University of TabrizComputational Methods for Differential Equations2345-398212320240501Numerical solution of Burgers' equation with nonlocal boundary condition: Use of Keller-Box scheme4254371716210.22034/cmde.2023.54380.2271ENAmirrezaAzadSchool of Mechanical Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran.Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, M5S 3G8, Canada.EhsanYaghoubiSchool of Mechanical Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran.AzadehJafariSchool of Mechanical Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran.0000-0002-1331-6640Journal Article20221206In this paper, we transform the given nonlocal boundary condition problem into a manageable local equation. By introducing an additional transformation of the variables, we can simplify this equation into conformable Burgers’ equation. Thus, the Keller Box method is used as a numerical scheme to solve the equation. A comparison is made between numerical results and the analytic solution to validate the results of our proposed method.https://cmde.tabrizu.ac.ir/article_17162_17cc32965f92b1bb8c5bfac4f6e1e19a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501Upper and lower solutions for fractional integro-differential equation of higher-order and with nonlinear boundary conditions4384531716510.22034/cmde.2023.56512.2363ENAbdelatiEl AllaouiMISCOM, National School of Applied Sciences, Cadi Ayyad University, Marrakech, Morocco0000-0001-7197-4951YoussefAllaouiLMACS, Faculty of Sciences and Technics, Sultan Moulay Slimane University, Beni Mellal, MoroccoSaidMellianiLMACS, Faculty of Sciences and Technics, Sultan Moulay Slimane University, Beni Mellal, Morocco0000-0002-5150-1185HamidEl KhalfiLab. LMRI, FP of Khouribga, Sultan Moulay Slimane University, MorrocoJournal Article20230509This paper delves into the identification of upper and lower solutions for a high-order fractional integro-differential equation featuring non-linear boundary conditions. By introducing an order relation, we define these upper and lower solutions. Through a rigorous approach, we demonstrate the existence of these solutions as the limits of sequences derived from carefully selected problems, supported by the application of Arzel\`a-Ascoli's theorem. To illustrate the significance of our findings, we provide an illustrative example. This research contributes to a deeper understanding of solutions in the context of complex fractional integro-differential equations.https://cmde.tabrizu.ac.ir/article_17165_fac8a371c1e287d299703d5691fceff5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501Efficiency of vaccines for COVID-19 and stability analysis with fractional derivative4544701692610.22034/cmde.2023.56465.2359ENMohammad EsmaelSameiDepartment of Mathematics, Faulty of Basic Science, Bu-Ali Sina University, Hamedan 65178-38695, Iran.0000-0002-5450-3127LotfollahKarimiDepartment of Mathematics, Hamedan University of Technology, Hamedan, Iran.MohammedK. A. KaabarGofa Camp, Near Gofa Industrial College and German Adebabay, Nifas Silk-Lafto, 26649 Addis Ababa, Ethiopia.Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia.0000-0003-2260-0341RoyaRaeisiDepartment of Pediatrics, Hamadan University of Medical Science, Hamadan, Iran.JehadAlzabutDepartment of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia.Department of Industrial Engineering, OST˙IM Technical University, 06374 Ankara, Turkey.0000-0002-5262-1138Francisco MartinezGonzalezDepartment of Applied Mathematics and Statistics, Technological University of Cartagena, Cartagena 30203, Spain.Journal Article20230506The objectives of this study are to develop the SEIR model for COVID-19 and evaluate its main parameters such as therapeutic vaccines, vaccination rate, and effectiveness of prophylactic. Global and local stability of the model and numerical simulation are examined. The local stability of equilibrium points was classified. A Lyapunov function is constructed to analyze the global stability of the disease-free equilibrium. The simulation part is based on two situations, including the USA and Iran. Our results provide a good contribution to the current research on this topic.https://cmde.tabrizu.ac.ir/article_16926_fcc84bfd3bfc2236c23f1955c8d3660f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501An efficient algorithm for computing the eigenvalues of conformable Sturm-Liouville problem4714831711410.22034/cmde.2023.57436.2403ENHanifMirzaeiFaculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.MahmoodEmamiFaculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.KazemGhanbariFaculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.MohammadShahriariDepartment of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran.Journal Article20230708In this paper, Computing the eigenvalues of the Conformable Sturm-Liouville Problem (CSLP) of order $2 \alpha$, $\frac{1}{2}<\alpha \leq 1$, and dirichlet boundary conditions is considered. For this aim, CSLP is discretized to obtain a matrix eigenvalue problem (MEP) using finite element method with fractional shape functions. Then by a method based on the asymptotic form of the eigenvalues, we correct the eigenvalues of MEP to obtain efficient approximations for the eigenvalues of CSLP. Finally, some numerical examples to show the efficiency of the proposed method are given. Numerical results show that for the $n$th eigenvalue, the correction technique reduces the error order from $O(n^4h^2)$ to $O(n^2h^2)$.https://cmde.tabrizu.ac.ir/article_17114_243b85e9d190741b80f2ad25cd182df0.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501Wong-Zakai approximation of stochastic Volterra integral equations4845011733110.22034/cmde.2023.58696.2485ENMinooKamraniDepartment of Mathematics, Faculty of Science, Razi university, Kermanshah, Iran.Journal Article20231002This study aims to investigate a stochastic Volterra integral equation driven by fractional Brownian motion with Hurst parameter $H\in (\frac 12, 1)$. We employ the Wong-Zakai approximation to simplify this intricate problem, transforming the stochastic integral equation into an ordinary integral equation. Moreover, we consider the convergence and the rate of convergence of the Wong-Zakai approximation for this kind of equation.https://cmde.tabrizu.ac.ir/article_17331_14c412acb7215b0acb9dfbf8a110f38b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501Some delta q−fractional linear dynamic equations and a generalized delta q−Mittag-Leffler function5025101701710.22034/cmde.2023.56037.2339ENNada K.MahdiDepartment of Mathematics, College of Science, University of Basrah, Basrah, Iraq.0009-0004-3399-503XAyad K.KhudairDepartment of Mathematics, College of Science, University of Basrah, Basrah, Iraq.0000-0001-8723-2223Journal Article20230404In this paper, we introduce a generalized delta q−Mittag-Leffler function. Also, we solve some Caputo delta q−fractional dynamic equations and these solutions are expressed by means of the newly introduced delta q−Mittag-Leffler function. https://cmde.tabrizu.ac.ir/article_17017_16d16a485a071ea845f518e2af4ae6fc.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501On fractional linear multi-step methods for fractional order multi-delay nonlinear pantograph equation5115221711510.22034/cmde.2023.58018.2444ENMoslemValizadehMathematics Department, Tabriz Branch, Islamic Azad University,
Tabriz, Iran.YaghoubMahmoudiMathematics Department, Tabriz Branch, Islamic Azad University,
Tabriz, Iran.FarhadDastmalchi SaeiMathematics Department, Tabriz Branch, Islamic Azad University,
Tabriz, Iran.Journal Article20230816This paper presents the development of a series of fractional multi-step linear finite difference methods (FLMMs) designed to address fractional multi-delay pantograph differential equations of order $0 < \alpha \leq 1$. These $p$ FLMMs are constructed using fractional backward differentiation formulas of first and second orders, thereby facilitating the numerical solution of fractional differential equations. Notably, we employ accurate approximations for the delayed components of the equation, guaranteeing the retention of stability and convergence characteristics in the proposed $p$-FLMMs. To substantiate our theoretical findings, we offer numerical examples that corroborate the efficacy and reliability of our approach.https://cmde.tabrizu.ac.ir/article_17115_3ffc601012b725cd33dd7edfbc6f987d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501A mathematical study on the non-linear boundary value problem of a porous fin5235431713210.22034/cmde.2023.57934.2435ENVembuAnanthaswamyResearch Centre and PG Department of Mathematics, The Madura College, Madurai, Tamil Nadu, India.0000-0002-2938-8745Ramaiya RamalingamSubanyaResearch Centre and PG Department of Mathematics, The Madura College, Madurai, Tamil Nadu, India.SathyamoorthySivasankariResearch Centre and PG Department of Mathematics, The Madura College, Madurai, Tamil Nadu, India.Journal Article20230810An analytical study of two different models of rectangular porous fins are investigated using a new approximate analytical method, the Ananthaswamy-Sivasankari method. The obtained results are compared with the numerical solution, which results in a very good agreement. The impacts of several physical parameters involved in the problem are interlined graphically. Fin efficiency and the heat transfer rate are also calculated and displayed. The result obtained by this method is in the most explicit and simple form. The convergence of the solution determined is more accurate as compared to various analytical and numerical methods.https://cmde.tabrizu.ac.ir/article_17132_20c532d3cd074e9f0dbec7d82b055178.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501A new perspective for the Quintic B-spline collocation method via the Lie-Trotter splitting algorithm to solitary wave solutions of the GEW equation5445601775910.22034/cmde.2024.58169.2454ENMelikeKartaDepartment of Mathematics, Faculty of Science and Arts, Ağrı İbrahim Çeçen University, Ağrı, Turkey.0000-0003-3412-4370Journal Article20230828A hybrid method utilizing the collocation technique with B-splines and Lie-Trotter splitting algorithm applied for 3 model problems which include a single solitary wave, two solitary wave interaction, and a Maxwellian initial condition is designed for getting the approximate solutions for the generalized equal width (GEW) equation. Initially, the considered problem has been split into 2 sub-equations as linear $U_t=\hat{A}(U)$ and nonlinear $U_t=\hat{B}(U)$ in the terms of time. After, numerical schemes have been constructed for these sub-equations utilizing the finite element method (FEM) together with quintic B-splines. Lie-Trotter splitting technique $\hat{A}o\hat{B}$ has been used to generate approximate solutions of the main equation. The stability analysis of acquired numerical schemes has been examined by the Von Neumann method. Also, the error norms $L_2$ and $L_\infty$ with mass, energy, and momentum conservation constants $I_1$, $I_2$, and $I_3$, respectively are calculated to illustrate how perfect solutions this new algorithm applied to the problem generates and the ones produced are compared with those in the literature. These new results exhibit that the algorithm presented in this paper is more accurate and successful, and easily applicable to other non-linear partial differential equations (PDEs) as the present equation.https://cmde.tabrizu.ac.ir/article_17759_39e7c9edc88bb5b24187f3d0365a31af.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501Plasma particles dispersion based on Bogoyavlensky-Konopelchenko mathematical model5615701774010.22034/cmde.2023.54839.2280ENAhmed S.Rashed1. Department of Physics and Engineering Mathematics Department, Faculty of Engineering, Zagazig University, Egypt.
2. Faculty of Engineering, Delta University for Science and Technology, Gamasa, Egypt.MokhtarMohamedFaculty of Engineering, Delta University for Science and Technology, Gamasa, Egypt.MustafaIncDepartment of Mathematics, Firat University, 23119 Elazig, Turkiye.Journal Article20230106An optimal system of Lie infinitesimals has been used in an investigation to find a solution to the (2+1)-dimensional Bogoyavlensky-Konopelchenko equation (BKE). This investigation was conducted to characterize certain fantastic characteristics of plasma-particle dispersion. A careful investigation into the Lie space with an unlimited number of dimensions was carried out to locate the relevant arbitrary functions. When developing accurate solutions for the BKE, it was necessary to establish an optimum system that could be employed in single, double, and triple combination forms.There were some fantastic wave solutions developed, and these were depicted visually. The Optimal Lie system demonstrates that it can obtain many accurate solutions to evolution equations.https://cmde.tabrizu.ac.ir/article_17740_bb1e39ebfbaf0b0e1ccbd30354c36c6b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501A numerical approach for solving Caputo-Prabhakar distributed-order time-fractional partial differential equation5715841741710.22034/cmde.2024.57844.2426ENMohsenKhastehDepartment of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran.Amir HoseinRefahi SheikhaniDepartment of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran.0000-0003-1664-5471FarhadShariffarDepartment of Applied Mathematics, Fouman and Shaft Branch, Islamic Azad University, Fouman, Iran.Journal Article20230803In this paper, we proposed a numerical method based on the shifted fractional order Jacobi and trapezoid methods to solve a type of distributed partial differential equations. The fractional derivatives are considered in the Caputo-Prabhakar type. By shifted fractional-order Jacobi polynomials our proposed method can provide highly accurate approximate solutions by reducing the problem under study to a set of algebraic equations which is technically simpler to handle. In order to demonstrate the error estimates, several lemmas are provided. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.https://cmde.tabrizu.ac.ir/article_17417_3bce8592bc6d36144cbc4fcd437ca9dd.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501Convolutional neural network-based high capacity predictor estimation for reversible data embedding in cloud network5855981773510.22034/cmde.2023.59134.2512ENPrasadCNResearch Scholar, Department of Computer Science, Chirashree Institute of Research and Development (CIRD), University of Mysore, Karnataka,
India.RSuchithraResearch Scholar, Department of Computer Science, Chirashree Institute of Research and Development (CIRD), University of Mysore, Karnataka,
India.Journal Article20231107This paper proposes a reversible data embedding algorithm in encrypted images of cloud storage where the embedding was performed by detecting a predictor that provides a maximum embedding rate. Initially, the scheme generates trail data which are embedded using the prediction error expansion in the encrypted training images to obtain the embedding rate of a predictor. The process is repeated for different predictors from which the predictor that offers the maximum embedding rate is estimated. Using the estimated predictor as the label the Convolutional neural network (CNN) model is trained with the encrypted images. The trained CNN model is used to estimate the best predictor that provides the maximum embedding rate. The estimation of the best predictor from the test image does not use the trail data embedding process. The evaluation of proposed reversible data hiding uses the datasets namely BossBase and BOWS-2 with the metrics such as embedding rate, SSIM, and PSNR. The proposed predictor classification was evaluated with the metrics such as classification accuracy, recall, and precision. The predictor classification provides an accuracy, recall, and precision of 92.63\%, 91.73\%, and 90.13\% respectively. The reversible data hiding using the proposed predictor selection approach provides an embedding rate of 1.955 bpp with a PSNR and SSIM of 55.58dB and 0.9913 respectively.https://cmde.tabrizu.ac.ir/article_17735_66ba4a4dfc426e0289bde97116daca2c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501Efficient family of three-step with-memory methods and their dynamics5996091702010.22034/cmde.2023.55761.2324ENValiTorkashvandDepartment of Mathematics, Shahr-e-Qods Branch, Islamic Azad University, Shahr-e-Qods, Iran.Department of Mathematics , Farhangian University , Tehran, Iran.0000-0001-8033-8279ManochehrKazemiDepartment of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, Iran.MasoudAzimiDepartment of Mathematics , Farhangian University , Tehran, Iran.Journal Article20230308In this work, we have proposed a general manner to extend some two-parametric with-memory methods to obtain simple roots of nonlinear equations. Novel improved methods are two-step without memory and have two self-accelerator parameters that do not have additional evaluation. The methods have been compared with the nearest competitions in various numerical examples. Anyway, the theoretical order of convergence is verified. The basins of attraction of the suggested methods are presented and corresponded to explain their interpretation.https://cmde.tabrizu.ac.ir/article_17020_21ab8685b4dcc441f4c07b2fe5a0cca2.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501Approximate solutions of inverse Nodal problem with conformable derivative6106231692410.22034/cmde.2023.56851.2380ENShahrbanooAkbarpoorDepartment of Mathematics, Jouybar Branch, Islamic Azad University, Jouybar, Iran.Abdol HadiDabbaghianDepartment of Mathematics, Neka Branch, Islamic Azad University, Neka, Iran.Journal Article20230527Our research is about the Sturm-Liouville equation which contains conformable fractional derivatives of order $\alpha \in (0,1]$ in lieu of the ordinary derivatives. First, we present the eigenvalues, eigenfunctions, and nodal points, and the properties of nodal points are used for the reconstruction of an integral equation. Then, the Bernstein technique was utilized to solve the inverse problem, and the approximation of solving this problem was calculated. Finally, the numerical examples were introduced to explain the results. Moreover, the analogy of this technique is shown in a numerical example with the Chebyshev interpolation technique .https://cmde.tabrizu.ac.ir/article_16924_219407085387722f9517ae85288fcddf.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212320240501The complex SEE transform technique in difference equations and differential difference equations6246371782810.22034/cmde.2024.60055.2561ENEman AjelMansourDepartment of Electrical Technologies, Southern Technical University, Technical Institute Nasiriyah, Iraq.
Iraq.Emad AbbasKuffiDepartment of Mathematics, College of Basic Education Mustansiriyah University, Baghdad, Iraq.Journal Article20240113Differential equations are used to represent different scientific problems are handled efficiently by integral transformations, where integral transforms represent an easy and effective tool for solving many problems in the mentioned fields. This work utilizes the integral transform of the Complex SEE integral transformation to provide an efficient solution method for the difference and differential-difference equations by benefiting from the properties of this complex transform to solve some problems related to difference and differential-difference equations. The 3D, contour and 2D surfaces, as well as the related density plot surfaces of some acquired data, are used to draw the physical aspect of the obtained findings. The proposed approach offers an efficient and rapid solution for addressing the inherent complexity of differential-difference problems with initial conditions.https://cmde.tabrizu.ac.ir/article_17828_a125749a01e9472b73dc62dfd46d6bde.pdf