University of TabrizComputational Methods for Differential Equations2345-398212220240301Boundary controller design for stabilization of stochastic nonlinear reaction-diffusion systems with time-varying delays1962061675210.22034/cmde.2023.50986.2124ENGokulakrishnanVeeraragavanDepartment of Mathematics, SRM Institute of Science and Technology, Ramapuram Campus, Chennai - 600 089, Tamilnadu, India.SrinivasanRadhakrishnanDepartment of Mathematics, SRM Institute of Science and Technology, Ramapuram Campus, Chennai - 600 089, Tamilnadu, India.Journal Article20220405This paper is focused on studying the stabilization problems of stochastic nonlinear reaction-diffusion systems (SNRDSs) with time-varying delays via boundary control. Firstly, the boundary controller was designed to stabilization for SNRDSs. By utilizing the Lyapunov functional method, Ito’s differential formula, Wirtinger’s inequality, Gronwall inequality, and LMIs, sufficient conditions are derived to guarantee the finite-time stability (FTS) of proposed systems. Secondly, the basic expressions of the control gain matrices are designed for the boundary controller. Finally, numerical examples are presented to verify the efficiency and superiority of the proposed stabilization criterion. https://cmde.tabrizu.ac.ir/article_16752_0d1d3bdc144836dc698c7dca2bdbcd05.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Prolific new M-fractional soliton behaviors to the Schrödinger type Ivancevic option pricing model by two efficient techniques2072251676310.22034/cmde.2023.56140.2345ENYeşimSaglam OzkanDepartment of Mathematics, Faculty of Arts and Sciences, Bursa Uludag
University, 16059 Bursa, Turkey.0000-0002-1364-5137EmrullahYasarDepartment of Mathematics, Faculty of Arts and Sciences, Bursa Uludag
University, 16059 Bursa, Turkey.0000-0003-4732-5753Journal Article20230410The principal purpose of this research is to study the M-fractional nonlinear quantum-probability grounded Schrödinger kind Ivancevic option pricing model (IOPM). This well-known economic model is an alternative of the standard Black-Scholes pricing model which represents a controlled Brownian motion in an adaptive setting with relation to nonlinear Schrödinger equation. The exact solutions of the underlying equation have been derived through the well-organized extended modified auxiliary equation mapping and generalized exponential rational function methods. Different forms of optical wave structures including dark, bright, and singular solitons are derived. To the best of our knowledge, verified solutions using Maple are new. The results obtained will contribute to the enrichment of the existing literature of the model under consideration. Moreover, some sketches are plotted to show more about the dynamic behavior of this model.https://cmde.tabrizu.ac.ir/article_16763_d81519facb788d7f9b5a4f21bcf3f2e8.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval2262351663810.22034/cmde.2023.54630.2275ENZahraKavoociFaculty of Sciences, Sahand University of Technology, Tabriz, Iran.KazemGhanbariFaculty of Sciences, Sahand University of Technology, Tabriz, Iran.School of Mathematics and Statistics, Carleton University, Ottawa, Canada.HanifMirzaeiFaculty of Sciences, Sahand University of Technology, Tabriz, Iran.Journal Article20221226Most of fractional differential equations are considered on a fixed interval. In this paper, we consider a typical fractional differential equation on a symmetric interval $[-\alpha,\alpha]$, where $\alpha$ is the order of fractional derivative. For a positive real number α we prove that the solutions are $T_{n,\alpha}(x)=(\alpha+x)^\frac{1}{2}Q_{n,\alpha}(x)$ where $Q_{n,\alpha}(x)$ produce a family of orthogonal polynomials with respect to the weight function$w_\alpha(x)=(\frac{\alpha+x}{\alpha-x})^{\frac{1}{2}}$ on $[-\alpha,\alpha]$. For integer case $\alpha = 1 $, we show that these polynomials coincide with classical Chebyshev polynomials of the third kind. Orthogonal properties of the solutions lead to practical results in determining solutions of some fractional differential equations. https://cmde.tabrizu.ac.ir/article_16638_a6531dd83f47c157d2e637b528ce71bc.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Rumor spread dynamics and its sensitivity analysis under the influence of the Caputo fractional derivatives2362651691110.22034/cmde.2023.55650.2316ENChandraliBaishyaDepartment of Studies and Research in Mathematics, Tumkur University, Tumkur-572103, Karnataka, India.0000-0002-9653-313XManisha KrishnaNaikDepartment of Studies and Research in Mathematics, Tumkur University, Tumkur-572103, Karnataka, India.0000-0002-5108-2697PremakumariR. N.Department of Studies and Research in Mathematics, Tumkur University, Tumkur-572103, Karnataka, India.Journal Article20230302Rumor spreading is the circulation of doubtful messages on the social network. Fact retrieving process that aims at preventing the spread of the rumor, appears to have a significant global impact. In this research, we have investigated a mathematical model projecting rumor spread by considering six groups of individuals namely ignorant, exposed, intentional rumor spreader, unintentional rumor spreader, stifler, and fact retriever. To represent the current abnormal and fast pattern of the message spread around various platforms, in the projected model, we have implemented the fractional derivative in the Caputo context. Using the existing theory of the fractional derivative, we have examined the theoretical aspects such as the existence and uniqueness of the solutions, the existence and stability of the rumor-free and rumor equilibrium points, and the global stability of the rumor-free equilibrium point. Computing basic reproduction numbers, we have analyzed the existence and stability of points of equilibrium. The sensitivity of basic reproduction numbers is also examined. Importance of the fact retrieving drive is highlighted by relating it to the basic reproduction number. Finally, by applying the Adams-Bashforth-Moulton method, we have presented the numerical results by capturing the profile of each of the groups under the influence of fractional derivative and investigated the impact of rumor verification rate and contact rate in controlling and preventing the rumor. With the Caputo fractional operator in the projected model, the current research highlights the significance of the fact retriever and the curb in individual contact and captures the relevant consequences.https://cmde.tabrizu.ac.ir/article_16911_b41c2f80d4f35b405dba214d6df665da.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions2662861692110.22034/cmde.2023.54283.2268ENAkbarDehghan NezhadSchool of Mathematics and Computer Science, Iran University of Science and Technology,
Narmak, Tehran, Iran.0000-0003-4200-6892MinaMoghaddam ZeabadiSchool of Mathematics and Computer Science, Iran University of Science and Technology,
Narmak, Tehran, Iran.Journal Article20221129We present a novel computational approach for computing invariant manifolds that correspond to equilibrium solutions of nonlinear parabolic partial differential equations (or PDEs). Our computational method combines Lie symmetry analysis with the parameterization method. The equilibrium solutions of PDEs and the solutions of eigenvalue problems are exactly obtained. As the linearization of the studied nonlinear PDEs at equilibrium solutions yields zero eigenvalues, these solutions are non-hyperbolic, and some invariant manifolds are center manifolds. We use the parameterization method to model the infinitesimal invariance equations that parameterize the invariant manifolds. We utilize Lie symmetry analysis to solve the invariance equations. We apply our framework to investigate the Fisher equation and the Brain Tumor growth differential equation. https://cmde.tabrizu.ac.ir/article_16921_27fa5a1ef04691fef4b60494658a6f54.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Exponentially fitted IMEX peer methods for an advection-diffusion problem2873031664110.22034/cmde.2023.53247.2248ENDajanaConteUniversity of Salerno, Fisciano, 84084, Salerno, Italy.0000-0001-8486-6861LeilaMoradiUniversity of Salerno, Fisciano, 84084, Salerno, Italy.0000-0002-1545-8263BeatricePaternosterUniversity of Salerno, Fisciano, 84084, Salerno, Italy.0000-0002-9712-1364Journal Article20220831In this paper, Implicit-Explicit (IMEX) Exponential Fitted (EF) peer methods are proposed for the numerical solution of an advection-diffusion problem exhibiting an oscillatory solution. Adapted numerical methods both in space and in time are constructed. The spatial semi-discretization of the problem is based on finite differences, adapted to both the diffusion and advection terms, while the time discretization employs EF IMEX peer methods. The accuracy and stability features of the proposed methods are analytically and numerically analyzed. https://cmde.tabrizu.ac.ir/article_16641_b69dfb5bd66d308205963120eeeec233.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Inverse coefficient problem in hyperbolic partial differential equations: An analytical and computational exploration3043131670910.22034/cmde.2023.56266.2352ENPariaSattari ShajariDepartment of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.AbdollahShidfarDepartment of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran.Behrouz ParsaMoghaddamDepartment of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.0000-0003-4957-9028Journal Article20230421This investigation centers on the analysis of an inverse hyperbolic partial differential equation, specifically addressing a coefficient inverse problem that emerges under the imposition of an over-determination condition. In order to address this challenging problem, we employ the well-established homotopy analysis technique, which has proven to be an effective and reliable approach in similar contexts. By utilizing this technique, our primary objective is to achieve an efficient and accurate solution to the inverse problem at hand. To substantiate the effectiveness and reliability of the proposed method, we present a numerical example as a practical illustration, demonstrating its applicability in real-world scenarios. https://cmde.tabrizu.ac.ir/article_16709_13bbe960e69a14a392e2324b4daa6990.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Finite element solution of a class of parabolic integro-differential equations with inhomogeneous jump conditions using FreeFEM++3143281663910.22034/cmde.2023.50871.2114ENMatthew OlayiwolaAdewoleDepartment of Computer Science and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeria.0000-0002-9130-6739Journal Article20220321The finite element solution of a class of parabolic integro–partial differential equations with interfaces is presented. The spatial discretization is based on the triangular element while a two-step implicit scheme together with the trapezoidal method is employed for time discretization. For the spatial discretization, the elements in the neighborhood of the interface are more refined such that the interface is at $\sigma$-distance from the approximate interface. The convergence rate of optimal order in L2-norm is analyzed with the assumption that the interface is arbitrary but smooth. Examples are given to support the theoretical findings with implementation on FreeFEM++.https://cmde.tabrizu.ac.ir/article_16639_aa0e530d3ca06a11d1d98b6fa5b5f8f2.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301A new Bernstein-reproducing kernel method for solving forced Duffing equations with integral boundary conditions3293371688310.22034/cmde.2023.57413.2401ENAzamGhasemiDepartment of Applied Mathematics, Faculty of Mathematical Sciences,
University of Kashan, Kashan 87317-53153, Iran.AbbasSaadatmandiDepartment of Applied Mathematics, Faculty of Mathematical Sciences,
University of Kashan, Kashan 87317-53153, Iran.0000-0002-7744-7770Journal Article20230705In the current work, a new reproducing kernel method (RKM) for solving nonlinear forced Duffing equations with integral boundary conditions is developed. The proposed collocation technique is based on the idea of RKM and the orthonormal Bernstein polynomials (OBPs) approximation together with the quasi-linearization method. In our method, contrary to the classical RKM, there is no need to use the Gram-Schmidt orthogonalization procedure and only a few nodes are used to obtain efficient numerical results. Three numerical examples are included to show the applicability and efficiency of the suggested method. Also, the obtained numerical results are compared with some results in the literature.https://cmde.tabrizu.ac.ir/article_16883_00bd02eeda68cf47d503b208a78be103.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Adaptive-grid technique for the numerical solution of a class of fractional boundary-value-problems3383491654710.22034/cmde.2023.55266.2296ENSandipMajiDepartment of Mathematics, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India.0000-0002-2220-1554SrinivasanNatesanDepartment of Mathematics, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India.0000-0001-7527-1989Journal Article20230206In this study, we numerically solve a class of two-point boundary-value-problems with a Riemann-Liouville-Caputo fractional derivative, where the solution might contain a weak singularity. Using the shooting technique based on the secant iterative approach, the boundary value problem is first transformed into an initial value problem, and the initial value problem is then converted into an analogous integral equation. The functions contained in the fractional integral are finally approximated using linear interpolation. An adaptive mesh is produced by equidistributing a monitor function in order to capture the singularity of the solution. A modified Gronwall inequality is used to establish the stability of the numerical scheme. To show the effectiveness of the suggested approach over an equidistributed grid, two numerical examples are provided.https://cmde.tabrizu.ac.ir/article_16547_603eebf1d379ea7f1966da4792be224e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Dynamics of an SEIR epidemic model with saturated incidence rate including stochastic influence3503601688510.22034/cmde.2023.56544.2365ENGKumarDepartment of Mathematics, Anurag University, Venkatapur, Hyderabad-500088, Telangana, India.KRameshDepartment of Mathematics, Anurag University, Venkatapur, Hyderabad-500088, Telangana, India.Kottakkaran SooppyNisarDepartment of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia.0000-0001-5769-4320Journal Article20230510This paper aims to develop a stochastic perturbation into SEIR (Susceptible-Exposed-Infected-Removed) epidemic model including a saturated estimated incidence. A set of stochastic differential equations is used to study its behavior, with the assumption that each population’s exposure to environmental unpredictability is represented by noise terms. This kind of randomness is considerably more reasonable and realistic in the proposed model. The current study has been viewed as strengthening the body of literature because there is less research on the dynamics of this kind of model. We discussed the structure of all equilibriums’ existence and the dynamical behavior of all the steady states. The fundamental replication number for the proposed method was used to discuss the stability of every equilibrium point; if $R_0<1$, the infected free equilibrium is resilient, and if $R_0>1$, the endemic equilibrium is resilient. The system’s value is primarily described by its ambient stochasticity, which takes the form of Gaussian white noise. Additionally, the suggested model can offer helpful data for comprehending, forecasting, and controlling the spread of various epidemics globally. Numerical simulations are run for a hypothetical set of parameter values to back up our analytical conclusions. https://cmde.tabrizu.ac.ir/article_16885_7f0c4e9ba4d70f79a08ae8939b61c773.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Synchronization a chaotic system with Quadratic terms using the contraction Method3613731688410.22034/cmde.2023.55987.2337ENMarziehPabastehDepartment of Mathematics, Payame Noor University, Tehran, Iran.BashirNaderiDepartment of Mathematics, Payame Noor University, Tehran, Iran.0000-0002-9345-5340HasanZareiDepartment of Mathematics, Payame Noor University, Tehran, IranJournal Article20230331In this article, Synchronization and control methods are discussed as essential topics in science. The contraction method is an exciting method that has been studied for the synchronization of chaotic systems with known and unknown parameters. The controller and the dynamic parameter estimation are obtained using the contraction theory to prove the stability of the synchronization error and the low parameter estimation. The control scheme does not employ the Lyapunov method. For demonstrate the ability of the proposed method, we performed a numerical simulation and compared the result with the previous literature.https://cmde.tabrizu.ac.ir/article_16884_ee0658a8b2c07311f312bc9b6dc27a7a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Numerical solution of fractional Volterra integro-differential equations using flatlet oblique multiwavelets3743911680710.22034/cmde.2023.56933.2382ENZahraShafinejhadDepartment of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil, Iran.MohammadZarebniaDepartment of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil, Iran.Journal Article20230602The presented paper investigates a new numerical method based on the characteristics of flatlet oblique multiwavelets for solving fractional Volterra integro-differential equations, in this method, first using the dual bases of the flatlet multiwavelets, the operator matrices are made for the derivative of fractional order and Volterra integral. Then, the fractional Volterra integro-differential equation reduces to a set of algebraic equations which can be easily solved. The error analysis and convergence of the presented method are discussed. Also, numerical examples will indicate the acceptable accuracy of the proposed method, which is compared with the methods used by other researchers. https://cmde.tabrizu.ac.ir/article_16807_6660b03a971ffa29ee35894c341d0226.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Novel traveling wave solutions of generalized seventh-order KdV equation and related equation3924121654610.22034/cmde.2023.53478.2254ENRiadhHedliDepartment of Mathematics
Faculty of Sciences
University of Ferhat Abbas Setif 1
Algeria.0000-0003-0988-3340FellaBerrimiDepartment of Computer Science
Faculty of Sciences
University of Ferhat Abbas Setif 1
Algeria.Journal Article20220914In this paper, we acquire novel traveling wave solutions of the generalized seventh-order Korteweg–de Vries equation and the seventh-order Kawahara equation as a special case with physical interest. Primarily, we use the advanced $\exp (-\varphi (\xi ))$-expansion method to find new exact solutions of the first equation, by considering two auxiliary equations. Then, we attain some exact solutions of the seventh-order Kawahara equation by using this method with another auxiliary equation, and also using the modified $(G^{'}/G) $-expansion method, where G satisfies a second-order linear ordinary differential equation. Additionally, utilizing the recent scientific instruments, the 2D, 3D, and contour plots are displayed. The solutions obtained in this paper include bright solitons, dark solitary wave solutions, and multiple dark solitary wave solutions. It is shown that these two methods provide an effective mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering.https://cmde.tabrizu.ac.ir/article_16546_f7411380cfa2b0e82eb35fada95a1c9f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212220240301Modified simple equation method (MSEM) for solving nonlinear (3+1)-dimensional space-time fractional equations4134241688610.22034/cmde.2023.51930.2168ENMohammad SaeedBarikbinDepartment of Mathematics, Takestan branch, Islamic Azad University, Takestan, IranJournal Article20220604In the present paper, modified simple equation method (MSEM) is implemented for obtaining exact solutions of three nonlinear (3 + 1)-dimensional space-time fractional equation, namely three types of modified Korteweg-de-Vries (mKdV) equations. Here, the derivatives are of the type of conformable fractional derivatives. The solving process produces a system of algebraic equations which is possible to be easily with no need of using software for determining unknown coefficients. Results show that this method can supply a powerful mathematical tool to construct exact solutions of mKdV equations and it can be employed for other nonlinear (3 + 1) - dimensional space-time fractional equations.https://cmde.tabrizu.ac.ir/article_16886_3907b478dcc84572a45d47e1c0bb930d.pdf