eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
152
162
8588
A numerical method for partial fractional Fredholm integro-differential equations
Mansureh Mojahedfar
m.mojahedfar@shahed.ac.ir
1
Abolfazl Tari Marzabad
tari@shahed.ac.ir
2
Department of Mathematics, Shahed University, Tehran, Iran
Department of Mathematics, Shahed University, Tehran, Iran
In this paper, an expansion method based on orthonormal polynomials is presented to find the numerical solution of partial fractional Fredholm integro-differential equations (PFFIDEs). A PFFIDE is converted to a system of linear algebraic equations, which is solved for the expansion coefficients of approximate solution based on orthonormal polynomials. An estimation error is discussed and some illustrative examples are given to demonstrate the validity and applicability of the proposed method.
https://cmde.tabrizu.ac.ir/article_8588_a74d9caf5eefcdfcfb28c78fa45725c2.pdf
Partial integro-differential equations
Fractional operators
Expansion method
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
163
176
8590
Numerical solution of nonlinear mixed Fredholm-Volterra integro-differential equations of fractional order by Bernoulli wavelets
Elham Keshavarz
keshavarz@alzahra.ac.ir
1
Yadollah Ordokhani
ordokhani@alzahra.ac.ir
2
Mohsen Razzaghi
razzaghi@math.msstate.edu
3
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762
In this paper, a numerical method for solving a class of nonlinear mixed Fredholm-Volterra integro-differential equations of fractional order is presented. The method is based upon Bernoulli wavelets approximations. The operational matrix of fractional order integration for Bernoulli wavelets is utilized to reduce the solution of the nonlinear fractional integro-differential equations to system of algebraic equations. Illustrative examples are included to demonstrate the efficiency and accuracy of the method.
https://cmde.tabrizu.ac.ir/article_8590_96db40726b0c0f3f3031c88ccf1ee410.pdf
Bernoulli wavelets
fractional calculus
Fredholm-Volterra integro-differential equations
Caputo derivative
Operational matrix
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
177
198
8675
Hermite wavelets method for the numerical solution of linear and nonlinear singular initial and boundary value problems
Siddu Channabasappa Shiralashetti
shiralashettisc@gmail.com
1
Kumbinarasaiah Srinivasa
kumbinarasaiah@gmail.com
2
Department of Mathematics, Karnatak University, Dharwad, India
Department of Mathematics, Karnatak University, Dharwad, India
In this article, Modified Hermite wavelets based numerical method is developed for the solution of singular initial and boundary value problems. It consists of reducing the differential equations with the associated initial and boundary conditions into system of algebraic equations by expanding the unknown function as a series of Hermite wavelets with unknown coefficients. Obtained system of equations are solved using Newton’s iterative method through Matlab. Illustrative examples are considered to demonstrate the applicability and accuracy of the proposed technique. Obtained results are compared favorably with the exact solutions. Also, we proved the theorem reveals that, when exact solution can be obtained by the proposed method.
https://cmde.tabrizu.ac.ir/article_8675_fa0cf37cf6a92c9b4d512602e40be5c3.pdf
Hermite wavelets
Singular initial and boundary value problems
Collocation method
Limit points
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
199
205
8592
Factorization method for fractional Schrödinger equation in D-dimensional fractional space and homogeneous manifold SL(2,c)/GL(1,c)
Hossein Jafari
jafari@umz.ac.ir
1
Jafar Sadeghi
pouriya@ipm.ir
2
Farzaneh Safari
f.safari@stu.umz.ac.ir
3
Amos Kubeka
kubekas@unisa.ac.za
4
Department of Mathematics, University of Mazandaran, Babolsar, Iran
Physics Department, University of Mazandaran, P. O. Box 4716-95447, Babolsar, Iran
Department of Mathematics, University of Mazandaran, Babolsar, Iran
Department of Mathematical Sciences, University of South Africa, P. O. Box 392, UNISA 0003, South Africa
In this paper, we consider a $D$-dimensional fractional Schr\"odinger equation with a Coulomb potential. By using the associated Laguerre and Jacobi equations, we obtain the wave function and energy spectrum and this then enable us to separate this equation in terms of the radial and angular momentum parts respectively. Also the associated Laguerre and Jacobi equations makes it possible to further factorize the $D$-dimensional fractional Schr\"odinger equation such that the resulting equations can be expressed in terms of the first order operators which are basically raising and lowering operators.
https://cmde.tabrizu.ac.ir/article_8592_d070d098b3ad5fea48bfcf615203dbe1.pdf
Factorization method
Fractional Schr"odinger equation
Laguerre equation
Jacobi equation
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
206
223
8666
Accurate splitting approach to characterize the solution set of boundary layer problems
Khosro Sayevand
ksayehvand@yahoo.com
1
Jose Antonio Tenreiro Machado
jtm@isep.ipp.pt
2
Faculty of Mathematical Sciences, Malayer University, P. O. Box 16846-13114, Malayer, Iran
Institute of Engineering of Polytechnic of Porto, Department of Electrical Engineering, Porto, Portugal
The boundary layer (BL) is an important concept and refers to the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. This paper studies singularly perturbed fractional differential equations where the fractional derivatives are defined in the Caputo sense. The solution of such equations, with appropriate boundary conditions, displays BL behavior. The solution out of the BL is estimated by the solution of the reduced problem and the layer solution is approximated by means of a modified truncated Chebyshev series. The coefficients of the truncated series are evaluated using a novel operational matrix technique. Moreover, the stability and the error analysis of the proposed method are analyzed. Several examples illustrate the validity and applicability of the method.
https://cmde.tabrizu.ac.ir/article_8666_d7b613d4c647b73430148d5432da5c39.pdf
Singular perturbation
Boundary value problem
Shifted Chebyshev polynomial
Operational matrix
Boundary laye
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
224
234
8589
On using topological degree theory to investigate a coupled system of non linear hybrid differential equations
Samina .
saminakhanqau@yahoo.com
1
Ibrar Ullah
ibrarullah.khan34@gmail.com
2
Rahmat Khan
rahmat_alipk@yahoo.com
3
Kamal Shah
kamalshah408@gmail.com
4
Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan
Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan
Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan
Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan
In this work we discuss the existence of solutions of nonlinear fractional differential equations. By using the topological degree theory, some results on the existence of solutions are obtained. Our analysis relies on the reduction of the problem considered to the equivalent system of Fredholm integral equations. As applications, an examples is also provided to illustrate our main results.
https://cmde.tabrizu.ac.ir/article_8589_5a58141088e21c177e94feb12c1489d7.pdf
Hybrid initial value problem
K-condensing, Existence of fixed point without compactness theorem
Caputo fractional derivative
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
235
251
8686
Interval structure of Runge-Kutta methods for solving optimal control problems with uncertainties
Navid Razmjooy
navid.razmjooy@hotmail.com
1
Mehdi Ramezani
ramezani@tafreshu.ac.ir
2
Department of Electrical and Control Engineering, Tafresh University, Tafresh, 39518 79611, Iran
Department of Mathematics, Tafresh University, Tafresh 39518 79611, Iran
In this paper, a new interval version of Runge-Kutta methods is proposed for time discretization and solving of optimal control problems (OCPs) in the presence of uncertain parameters. A new technique for interval arithmetic is introduced to evaluate the bounds of interval functions. The proposed approach is based on the new forward representation of Hukuhara interval differencing and combining it with Runge-Kutta method for solving the OCPs with interval uncertainties. To perform the proposed method on OCPs, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, using some algebraic manipulations, they are converted to an ordinary differential equation to achieve the interval optimal solution for the considered OCP with uncertain parameters. Shooting method is also employed to cover the Runge-Kutta methods restrictions in solving the OCPs with boundary values. The simulation results are applied to some practical case studies to demonstrate the effectiveness of the proposed method.
https://cmde.tabrizu.ac.ir/article_8686_cc55d77167c9589882629fd78632b441.pdf
Optimal control
Interval analysis
Lagrange multiplier method
Runge-Kutta methods
Hukuhara difference
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
252
265
8685
Linear fractional fuzzy differential equations with Caputo derivative
Rahman Abdollahi
r.abdollahimath@gmail.com
1
Mohammad Bagher Farshbaf Moghimi
m.b.moghimi56@gmail.com
2
Alireza Khastan
khastan@iasbs.ac.ir
3
Mohammad Reza Hooshmandasl
hooshmandasl@yazd.ac.ir
4
Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran
Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran
Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran
Department of Computer Science, Yazd University, Yazd, Iran
In this paper, we study linear fractional fuzzy differential equations involving the Caputo generalized Hukuhara derivative. Using the fuzzy Laplace transform, we present the general form of solutions in terms of Mittag-Leffler functions. Finally, some examples are provided to illustrate our results.
https://cmde.tabrizu.ac.ir/article_8685_35dfe469019b2f6ba8587082b3abdd11.pdf
Caputo fractional derivative
Fuzzy fractional differential equation
Fuzzy initial value problem
Fuzzy Laplace transform, Generalized Hukuhara differentiability
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
266
275
8562
Approximate symmetry and exact solutions of the perturbed nonlinear Klein-Gordon equation
Mohammad Rahimian
rahimian.mohammad@gmail.com
1
Mehdi Nadjafikhah
m-nadjafikhah@iust.ac.ir
2
Department of Mathematics, Masjed-Soleiman Branch, Islamic Azad University, Masjed-Soleiman, Iran.
Department of Pure Mathematics, School of Mathematics, Iran University of Science and Technology,Narmak, Tehran, 1684613114, Iran
In this paper, the Lie approximate symmetry analysis is applied to investigate new exact solutions of the perturbed nonlinear Klein-Gordon equation. The nonlinear Klein-Gordon equation is used to model many nonlinear phenomena. The tanh-coth method, is employed to solve some of the obtained reduced ordinary differential equations. As a result, we construct new analytical solutions with small parameter which is effectively obtained by the proposed method.
https://cmde.tabrizu.ac.ir/article_8562_03583e084a8781e6b0ed51181e1c5a5f.pdf
Perturbed Klein-Gordon equation
Exact solutions
Approximate symmetry
Approximate invariant solutions
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
276
288
8591
Analysis of meshless local radial point interpolant on a model in population dynamics
Elyas Shivanian
shivanian@sci.ikiu.ac.ir
1
Hedayat Fatahi
fatahi_iau@yahoo.com
2
Department of Applied Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran
Department of Mathematics, Baneh Branch, Islamic Azad University, Baneh, Iran
In this work, we present an improvement of the spectral meshless radial point interpolation (SMRPI) method to uncover a simulation behaviour of the population dynamic model which mathematically is the nonlinear partial integro-differential equation. This PDE is a kind of competition strategy in which equivalent individuals match for the same supplies. oreover, this boundary value problem is a particular type of reaction-diffusion problem augmented to an integral term corresponding to the nonlocal consumption of resources. As a result of applying meshless method, it does not matter how the geometry of the domain is complicated because the method enjoys the element free adoption. Applying the SMRPI on the two-dimensional integral equation leads to a linear system of algebraic equations which is easy to treat. Finally, some numeric experiments are presented to show the reliable results.
https://cmde.tabrizu.ac.ir/article_8591_fb05fd4443582dedfcf9477f2ed97ef1.pdf
Spectral meshless radial point interpolation (SMRPI) method
Radial basis function
Partial integro-differential equation
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
289
301
8691
Spectral solution of fractional fourth order partial integro-differential equations
Hamed Bazgir
bazgir.ha@fs.lu.ac.ir
1
Bahman Ghazanfari
ghazanfari.b@gmail.com
2
Department of Mathematics, Lorestan University, Khorramabad, Iran
Department of Mathematics, Lorestan University, Khorramabad, Iran
In this paper, mixed spectral method is applied to solve the fractional fourth order partial integro-differential equations together with weak singularity. Eigenfunctions of the fourth order self-adjoint positive-definite differential operator are used for the discretization of spatial variable and its derivatives. Also, shifted Legendre polynomials are applied to the discretization of time variable. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. The method is easy to apply and produces very accurate numerical results.
https://cmde.tabrizu.ac.ir/article_8691_1a42c129f42d1cc7ddccf2925ddd498d.pdf
Integro-differential equation
Weakly singular kernel
Collocation method
Operational matrix
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-04-01
7
2
302
318
8553
Stabilization of linear systems of delay differential equations by the delayed feedback method
Mohammad Mousa-Abadian
m.abadian@shahed.ac.ir
1
Sayed Hodjatollah Momeni-Masuleh
momeni@shahed.ac.ir
2
Mohammad Haeri
haeri@sharif.ir
3
Department of Mathematics, Shahed University, P.O. Box 18151-159, Tehran, Iran
Department of Mathematics, Shahed University, P.O. Box 18151-159, Tehran, Iran
Advanced Control Systems Lab, Electrical Engineering, Sharif University of Technology, Tehran, Iran
This paper consists of two folds. At first, we deal with the stability analysis of a linear system of delay differential equations. It is shown that the direct and cluster treatment methods are not applicable if there are some purely imaginary roots of the characteristic equation with multiplicity greater than one. To overcome the above difficulty, the system is decomposed into several subsystems. For the decomposition of a system, an invertible transformation is required to convert the matrices of the system into a block triangular (diagonal) form simultaneously. To achieve this goal, a necessary and sufficient condition is established. The second part concerns the stabilization of a linear system of delay differential equations using the delayed feedback method and design a controller for generating the desired response. More precisely, the unstable poles of the linear system of delay differential equations are moved to the left-half of the complex plane by the delayed feedback method. It is shown that the performance of the linear system of delay differential equations can be improved by applying the delayed feedback method.
https://cmde.tabrizu.ac.ir/article_8553_49a4ed2b656f3fe5a2de5831c03e70ef.pdf
Linear time delay system
Stability analysis
Simultaneous block triangularization
Delayed feedback method
Stabilization
Controller design