فهرست مطالب

• Volume:8 Issue: 1, Winter 2020
• تاریخ انتشار: 1398/10/11
• تعداد عناوین: 15
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• Samaneh Akbarpour, Abdollah Shidfar, Hashem Saberinajafi * Pages 1-13
This paper investigates the inverse problem of determining the time-dependent heat source and the temperature for the heat equation with Dirichlet boundary conditions and an integral over determination conditions. The numerical method is presented for solving the Inverse problem. Shifted Chebyshev polynomial is used to approximate the solution of the equation as a base of the tau method which is based on the Chebyshev operational matrices. The main advantage of this method is based upon reducing the partial differential equation into a system of algebraic equations of the solution. Numerical results are presented and discussed.
Keywords: Inverse source problem, Heat equation, Shifted Chebyshev polynomial, Operational matrix, Shifted Chebyshev-Tau method
• Özkan Ökalan *, Nurten Kılıç, Umut Özkan, Sermin Öztürk Pages 14-27
This paper is devoted to obtaining some new sufficient conditions for the oscillation of all solutions of first order nonlinear differential equations with several deviating arguments. Finally, an illustrative example related to our results is given.
Keywords: Nonlinear, Delay differential equation, Non-monotone argument, Oscillatory solutions, Nonoscillatory solutions

In this study, the new extended direct algebraic method is exerted for constructing more general exact solutions of the three nonlinear evolution equations with physical interest namely, the Tzitzeica equation, the Dodd-Bullough-Mikhailor equation and the Liouville equation. By using of an appropriate traveling wave transformation reduces these equations to ODE. We state that this method is excellently a generalized form to obtain solitary wave solutions of the nonlinear evolution equations that are widely used in theoretical physics. The method appears to be easier and faster by means of symbolic computation system.

Keywords: Nonlinear evolution equation, Tzitzica type evolution equations, New extended direct algebraic method, traveling wave solutions
• Mousa Ilie, Jafar Biazar *, Zainab Ayati Pages 54-68
This paper deals with the solution of a class of Volterra integral equations in the sense of the conformable fractional derivative. For this goal, the well-organized Neumann method is developed and some theorems related to existence, uniqueness, and sufficient condition of convergence are presented. Some illustrative examples are provided to demonstrate the efficiency of the method in solving conformable fractional Volterra integral equations.
Keywords: Volterra integral equations, Conformable fractional derivative, Neumann method, Existence, uniqueness, sufficient condition of convergence
• Razieh Ghorbanpoor, Jafar Saberi Nadjafi *, Nik Mohd Assri Nik Long, Majid Erfanian Pages 69-84

In this paper, numerical solutions of multiple cracks problems in an infinite plate are studied. Hypersingular integral equations (hieq) for the cracks are formulated using the complex potential method. For all kernels such as regular or hypersingular kernels, we are using the appropriate quadrature formulas to solve and evaluate the unknown functions numerically. Furthermore, by using this equation the stress intensity factor (SIF) was calculated for crack tips. For two serial cracks (horizontal) and two dissimilar cracks (horizontal and inclined), our numerical results agree with the previous works.

Keywords: ‎Hypersingular integral equation‎, ‎Multiple cracks‎ ‎problem‎, ‎Inclined cracks‎, ‎Double stress intensity factors‎
• Saeede Rashidi *, SeyedReza Hejazi Pages 85-98

The present paper considers the group analysis of extended (1 + 1)-dimensional Buckmaster equation and its conservation laws. Symmetry operators of Buckmaster equation are found via Lie algorithm of differential equations. The method of non-linear self-adjointness is applied to the considered equation. The infinite set of conservation laws associated with the finite algebra of Lie point symmetries of the Buckmaster equation is computed. The corresponding conserved quantities are obtained from their respective densities. Furthermore, the similarity reductions corresponding to the symmetries of the equation are constructed.

Keywords: Buckmaster equation‎, ‎Lie point symmetry‎‎‎, ‎Direct method‎, ‎Homotopy operator‎, ‎Similarity Reductions
• Ramy Hafez, Youssri Youssri * Pages 99-110
A numerical method for the variable-order fractional functional differential equations (VO-FFDEs) has been developed. This method is based on approximation with shifted Legendre polynomials. The properties of the latter were stated, first. These properties, together with the shifted Gauss-Legendre nodes were then utilized to reduce the VO-FFDEs into a solution of matrix equation. Sequentially, the error estimation of the proposed method was investigated. The validity and efficiency of our method were examined and verified via numerical examples.
Keywords: Variable-order fractional functional differential equations, Shifted Legendre polynomials, Gauss-Legendre nodes, Matrix equation
• Ramin Najafi * Pages 111-118
In this paper, the Lie symmetry analysis is presented for the time-fractional KdV equation with the Riemann-Liouville derivative. We introduce a generalized approximate nonclassical method that is applied to differential equations with fractional order. In the sense of this symmetry, the vector fields of fractional KdV equation are obtained. The similarity reduction corresponding to the symmetries of the equation is constructed.
Keywords: Approximate Lie symmetry analysis, Time-fractional KdV equation, Group-invariant solution
• Ghulam Farid *, Atiq Rehman, Qurat Ain Pages 119-140
In this paper, we establish Hadamard type fractional integral inequalities for a more general class of functions that is the class of (h _ m)_convex functions. These results are due to Riemann-Liouville (RL) k-fractional integrals: a generalization of RL fractional integrals. Several known results are special cases of proved results.
Keywords: Riemann-Liouville fractional integrals_Riemann-Liouville k−fractional integrals_(h − m)−convex functions
• Amjad Shaikh *, Bhausaheb Sontakke Pages 141-154
In this article we consider, impulsive initial value problems for a class of implicit fractional differential equations involving the Caputo fractional derivative of order β in (1,2]. The solutions of this nonlinear equation are analyzed by establishing sufficient conditions for existence and uniqueness using Banach's contraction mapping principle and the Schaefer's fixed point theorem. In addition, using the Banach contraction principle, we establish uniqueness result. To demonstrate main results two examples are presented.
Keywords: Initial value problem, Implicit fractional differential equations, Impulses, fixed point theorem
• Elyas Shivanian *, Mostafa Hosseini, Asghar Rahimi Pages 155-172
In this article, the meshless local radial point interpolation (MLRPI) methods are applied to simulate two dimensional wave equation subject to given appropriate initial and Neumann’s boundary conditions. In MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as square or circle. The radial point interpolation method is proposed to construct shape functions for MLRPI. A weak formulation with a Heaviside step function transforms the set of governing equations into local integral equations on local sub domains where Neumann’s boundary condition is imposed naturally. A two-step time discretization method with the help of Crank-Nicolson technique is employed to approximate the time derivatives. Convergence studies in the numerical example show that MLRPI method possesses excellent rates of convergence.
Keywords: Meshless local radial point interpolation (MLRPI), Local weak formulation, Radial basis function, 2-D wave equation, Neumann’s boundary conditions, Finite difference
• Leila Saeedi, Abolfazl Tari Marzabad *, Esmail Babolian Pages 173-193
In this paper, functional Hammerstein integro-differential equations of fractional order is studied. Here the existence and uniqueness of the solution is proved. A numerical method to approximate the solution of problem is also presented which is based on an improvement of the successive approximations method. Error estimation of the method is analyzed and error bound is obtained. The convergence and stability of the method are proved. At the end, application of the method is revealed by presenting some examples.
Keywords: Functional Hammerstein integro-differential equations, Fractional order, Successive approximations‎‎, ‎Spline interpolation‎, Trapezoidal quadrature rule
• Elham Lashkarian *, SeyedReza Hejazi, Noora Habibi Pages 194-204

In the present paper the process of finding new solutions from previous solutions of a given fractional differential equation (FDE) is considered. For this issue, first we should construct an exact solution by using the symmetry operators of the equation. Then, the commutator brackets of the obtained operators give new solutions from old ones via a systematic method.

Keywords: Lie point symmetry, fractional calculus, fractional differential equation, Exact solution
• Ali Khani *, Saeid Panahy Pages 205-211
In this paper, an integration method is presented based on using ultraspherical polynomials for solving a class of linear fractional integro-di_erential equations of Volterra types. This method is based on a new investigation of ultrasphreical integration to approximate the highest order derivative in the equations and generate approximations to the lower order derivatives through integration of the higher-order derivatives. Numerical example illustrate the efficiency and accuracy of the method.
Keywords: Fractional integro-differential equation, Ultraspherical functions, Caputo derivative
• Fikret Aliev *, Nihan Aliev, Nargiz Safarova, Naila Velieva Pages 212-221
A new simplified analytical formula is given for solving the Cauchy problem for a homogeneous system of fractional order linear differential equations with constant coefficients (SFOLDECC). The matrix exponential function in this formula is re- placed by a Taylor series. Next, an analytical expression of the integral is obtained, with the help of which, for the transition matrix, a relation is obtained that allows one to obtain a solution of the Cauchy problem with high accuracy. The results also apply to the case of inhomogeneous systems with constant perturbations and are illustrated by numerical examples.
Keywords: Cauchy problem, Linear fractional derivative system, Mittag-Leffler function, Constant matrix coefficients