University of TabrizComputational Methods for Differential Equations2345-39829120210101First order linear fuzzy differential equations with fuzzy variable coefficients121993910.22034/cmde.2020.34127.1568ENRobabAlikhaniFaculty of Mathematical science,
University of Tabriz, Tabriz, Iran.0000-0002-4139-9834MahdiMostafazadehFaculty of Mathematical science,
University of Tabriz, Tabriz, Iran.Journal Article20190624In this study, we investigate the first order linear fuzzy differential equations with fuzzy variable coefficients. Appearance of the multiplication of a fuzzy variable coefficient by an unknown fuzzy function in linear differential equations persuades us to employ the concept of the cross product of fuzzy numbers. Mentioned product overcomes to some difficulties we face to in the case of the usual product obtained by Zadeh’s extension principle. Under the cross product, we obtain the explicit fuzzy solutions for a fuzzy initial value problem applying the concept of the strongly generalized differentiability. Finally, some examples are given to illustrate the theoretical results. The obtained numerical results are compared with other approaches in the literature for similar parameters.https://cmde.tabrizu.ac.ir/article_9939_5a4ede5c8703c2914d4cb588d3f27dff.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Application of cubic B-spline collocation method for reaction diffusion Fisher's equation22351029910.22034/cmde.2020.35430.1605ENMahboubehMolavi-ArabshahiSchool of Mathematics,
Iran University of Science and Technology,
Narmak 16844, Tehran, Iran.0000-0003-1242-4989KowsarShavali-koohshooriSchool of Mathematics,
Iran University of Science and Technology,
Narmak 16844, Tehran, Iran.Journal Article20190903In this study, an effective collocation method based on cubic B-spline has been implemented to get the numerical solutions for the non-linear Fisher’s equation. After separating this scheme with this method, the stability of the method was proven. To check the efficiency and accuracy of the proposed method, some numerical problems have been considered. The numerical results are found in good agreement with the exact solutions.https://cmde.tabrizu.ac.ir/article_10299_387da6bb836e892b8270550c038a6024.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101On the existence of positive solutions for a non-autonomous fractional differential equation with integral boundary conditions3651994010.22034/cmde.2020.29444.1420ENAzizollahBabakhaniDepartment of Mathematics,
Babol Noshirvani University of Technology,
Shariati Ave, Babol 47148-71167, Iran.QasemAl-MdallalDepartment of Mathematical Sciences , United Arab Emirates University
Al-Ain, UAE.Journal Article20180919By using the Guo-Krasnoselskii’s fixed point theorem, we investigate the existence of positive solutions for a non-autonomous fractional differential equations with integral boundary conditions of fractional order α ∈ (2, 3] in an ordered Banach space. The Fredholm integral equation has an important role in this article. Some examples are presented to illustrate the efficiency of the obtained results.https://cmde.tabrizu.ac.ir/article_9940_dcbd56c554c5be9a0970baa6662c13bb.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101On traveling wave solutions: the decoupled nonlinear Schrödinger equations with inter modal dispersion52621032610.22034/cmde.2019.34431.1574ENAhmetBekirNeighbourhood of Akcaglan,
Imarli Street, Number: 28/4, 26030, Eskisehir, Turkey.0000-0001-9394-4681MuhammadYounisPunjab University College of Information Technology,
University of the Punjab, 54590 Lahore, Pakistan.Syed TahirRizviMathematics Department,
COMSATS University Islamabad, Lahore Campus, Pakistan.AliSardarMathematics Department,
National University of Computer and Emerging Sciences, Lahore, Pakistan.Syed AmerMahmoodDepartment of Space Science,
University of the Punjab, 54590 Lahore, Pakistan.Journal Article20190705In this article, the decoupled nonlinear Schrdingers equations have been considered that describe the model of dual-core fibers with group velocity mismatch, group velocity dispersion, and spatio-temporal dispersion. These equations are analyzed using two different integrations schemes, namely, extended tanh-function and sinecosine schemes. The different kind of traveling wave solutions: solitary, topological, periodic and rational, fall out as by-product of these schemes. Finally, the existence of the solutions for the constraint conditions is also shown.https://cmde.tabrizu.ac.ir/article_10326_e660a80b12f31ce1abbf6697163dbd36.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Asymptotic decomposition method for fractional order Riccati differential equation6378991110.22034/cmde.2020.28235.1386ENBijanHasani LichaeDepartment of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.JafarBiazarDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box. 41335-1914, Guilan, Rasht, Iran.ZainabAyatiDepartment of Engineering sciences, Faculty of Technology and Engineering East of Guilan, University of Guilan P.C.44891-Rudsar-Vajargah,Iran.0000-0001-5854-1051Journal Article20180706In this study, the asymptotic Adomian decomposition method (AADM) is implemented to solve fractional order Riccati differential equations. The product integration method is used to solve the singular integrals, resulted from fractional derivative. Some fractional order Riccati differential equations are presented as examples to illustrate the ability and efficiency of the proposed approach. The approximate solutions of AADM are compared with the results of the Laplace Adomian Pade method (LAPM). Generalizing AADM for solving fractional Riccati differential equations by the far-field approximation indicates the novelty of the paper.https://cmde.tabrizu.ac.ir/article_9911_b920980f783bdc86c8cba4131678296c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Solving fractional optimal control problems using Genocchi polynomials79931093210.22034/cmde.2020.40292.1759ENMaryamArablouye MoghaddamDepartment of Mathematics,
Payame Noor University, Tehran, Iran.YousefEdrisi TabrizDepartment of Mathematics,
Payame Noor University, Tehran, Iran.MehrdadLakestaniDepartment of Applied Mathematics,
Faculty of Mathematical Sciences,
University of Tabriz, Tabriz, Iran.0000-0002-2752-0167Journal Article20200212In this paper, we solve a class of fractional optimal control problems in the sense of Caputo derivative using Genocchi polynomials. At first we present some properties of these polynomials and we make the Genocchi operational matrix for Caputo fractional derivatives. Then using them, we solve the problem by converting it to a system of algebraic equations. Some examples are presented to show the efficiency and accuracy of the method.https://cmde.tabrizu.ac.ir/article_10932_d537cb07244f8755a7d688aaaf081af1.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Exact solutions of the combined Hirota-LPD equation with variable coefficients941161147710.22034/cmde.2020.31022.1466ENMehdiFazli AghdaeiDepartment of Mathematics,
Payame Noor University,
PO BOX 19395-3697, Tehran, Iran.HojatollahAdibiDepartment of Applied Mathematics,
faculty of Mathematics and Computer Sciences,
Amirkabir University of Technology,
No.424, Hafez Avenue, Tehran 15914, Iran.0000-0002-8167-5020Journal Article20181221In this paper, we construct exact families of traveling wave (periodic wave, singular wave, singular periodic wave, singular-solitary wave and shock wave) solutions of a well-known equation of nonlinear PDE, the variable coefficients combined HirotaLakshmanan-Porsezian-Daniel (Hirota-LPD) equation with the fourth nonlinearity, which describes an important development, and application of soliton dispersion management experiment in nonlinear optics is considered, and as an achievement, a series of exact traveling wave solutions for the aforementioned equation is formally extracted. This nonlinear equation is solved by using the extended trial equation method (ETEM) and the improved tan(ϕ/2)-expansion method (ITEM). Meanwhile, the mechanical features of some families are explained through offering the physical descriptions. Analytical treatment to find the nonautonomous rogue waves are investigated for the combined Hirota-LPD equation.https://cmde.tabrizu.ac.ir/article_11477_0d9481ffc78cd9c23911074e9a6501c3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101A mathematical analysis of Zika virus transmission with optimal control strategies1171451032010.22034/cmde.2019.34715.1585ENNabaGoswamiDepartment of Mathematics, PET Research Center, University of Mysore, Mysore, India.0000-0001-8213-0631BShanmukhaDepartmant of Mathematics, PES College of Engineering, Mandya, India.Journal Article20190719This paper presents a mathematical model for transmission dynamics of Zika virus by considering standard incidence type interaction for the human to human transmission. The model involves the transmission through the bite of infected Aedes mosquitoes and human to human sexual transmission. The equilibria of the proposed model are found and the basic reproduction number R0 is computed. If R0 < 1, the disease-free equilibrium point is locally asymptotically stable and it is also globally asymptotically stable under certain conditions. The analysis shows that the model exhibits the occurrence of backward bifurcation, which suggests that when R0 < 1 is not completely sufficient for eradicating the disease where the stable disease-free equilibrium co-exists with a stable endemic equilibrium. The endemic equilibrium point of the system exists and locally asymptotically stable under some restriction on parameters, whenever R0 > 1. The sensitivity analysis is performed to identify the key parameters that affect the basic reproduction number, which can be regulated to control the transmission dynamics of the Zika. Further, this model is extended to the optimal control model and to reveals the optimal control strategies we used the Pontryagin’s Maximum Principle. It has been noticed that the optimal control gives better result than without the optimal control model. Numerical simulation is presented to support our mathematical findings.https://cmde.tabrizu.ac.ir/article_10320_eb4f5874a72b022ac83e9fd94c85605c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Semi-Algebraic mode analysis for finite element discretisations of the heat equation146158994610.22034/cmde.2019.32018.1549ENNooraHabibiFaculty of Applied Mathematics,
Shahrood University Of Technology,
P.O. Box 3619995161, Shahrood, Iran.0000-0001-7106-1052AliMesforushFaculty of Applied Mathematics,
Shahrood University Of Technology,
P.O. Box 3619995161, Shahrood, Iran.0000-0001-9098-8953FranciscoGaspar J. L.Department of Applied Mathematics,
Science Faculty, University of Zaragoza,
Pedro Cerbuna, 12, 50009 Zaragoza, Spain.CarmenRodrigoDepartment of Applied Mathematics,
School of Engineering and Architecture,
University of Zaragoza, Maria de Luna, 3, 50018, Zaragoza, Spain.Journal Article20190527In this work, a semialgebraic mode analysis (SAMA) is proposed for investigating the convergence of a multigrid waveform relaxation method applied to the Finite Element (FE) discretization of the heat equation in two and three dimensions. This analysis for finite element methods is more involved and more general than that for Finite Difference (FD) discretizations, since mass matrix must be considered. The proposed analysis results in a very useful tool to study the behaviour of the multigrid waveform relaxation method depending on the parameters of the problem.https://cmde.tabrizu.ac.ir/article_9946_f1ff5049dbae829a3ee717e84f69f921.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101A new Monte Carlo method for solving systems of linear algebraic equations159179994110.22034/cmde.2020.30640.1453ENBehrouzFathi-vajargahDepartment of Statistics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box:
41335-19141, Rasht, IranZeinabHassanzadehDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box:
41335-19141, Rasht, IranJournal Article20181201In this paper, we firstly study the employing of the Monte Carlo method for solving system of linear algebraic equations and then analyze on convergence of this method. We propound new results related to the convergence of the Monte Carlo method. Additionally, we introduce a new Monte Carlo algorithm with effective techniques. Finally, we compare the efficiency of new Monte Carlo algorithm with its old version in the numerical experiments.https://cmde.tabrizu.ac.ir/article_9941_5505de9e6ed44357cc1d84a45c2d0259.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Approximate distributed controllability of nonlocal Rayleigh beam180186993510.22034/cmde.2020.29618.1434ENHanifHeidariSchool of Mathematics and Computer Sciences,
Damghan University, Damghan,
P.O. Box 36715-364, Iran.ParisaAlasvand HadiSchool of Mathematics and Computer Sciences,
Damghan University, Damghan,
P.O. Box 36715-364, Iran.RezaNazemnezhadSchool of Engineering, Damghan University,
Damghan, Iran.Journal Article20181013This paper investigates the distributed controllability of nonlocal Rayleigh beam. The mathematical problem is formulated as an abstract differential equation. It is shown that a sequence of eigenfunction of nonlocal Rayleigh beam forms Riesz basis. Based on Riesz basis properties and theory of abstract differential equation, it is proved that a vibrating nonlocal Rayleigh beam is approximately controllable under suitable distributed control force while it is not exponentially stable.https://cmde.tabrizu.ac.ir/article_9935_18808f781639c56ab6964be04adc9ae7.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Numerical solution of fractional Riesz space telegraph equation187210994710.22034/cmde.2019.33771.1551ENMohammadJavidiFaculty of Mathematical Sciences,
University of Tabriz, Tabriz, Iran.0000-0002-1892-1647MalekAhmadian AslDepartement of Mathematics,
Islamic Azad University Tabriz Branch, Tabriz, Iran.FarhadDastmalci SaeiDepartement of Mathematics,
Islamic Azad University Tabriz Branch, Tabriz, Iran.YaghoubMahmoudiDepartement of Mathematics,
Islamic Azad University Tabriz Branch, Tabriz, Iran.Journal Article20190529In this paper, a numerical method based on polynomial approximation is presented for the Riesz fractional telegraph equation. First, a system of fractional differential equations are obtained from the telegraph equation with respect to the time variable by using the method of lines. Then a new numerical algorithm, as well as its modification for solving fractional differential equations (FDEs) based on the polynomial interpolation, is proposed. The algorithms are designed to estimate to linear fractional systems. The convergence order and stability of the fractional order algorithms are proved. At the end three examples with known exact solutions are chosen. Numerical results show that accuracy of present scheme is of order O(∆t 2 ).https://cmde.tabrizu.ac.ir/article_9947_6fee74e3c4230465d2608652cf25aa1b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Quintic Spline functions and Fredholm integral equation211224994810.22034/cmde.2019.31983.1492ENKhosrowMaleknejadSchool of Mathematics, Iran University of Science and Technology Narmak, Tehran 16844, Iran.JalilRashidiniaSchool of Mathematics, Iran University of Science and Technology Narmak, Tehran 16844, Iran.0000-0002-9177-900XHamedJalilianSchool of Mathematics, Iran University of Science and Technology Narmak, Tehran 16844, Iran.Journal Article20190209A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we first formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.https://cmde.tabrizu.ac.ir/article_9948_d01f26099943ca195f6b0f7a0a05fbbd.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101An efficient numerical solution for time switching optimal control problems225243994410.22034/cmde.2020.33529.1542ENMohammad AliMehrpouyaDepartment of Mathematics, Tafresh University,
39518-79611, Tafresh, Iran.0000-0001-9598-4943MahmoodKhaksar-e OshaghMosaheb Institute of Mathematics, Kharazmi Universtity,
No. 50 Taleghani Ave., Tehran, Iran.Journal Article20190516In this paper, an efficient computational algorithm for the solution of Hamiltonian boundary value problems arising from bang-bang optimal control problems is presented. For this purpose, at first, based on the Pontryagin’s minimum principle, the first order necessary conditions of optimality are derived. Then, an indirect shooting method with control parameterization, in which the control function is replaced with a piecewise constant function with values and switching points taken as unknown parameters, is presented. Thereby, the problem is converted to the solution of the shooting equation, in which the values of the control function and the switching points as well the initial values of the costate variables are unknown parameters. The important advantages of this method are that, the obtained solution satisfies the first order optimality conditions, further the switching points can be captured accurately which is led to an accurate solution of the bangbang problem. However, solving the shooting equation is nearly impossible without a very good initial guess. So, in order to cope with the difficulty of the initial guess, a homotopic approach is combined with the presented method. Consequently, no priori assumptions are made on the optimal control structure and number of the switching points, and sensitivity to the initial guess for the unknown parameters is resolved too. Illustrative examples are included at the end and efficiency of the method is reported.https://cmde.tabrizu.ac.ir/article_9944_460111d27afba80880de578557da969f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Inverse Sturm-Liouville problems with two supplementary discontinuous conditions on two symmetric disjoint intervals244257993610.22034/cmde.2020.32543.1509ENSeyfollahMosazadehDepartment of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, 87317-53153, Iran.0000-0002-4599-3697Journal Article20190316In this paper, we consider Sturm-Liouville problems on two symmetric disjoint intervals with two supplementary discontinuous conditions at an interior point. First, we investigate some spectral properties of boundary value problems, and obtain the asymptotic form of the eigenvalues and the eigenfunctions. Then, the eigenfunction expansion of Green’s function is presented and we prove the uniqueness theorems for the solution of the inverse problem, and reconstruct the Sturm-Liouville operator and the coefficients of boundary conditions using the Weyl m-function and spectral data. Also, numerical examples are presented.https://cmde.tabrizu.ac.ir/article_9936_9182fa2174d9d100037dfe0b66fd8fc9.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Exact solutions and numerical simulations of time-fractional Fokker-Plank equation for special stochastic process2582721032110.22034/cmde.2019.30717.1458ENRezaHejaziFaculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.AzadehNaderifardFaculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.SoleimanHosseinpourFaculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.ElhamDastranjFaculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.Journal Article20181204In this paper, a type of time-fractional Fokker-Planck equation (FPE) of the OrnsteinUhlenbeck process is solved via Riemann-Liouville and Caputo derivatives. An analytical method based on symmetry operators is used for finding reduced form and exact solutions of the equation. A numerical simulation based on the M¨untz-Legendre polynomials is applied in order to find some approximated solutions of the equation.https://cmde.tabrizu.ac.ir/article_10321_f3c588d6b4e5567e7503c9d4ba81bb28.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Application of the method of fundamental solutions for designing the optimal shape in heat transfer2732881031910.22034/cmde.2020.35593.1611ENKamalRashediDepartment of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran0000-0001-7826-1954AkbarHashemiDepartment of Mathematics,
University of Science and Technology of Mazandaran, Behshahr, IranMaryamZarhounDepartment of Mathematics,
University of Science and Technology of Mazandaran, Behshahr, IranJournal Article20190913In this paper, we propose a meshless regularization technique for solving an optimal shape design problem (OSD) which consists of constructing the optimal configuration of a conducting body subject to given boundary conditions to minimize a certain objective function. This problem also can be seen as the problem of building a support for a membrane such that its deflection is as close as possible to 1 in the subset D of the domain. We propose a numerical technique based on the combination of the method of fundamental solutions and application of the Tikhonov’s regularization method to obtain stable solution. Numerical experiments while solving several test examples are included to show the applicability of the proposed method for obtaining the satisfactory results.https://cmde.tabrizu.ac.ir/article_10319_3dcd256b467dfd3f322e8f227f965331.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Existence and uniqueness of solutions of uncertain linear systems289299993810.22034/cmde.2020.33483.1541ENVahidRoomiDepartment of Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.Hamid RezaAhmadiDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.Journal Article20190514This paper presents some new definitions and theorems about a system of linear uncertain differential equations. An existence and uniqueness of solutions of the system with initial condition will be proven. Also, it will be shown that the collection of solutions of the homogeneous uncertain system is a linear space.https://cmde.tabrizu.ac.ir/article_9938_c3c76751578becb93f7067712ef775f3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Generalized symmetries and conservation laws of (3+1)-dimensional variable coefficient Zakharov-Kuznetsov equation3003121032310.22034/cmde.2020.35574.1610ENManjitSinghYadavindra College of Engineering, Punjabi University Guru Kashi Campus, Talwandi Sabo 151302, Punjab, India.Journal Article20190911The nonlinear variable coefficient Zakharov-Kuznetsov (Vc-ZK) equation is derived using reductive perturbation technique for ion-acoustic solitary waves in magnetized three-component dusty plasma having negatively charged dust particles, isothermal ions, and electrons. The equation is investigated for generalized symmetries using a recently proposed compatibility method. Some more general symmetries are obtained and group invariant solutions are also constructed for these symmetries. Besides this, the equation is also investigated for nontrivial local conservation lawshttps://cmde.tabrizu.ac.ir/article_10323_cc57fc758759c3da3059edcd72c90cfc.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829120210101Regularization of a nonlinear inverse problem by discrete mollification method313326994210.22034/cmde.2020.30808.1461ENSoheilaBodaghiFaculty of Mathematics,
K. N. Toosi University of Technology,
Tehran, Iran.AliZakeriFaculty of Mathematics,
K. N. Toosi University of Technology,
Tehran, Iran.AmirAmiraslaniSTEM department, University of Hawaii-Maui collegeJournal Article20181209In this article, the application of discrete mollification as a regularization procedure for solving a nonlinear inverse problem in one dimensional space is considered. Illposedness is identified as one of the main characteristics of inverse problems. It is clear that if we have a noisy data, the inverse problem becomes unstable. As such, a numerical procedure based on discrete mollification and space marching method is applied to address the ill-posedness of the mentioned problem. The regularization parameter is selected by generalized cross validation (GCV) method. The numerical stability and convergence of the proposed method are investigated. Finally, some test problems, whose exact solutions are known, are solved using this method to show the efficiency.https://cmde.tabrizu.ac.ir/article_9942_663995c58072ce22b1f2cc3705c7b456.pdf