H.; Baleanu, D.; 56389; Matematik
In this manuscript, a fractional Boussinesq equation is obtained by assuming power-law changes of flux in a control volume and using a fractional Taylor series. Furthermore, it was assumed that the average thickness of the watery layer of an aquifer is constant, and the linear fractional Boussinesq equation was derived. Unlike classical Boussinesq equation, due to the non-locality property of fractional derivatives, the parameters of the fractional Boussinesq equation are constant and scale-invariant. In addition, the fractional Boussinesq equation has two various fractional orders of differentiation with respect to x and y that indicate the degree of heterogeneity in the x and y directions, respectively.
Citation - Scopus: 10
Fractional calculus for modeling unconfined groundwater
(De Gruyter, 2019) Jafari, H.; Baleanu, Dumitru; Mehdinejadiani, B.; Baleanu, D.; 56389; Matematik
The porous medium which groundwater flows in is heterogeneous at all scales. This complicates the simulation of groundwater flow. Fractional derivatives, because of their non-locality property, can reduce the scale effects on the parameters and, consequently, better simulate the hydrogeological processes. In this chapter a fractional governing partial differential equation on unconfined groundwater (fractional Boussinesq equation [FBE]) is derived using the fractional mass conservation law. The FBE is a generalization of the Boussinesq equation (BE) that can be used in both homogeneous and heterogeneous unconfined aquifers. Compared to the BE, the FBE includes an additional parameter which represents the heterogeneity degree of the porous medium. This parameter changes within the range of 0 to 1 in the non-linear form of the FBE. The smaller the value of the heterogeneity degree, the more heterogeneous the aquifer is, and vice versa. To investigate the applicability of the FBE to real problems in groundwater flow, a fractional Glover-Dumm equation (FGDE) was obtained using an analytical solution of the linear form of the FBE for onedimensional unsteady flow towards parallel subsurface drains. The FGDE was fitted to water table profiles observed at laboratory and field scales, and its performance was compared to that of the Glover-Dumm equation (GDE). The parameters of the FGDE and the GDE were estimated using the inverse problem method. The results indicate that one can recognize the heterogeneity degree of porous media examined according to the obtained values for the indicator of the heterogeneity degree. The FGDE and the GDE showed similar performances in homogeneous soil, while the performance of the FGDE was significantly better than that of the GDE in heterogeneous soil. In summary, the FBE can be used as a highly general differential equation governing groundwater flow in unconfined aquifers. © 2019 Walter de Gruyter GmbH, Berlin/Boston.
Citation - WoS: 50
Citation - Scopus: 63
Fractional Complex Transform Method For Wave Equations On Cantor Sets Within Local Fractional Differential Operator
(Springer, 2013) Su, Wei-Hua; Baleanu, Dumitru; Yang, Xiao-Jun; Jafari, H.; Baleanu, Dumitru; 56389; Matematik
This paper points out the fractional complex transform method for wave equations on Cantor sets within the local differential fractional operators. The proposed method is efficient to handle differential equations on Cantor sets.
Citation - WoS: 43
Citation - Scopus: 46
Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
(Sciendo, 2016) Johnston, S. J.; Baleanu, Dumitru; Jafari, H.; Moshokoa, S. P.; Ariyan, V. M.; Baleanu, D.; 56389; Matematik
The fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.
Citation - Scopus: 6
Lie Group Theory for Nonlinear Fractional K(m, n) Type Equation with Variable Coefficients
(Springer Science and Business Media Deutschland GmbH, 2022) Jafari, H.; Baleanu, Dumitru; Kadkhoda, N.; Baleanu, D.; 56389; Matematik
We investigated the analytical solution of fractional order K(m, n) type equation with variable coefficient which is an extended type of KdV equations into a genuinely nonlinear dispersion regime. By using the Lie symmetry analysis, we obtain the Lie point symmetries for this type of time-fractional partial differential equations (PDE). Also we present the corresponding reduced fractional differential equations (FDEs) corresponding to the time-fractional K(m, n) type equation. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
Citation - WoS: 62
Citation - Scopus: 73
Numerical Approach of Fokker-Planck Equation With Caputo-Fabrizio Fractional Derivative Using Ritz Approximation
(Elsevier, 2018) Firoozjaee, M. A.; Baleanu, Dumitru; Jafari, H.; Lia, A.; Baleanu, D.; 56389; Matematik
In this manuscript, a type of Fokker-Planck equation (FPE) with Caputo-Fabrizio fractional derivative is considered. We present a numerical approach which is based on the Ritz method with known basis functions to transform this equation into an optimization problem. It leads to a nonlinear algebraic system. Then, we obtain the coefficients of basis functions by solving the algebraic system. The convergence of this technique is discussed extensively. Three examples are included to show the applicability and validity of this method. (C) 2017 Elsevier B.V. All rights reserved.
Citation - WoS: 13
Citation - Scopus: 15
On a numerical approach to solve multi-order fractional differential equations with initial/boundary conditions
(Asme, 2015) Firoozjaee, M. A.; Baleanu, Dumitru; Yousefi, S. A.; Jafari, H.; Baleanu, D.; Matematik
In this manuscript, a new method is introduced for solving multi-order fractional differential equations. By transforming the fractional differential equations into an optimization problem and using polynomial basis functions, we obtain the system of algebraic equation. Then, we solve the system of nonlinear algebraic equation and obtain the coefficients of polynomial expansion. Also, we show the convergence of the method. Some numerical examples are presented which illustrate the theoretical results and the performance of the method.
Citation - WoS: 12
Citation - Scopus: 13
On the exact solutions of nonlinear long-short wave resonance equations
(Editura Acad Romane, 2015) Jafari, H.; Baleanu, Dumitru; Soltani, R.; Khalique, C. M.; Baleanu, D.; 56389; Matematik
The long-short wave resonance model arises when the phase velocity of a long wave matches the group velocity of a short wave. In this paper, the first integral method is used to construct exact solutions of the nonlinear long-short wave resonance equations. One-soliton solutions are also obtained using the travelling wave hypothesis.
Citation - Scopus: 12
Optimal system and symmetry reduction of the (1+1) dimensional Sawada-Kotera equation
(Academic Press, 2016) Kadkhoda, N.; Jafari, H.; Moremedi, G.M.; Baleanu, D.; 56389
We study the nonlinear fifth order (1 + 1) dimensional Sawada-Kotera equation using Lie symmetry group. For this equation Lie point symmetry operators and optimal system are obtained. We determine the corresponding invariant solutions and reduced equations using obtained infinitesimal generators. © 2016 Academic Publications, Ltd.
Citation - WoS: 3
Citation - Scopus: 8
Revised Variational Iteration Method for Solving Systems Of Nonlinear Fractional-Order Differential Equations
(Hindawi Ltd, 2013) Unlu, C.; Baleanu, Dumitru; Jafari, H.; Baleanu, D.; 56389; Matematik
A modification of the variational iteration method (VIM) for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE) obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.
Citation - Scopus: 60
Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis
(2013) Rostamy, D.; Baleanu, Dumitru; Alipour, M.; Jafari, H.; Baleanu, D.; 56389; Matematik
In this paper, we present a numerical method for solving a class of fractional differential equations (FDEs). Based on Bernstein Polynomials (BPs) basis, new matrices are utilized to reduce the multi-term orders fractional differential equation to a system of algebraic equations. Convergence analysis is shown by several theorems. Illustrative examples are included to demonstrate the validity and applicability of this method.
Citation - WoS: 16
Citation - Scopus: 23
Solving partial q-differential equations within reduced q-differential transformation method
(Editura Acad Romane, 2014) Jafari, H.; Baleanu, Dumitru; Haghbin, A.; Hesam, S.; Baleanu, D.; 56389; Matematik
In this paper, the reduced q-differential transform method is presented for solving partial differential equations. In this method, the solution is calculated in the form of convergent power series with easily computable components. Three test problems are discussed to illustrate the effectiveness and performance of the proposed method. The results show that the proposed iteration technique is very effective and convenient.
Citation - WoS: 7
Citation - Scopus: 8
Stability of dirac equation in four-dimensional gravity
(Iop Publishing Ltd, 2017) Safari, F.; Baleanu, Dumitru; Jafari, H.; Sadeghi, J.; Johnston, S. J.; Baleanu, D.; 56389; Matematik
We introduce the Dirac equation in four-dimensional gravity which is a generally covariant form. We choose the suitable variable and solve the corresponding equation. To solve such equation and to obtain the corresponding bispinor, we employ the factorization method which introduces the associated Laguerre polynomial. The associated Laguerre polynomials help us to write the Dirac equation of four-dimensional gravity in the form of the shape invariance equation. Thus we write the shape invariance condition with respect to the secondary quantum number. Finally, we obtain the spinor wave function and achieve the corresponding stability of condition for the four-dimensional gravity system.