University of TabrizComputational Methods for Differential Equations2345-398210220220401Analysis of non-hyperbolic equilibria for Caputo fractional system2983061251010.22034/cmde.2021.41486.1799ENMarvinHotiDepartment of Mathematics, Ryerson University, Toronto, Canada.Journal Article20200829In this manuscript, a center manifold reduction of the flow of a non-hyperbolic equilibrium point on a planar dynamical system with the Caputo derivative is proposed. The stability of the non-hyperbolic equilibrium point is shown to be locally asymptotically stable, under suitable conditions, by using the fractional Lyapunov direct method.https://cmde.tabrizu.ac.ir/article_12510_78e1cfe36fbe222efcf7023c745cd3c8.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401An efficient approximate solution of Riesz fractional advection-diffusion equation3073191272110.22034/cmde.2021.41690.1815ENSiavashMockaryDepartment of Mathematics, College of Science, Yadegar-e-Imam Khomeini (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran.0000-0002-1026-8874AlirezaVahidiDepartment of Mathematics, College of Science, Yadegar-e-Imam Khomeini (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran.EsmailBabolianFaculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.Journal Article20200925The Riesz fractional advection-diffusion is a result of the mechanics of chaotic dynamics. It’s of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [−1, 1] × [−1, 1]. Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical solution. Examples show the effectiveness of the method.https://cmde.tabrizu.ac.ir/article_12721_1cb3cc01abce405e339fc8c370263ba6.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401PDTM approach to solve Black Scholes equation for powered ML-Payoff function3203261268710.22034/cmde.2021.37944.1675ENSanjay JGhevariyaDepartment of Mathematics, Sardar Patel University, Vallabh Vidyanagar, Gujarat, India.0000-0002-8227-6689Journal Article20200117In this paper, the Projected Differential Transform Method (PDTM) has been used to solve the Black Scholes differential equation for powered Modified Log Payoff (ML-Payoff) functions,$\max\{S^k\ln\big(\frac{S}{K}\big),0\}$ and $\max\{S^k\ln\big(\frac{K}{S}\big),0\}, (k\in \mathbb{R^{+}}\cup \{0\})$. It is the generalization of Black Scholes model for ML-Payoff functions. It can be seen that values from PDTM are quite accurate to the closed form solutions.https://cmde.tabrizu.ac.ir/article_12687_bd0fb67a478cc54c9685c69a53ce6fbf.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Numerical solution of optimal control problem for economic growth model using RBF collocation method3273371268810.22034/cmde.2021.40223.1757ENAhmadGolbabaiSchool of Mathematics, Iran University of
Science and Technology, Narmak, Tehran, Iran.NimaSafaeiSchool of Mathematics, Iran University of
Science and Technology, Narmak, Tehran, Iran.MahboubehMolavi-ArabshahiSchool of Mathematics, Iran University of
Science and Technology, Narmak, Tehran, Iran.0000-0003-1242-4989Journal Article20200608In the current paper, for the economic growth model, an efficient numerical approach on arbitrary collocation points is described according to Radial Basis Functions (RBFs) interpolation to approximate the solutions of optimal control problems. The proposed method is based on parametrizing the solutions with any arbitrary global RBF and transforming the optimal control problem into a constrained optimization problem using arbitrary collocation points. The superiority of the method is its flexibility to select between different RBF functions for the interpolation and also parametrization an extensive range of arbitrary nodes. The Lagrange multipliers method is employed to convert the constrained optimization problem into a system of algebraic equations. Numerical results approve the accuracy and performance of the presented method for solving optimal control problems in the economic growth model. https://cmde.tabrizu.ac.ir/article_12688_cef306a8c797e612ec8f4374c8652e0c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Studying the thermal analysis of rectangular cross section porous fin: A numerical approach3383501251110.22034/cmde.2021.37458.1669ENWaleedAdelDepartment of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Egypt.Université Française d’Egypte, Ismailia Desert Road, El Shorouk, Cairo, Egypt.0000-0002-0557-8536AhmetYildirimDepartment of Mathematics, Faculty of Science, Ege University, 35100 Izmir, Bornova, Turkey.0000-0001-8989-4271Journal Article20200109In this work, a direct computational method has been developed for solving the thermal analysis of porous fins with a rectangular cross-section with the aid of Chebyshev polynomials. The method transforms the nonlinear differential equation into a system of nonlinear algebraic equations and then solved using a novel technique. The solution of the system gives the unknown Chebyshev coefficients. An algorithm for solving this nonlinear system is presented. The results are obtained for different values of the variables and a comparison with other methods is made to demonstrate the effectiveness of the method. https://cmde.tabrizu.ac.ir/article_12511_d07029a4829d907065e534dadd823f2a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Application of fuzzy systems on the numerical solution of the elliptic PDE-constrained optimal control problems3513711276610.22034/cmde.2021.39351.1725ENMasoomehAziziDepartment of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.MajidAmirfakhrianDepartment of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.Department of Computer Sciences, University of Calgary, Calgary, Canada.Mohammad AliFariborzi AraghiDepartment of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.Journal Article20200424This paper presents a numerical fuzzy indirect method based on the fuzzy basis functions technique to solve an optimal control problem governed by Poisson’s differential equation. The considered problem may or may not be accompanied by a control box constraint. The first-order necessary optimality conditions have been derived, which may contain a variational inequality in function space. In the presented method, the obtained optimality conditions have been discretized using fuzzy basis functions and a system of equations introduced as the discretized optimality conditions. The derived system mostly contains some nonsmooth equations and conventional system solvers fail to solve them. A fuzzy system-based semi-smooth Newton method has also been introduced to deal with the obtained system. Solving optimality systems by the presented method gets us unknown fuzzy quantities on the state and control fuzzy expansions. Finally, some test problems have been studied to demonstrate the efficiency and accuracy of the presented fuzzy numerical technique.https://cmde.tabrizu.ac.ir/article_12766_01cc12fed9c21ba63ee80b3fd2fac58c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Fractional study on heat and mass transfer of MHD Oldroyd-B fluid with ramped velocity and temperature3723951228810.22034/cmde.2021.39703.1739ENNazishIftikharDepartment of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Pakistan.Syed TauseefSaeedDepartment of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Pakistan.0000-0002-0971-8364Muhammad BilalRiazDepartment of Mathematics, University of Management and Technology, Pakistan.Institute of Grounderwater Studies, University of the Free State, South Africa.Journal Article20200508This study explores the time-dependent flow of MHD Oldroyd-B fluid under the effect of ramped wall velocity and temperature. The flow is confined to an infinite vertical plate embedded in a permeable surface with the impact of heat generation and thermal radiation. Solutions of velocity, temperature, and concentration are derived symmetrically by applying non-dimensional parameters along with Laplace transformation $(LT)$ and numerical inversion algorithm. Graphical results for different physical constraints are produced for the velocity, temperature, and concentration profiles. Velocity and temperature profile decrease by increasing the effective Prandtl number. The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity. Velocity is decreasing for $\kappa$, $M$, $Pr_{reff,}$ and $S_{c}$ while increasing for $G_{r}$ and $G_{c}$. Temperature is an increasing function of the fractional parameter. Additionally, Atangana-Baleanu $(ABC)$ model is good to explain the dynamics of fluid with better memory effect as compared to other fractional operators.https://cmde.tabrizu.ac.ir/article_12288_b8aacd078bc8ae25b736b0c1fc6a9837.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401New analytical methods for solving a class of conformable fractional differential equations by fractional Laplace transform3964071269010.22034/cmde.2021.40834.1775ENMohammadMolaeiDepartment of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.FarhadDastmalchi SaeiDepartment of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.MohammadJavidiFaculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.YaghoubMahmoudiDepartment of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.Journal Article20200718In this paper, new analytical solutions for a class of conformable fractional differential equations (CFDEs) and some more results about Laplace transform introduced by Abdeljawad are investigated. The Laplace transform method is developed to get the exact solution of CFDEs. The aim of this paper is to convert the CFDEs into ordinary differential equations (ODEs), this is done by using the fractional Laplace transform of (α + β) order.https://cmde.tabrizu.ac.ir/article_12690_1865b21558efdf28ea0e8ace634e010e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Shifted Jacobi collocation method for Volterra-Fredholm integral equation4084181279610.22034/cmde.2021.38146.1680ENAmany SaadMohamedDepartment of Mathematics,
Faculty of Science, Helwan University, Egypt.0000-0001-9517-0296Journal Article20200126In this paper, we compute the approximate numerical solution for the Volterra-Fredholm integral equation (VFIE) by using the shifted Jacobi collocation (SJC) method which depends on the operational matrices. Some properties of the shifted Jacobi polynomials are introduced. These properties allow us to transform the VolterraFredholm integral equation into a system of algebraic equations in a nice form with the expansion coefficients of the solution. Also, the convergence and error analysis are studied extensively. Finally, some examples which verify the efficiency of the given method are supplied and compared with other methods. https://cmde.tabrizu.ac.ir/article_12796_69b686eb40e07349c7852844355baaa3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Stochastic analysis and invariant subspace method for handling option pricing with numerical simulation4194301270710.22034/cmde.2021.38468.1692ENRezaHejaziFaculty of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran.ElhamDastranjFaculty of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran.NooraHabibiFaculty of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran.0000-0001-7106-1052AzadehNaderifardFaculty of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran.Journal Article20200218In this paper, option pricing is given via stochastic analysis and invariant subspace method. Finally numerical solutions is driven and shown via diagram. The considered model is one of the most well known non-linear time series model in which the switching mechanism is controlled by an unobservable state variable that follows a first-order Markov chain. Some analytical solutions for option pricing are given under our considered model. Then numerical solutions are presented via finite difference method. https://cmde.tabrizu.ac.ir/article_12707_f94c97ceb397384b05b4a404ceb7e6d0.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Numerical solution of space fractional diffusion equation using shifted Gegenbauer polynomials4314441222110.22034/cmde.2020.42106.1818ENKazeemIssaDepartment of Statistics and Mathematical Sciences, Kwara State University, Malete, Nigeria.Babatunde M.YisaDepartment of Mathematics, University of Ilorin, Ilorin, Nigeria.JafarBiazarDepartment of Mathematical Sciences, University of Guilan, Rasht, Iran.0000-0001-8026-2999Journal Article20201004This paper is concerned with numerical approach for solving space fractional diffusion equation using shifted Gegenbauer polynomials, where the fractional derivatives are expressed in Caputo sense. The properties of Gegenbauer polynomials are exploited to reduce space fractional diffusion equation to a system of ordinary differential equations, that are then solved using finite difference method. Some selected numerical simulations of space fractional diffusion equations are presented and the results are compared with the exact solution, also with the results obtained via other methods in the literature. The comparison reveals that the proposed method is reliable, effective and accurate. All the computations were carried out using Matlab package. https://cmde.tabrizu.ac.ir/article_12221_076547f1cc5f7ba9dcca97c63c0840ec.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Some new soliton solutions for the nonlinear the fifth-order integrable equations4454601221810.22034/cmde.2020.30833.1462ENMehrdadLakestaniDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.0000-0002-2752-0167JalilManafianDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.0000-0001-7201-6667Ali RezaNajafizadehDepartment of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.MohammadPartohaghighiDepartment of Mathematics, University of Bonab, Bonab, Iran.Journal Article20181210In this work, we established some exact solutions for the $(1+1)$-dimensional and $(2+1)$-dimensional fifth-order integrable equations ($(1+1)$D and $(2+1)$D FOIEs) which is considered based on the improved $\tanh(\phi(\xi)/2)$ expansion method (IThEM), by utilizing Maple software. We obtained new periodic solitary wave solutions. The obtained solutions include soliton, periodic, kink, kink-singular wave solutions. Comparing our new results with Wazwaz results, namely, the Hereman-Nuseri method shows that our results give further solutions. Many other such types of nonlinear equations arise in fluid dynamics, plasma physics, and nonlinear physics.https://cmde.tabrizu.ac.ir/article_12218_662e8f57484e2b59730502a4b3c2b042.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Exact solutions and numerical simulation for Bakstein-Howison model4614741276410.22034/cmde.2021.42640.1834ENElhamDastranjFaculty of Mathematical Sciences, Shahrood university of technology, Shahrood,
Semnan, Iran.HosseinSahebi FardFaculty of Mathematical Sciences,
Shahrood university of technology, Shahrood, Semnan, IranJournal Article20201104In this paper, European options with transaction cost under some Black-Scholes markets are priced. In fact, stochastic analysis and Lie group analysis are applied to find exact solutions for European options pricing under considered markets. In the sequel, using the finite difference method, numerical solutions are presented as well. Finally, European options pricing are presented in four maturity times under some Black-Scholes models equipped with the gold asset as underlying asset. For this, the daily gold world price has been followed from Jan 1, 2016 to Jan 1, 2019 and the results of the profit and loss of options under the considered models indicate that call options prices prevent arbitrage opportunity but put options create it. https://cmde.tabrizu.ac.ir/article_12764_4d68a8f1607e125472eceffd9b129b95.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401A Robust computational method for singularly perturbed delay parabolic convection-diffusion equations arising in the modeling of neuronal variability4754881279710.22034/cmde.2021.44306.1873ENImiru TakeleDabaDepartment of Mathematics, Wollega University, Nekemte, Ethiopia.0000-0002-7822-9679Gemechis FileDuressaDepartment of Mathematics, Jimma University, Jimma, Ethiopia.0000-0003-1889-4690Journal Article20210130In this study, a robust computational method involving exponential cubic spline for solving singularly perturbed parabolic convection-diffusion equations arising in the modeling of neuronal variability has been presented. Some numerical examples are considered to validate the theoretical findings. The proposed scheme is shown to be an $\varepsilon-$uniformly convergent accuracy of order $ O\left( \left( \Delta t\right) +h^2 \right) $.https://cmde.tabrizu.ac.ir/article_12797_0ce1b865e3f8ad5955d7c1541d608a5c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401An adaptive Monte Carlo algorithm for European and American options4895011276510.22034/cmde.2021.37369.1654ENMahboubehAalaeiInsurance Research Center,
Saadat Abad, Tehran, Iran.0000-0002-6138-3186MahnazManteqipourInsurance Research Center,
Saadat Abad, Tehran, Iran.Journal Article20191218In this paper, a new adaptive Monte Carlo algorithm is proposed to solve systems of linear algebraic equations (SLAEs). The corresponding properties of the algorithm and its advantages over the conventional and previous adaptive Monte Carlo algorithms are discussed and theoretical results are established to justify the convergence of the algorithm. Furthermore, the algorithm is used to solve the SLAEs obtained from finite difference method for the problem of European and American options pricing. Numerical tests are performed on examples with matrices of different sizes and on SLAEs coming from option pricing problems. Comparisons with standard numerical and stochastic algorithms are also done which demonstrate the computational efficiency of the proposed algorithm. https://cmde.tabrizu.ac.ir/article_12765_f6369f11391e2c9862d43d0e0ccfcf0e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Uniformly convergent fitted operator method for singularly perturbed delay differential equations5025181297710.22034/cmde.2021.41166.1789ENMesfin MekuriaWoldaregayDepartment of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia.0000-0002-6555-7534Habtamu GaromaDebelaDepartment of Mathematics, Jimma University, Jimma, Ethiopia.0000-0003-1033-3602-Gemechis FileDuressaDepartment of Mathematics, Jimma University, Jimma, Ethiopia.Journal Article20200814This paper deals with the numerical treatment of singularly perturbed delay differential equations having a delay on the first derivative term. The solution of the considered problem exhibits boundary layer behavior on the left or right side of the domain depending on the sign of the convective term. The term with the delay is approximated using Taylor series approximation, resulting in an asymptotically equivalent singularly perturbed boundary value problem. The uniformly convergent numerical scheme is developed using exponentially fitted finite difference method. The stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. Numerical examples are considered to validate the theoretical analysis. https://cmde.tabrizu.ac.ir/article_12977_8b19ddf041d364852e4352dfa5cd2e42.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401The interior inverse boundary value problem for the impulsive Sturm-Liouville operator with the spectral boundary conditions5195251251210.22034/cmde.2021.34215.1567ENYasserKhaliliDepartment of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, 578 Sari, Iran.MohsenKhaleghi MoghadamDepartment of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, 578 Sari, Iran.Journal Article20190624In this study, we discuss the inverse problem for the Sturm-Liouville operator with the impulse and with the spectral boundary conditions on the finite interval (0, π). By taking the Mochizuki-Trooshin’s method, we have shown that some information of eigenfunctions at some interior point and parts of two spectra can uniquely determine the potential function q(x) and the boundary conditions.https://cmde.tabrizu.ac.ir/article_12512_e3229972f960011cca2d9282270e602f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401A meshless technique based on the radial basis functions for solving systems of partial differential equations5265371270610.22034/cmde.2021.39707.1740ENMehranNematiDepartment of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.MahmoudShafieeDepartment of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.0000-0001-8974-3895HamidehEbrahimiDepartment of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.Journal Article20200614The radial basis functions (RBFs) methods were first developed by Kansa to approximate partial differential equations (PDEs). The RBFs method is being truly meshfree becomes quite appealing, owing to the presence of distance function, straight-forward implementation, and ease of programming in higher dimensions. Another considerable advantage is the presence of a tunable free shape parameter, contained in most of the RBFs that control the accuracy of the RBFs method. Here, the solution of the two-dimensional system of nonlinear partial differential equations is examined numerically by a Global Radial Basis Functions Collocation Method (GRBFCM). It can work on a set of random or uniform nodes with no need for element connectivity of input data. For the timedependent partial differential equations, a system of ordinary differential equations (ODEs) is derived from this scheme. Like some other numerical methods, a comparison between numerical results with analytical solutions is implemented confirming the efficiency, accuracy, and simple performance of the suggested method.https://cmde.tabrizu.ac.ir/article_12706_95aabe31aadbfad3e850c2daa243f89a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401The convergence of exponential Euler method for weighted fractional stochastic equations5385481279410.22034/cmde.2021.41430.1795ENFatemehMahmoudiDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.MahdiehTahmasebiDepartment of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran.Journal Article20200824In this paper, we propose an exponential Euler method to approximate the solution of a stochastic functional differential equation driven by weighted fractional Brownian motion $ B^{ a, b}$ under some assumptions on $a$ and $b$. We obtain also the convergence rate of the method to the true solution after proving an $L^{ 2}$-maximal bound for the stochastic integrals in this case.https://cmde.tabrizu.ac.ir/article_12794_63e5ebfb2f5b736dd0e94eb97dd84996.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210220220401Dynamics of combined soliton solutions of unstable nonlinear fractional-order Schrödinger equation by beta-fractional derivative5495661268910.22034/cmde.2021.40523.1766ENMajidBagheriFaculty of Science, Department of Applied Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.AliKhaniFaculty of Science, Department of Applied Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.0000000289284235Journal Article20200627In this article, a new version of the trial equation method is suggested. This method allows new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear fractionalorder Schrödinger equation in fractional time derivative form of order α. Some exact solutions of the fractionalorder fractional PDE are attained by employing the new powerful expansion approach using by beta-fractional derivatives which are used to get many solitary wave solutions by changing various parameters. New exact solutions are expressed with rational hyperbolic function solutions, rational trigonometric function solutions, 1-soliton solutions, dark soliton solitons, and rational function solutions. We can say that unstable nonlinear Schrödinger equation exist different dynamical behaviors. In addition, the physical behaviors of these new exact solutions are given with two and three dimensional graphs.https://cmde.tabrizu.ac.ir/article_12689_ce2d4326c4cd5a7ce82c118b442d2f91.pdf