University of TabrizComputational Methods for Differential Equations2345-398212420240901Lie symmetry analysis, and exact solutions to the time–fractional Black–Scholes equation of the Caputo–type6386501817410.22034/cmde.2024.59096.2510ENRaminNajafiDepartment of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran.0000-0003-2432-2947FaribaBahramiFaculty of Mathematics and Computer Science, University of Tabriz, Tabriz, Iran.ParisaVafadarFaculty of Mathematics and Computer Science, University of Tabriz, Tabriz, Iran.Journal Article20231106In this study, the Lie symmetry analysis, and exact solutions are investigated to the fractional Black–Scholes(B-S) equations of the Caputo–type modeling of the pricing options under the absence of arbitrage and self-financing portfolio assumptions. A class of exact invariant and solitary solutions are given to B-S equations. Some examples are presented in which we use the obtained reductions to find their exact solutions.https://cmde.tabrizu.ac.ir/article_18174_a1c0ab95666569fe9a4ad2c8a006b26b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901Direct and inverse problems of ROD equation using finite element method and a correction technique6516681778910.22034/cmde.2024.57676.2417ENHanifMirzaeiFaculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.KazemGhanbariFaculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.VahidAbbasnavazFaculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.AngeloMingarelliSchool of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada.Journal Article20230723The free vibrations of a rod are governed by a differential equation of the form $(a(x)y^\prime)^\prime+\lambda a(x)y(x)=0$, where $a(x)$ is the cross sectional area and $\lambda$ is an eigenvalue parameter. Using the finite element method (FEM) we transform this equation to a generalized matrix eigenvalue problem of the form $(K-\Lambda M)u=0$ and, for given $a(x)$, we correct the eigenvalues $\Lambda$ of the matrix pair $(K,M)$ to approximate the eigenvalues of the rod equation. The results show that with step size $h$ the correction technique reduces the error from $O(h^2i^4)$ to $O(h^2i^2)$ for the $i$-th eigenvalue. We then solve the inverse spectral problem by imposing numerical algorithms that approximate the unknown coefficient $a(x)$ from the given spectral data. The cross section is obtained by solving a nonlinear system using Newton's method along with a regularization technique. Finally, we give numerical examples to illustrate the efficiency of the proposed algorithms.https://cmde.tabrizu.ac.ir/article_17789_c0b92eca11b51e1221f68e8f71aae686.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901A numerical investigation for the COVID-19 spatiotemporal lockdown-vaccination model6696861780810.22034/cmde.2024.57085.2388ENAhmed F.KouraBasic Science Department, Al-Safwa High Institute of Engineering, Egypt.Kamal R.RaslsnMathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt.KhalidK. AliMathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt.Mohamed AbozeidShaalanHigher Technological Institute, Tenth of Ramadan City, Egypt.Journal Article20230612The present article investigates a numerical analysis of COVID-19 (temporal and spatio-tempora) lockdown-vaccination models. The proposed models consist of six nonlinear ordinary differential equations as a temporal model and six nonlinear partial differential equations as a spatio-temporal model. The evaluation of reproduction number is a forecast spread of the COVID-19 pandemic. Sensitivity analysis is used to emphasize the importance of pandemic parameters. We show the stability regions of the disease-free equilibrium point and pandemic equilibrium point. We use effective methods such as central finite difference (CFD) and Runge-Kutta of fifth order (RK-5). We apply Von-Neumann stability and consistency of the numerical scheme for the spatio-temporal model. We examine and compare the numerical results of the proposed models under various parameters.https://cmde.tabrizu.ac.ir/article_17808_9b62db862b565728afd33ce3642690a4.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901A generalized adaptive Monte Carlo algorithm based on a two-step iterative method for linear systems and its application to option pricing6877021774310.22034/cmde.2024.51747.2159ENMahboubehAalaeiInsurance Research Center, Saadat Abad, Tehran, Iran.0000-0002-6138-3186Journal Article20220524In this paper, we present a generalized adaptive Monte Carlo algorithm using the Diagonal and Off-Diagonal Splitting (DOS) iteration method to solve a system of linear algebraic equations (SLAE). The DOS method is a generalized iterative method with some known iterative methods such as Jacobi, Gauss-Seidel, and Successive Overrelaxation methods as its special cases. Monte Carlo algorithms usually use the Jacobi method to solve SLAE. In this paper, the DOS method is used instead of the Jacobi method which transforms the Monte Carlo algorithm into the generalized Monte Carlo algorithm. we establish theoretical results to justify the convergence of the algorithm. Finally, numerical experiments are discussed to illustrate the accuracy and efficiency of the theoretical results. Furthermore, the generalized algorithm is implemented to price options using the finite difference method. We compare the generalized algorithm with standard numerical and stochastic algorithms to show its efficiency.https://cmde.tabrizu.ac.ir/article_17743_54ccc0f40ccfc9ead7d2266d6e691429.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901Quality theorems on the solutions of quasilinear second-order parabolic equations with discontinuous coefficients7037091780110.22034/cmde.2024.57193.2391ENSarvanHuseynovBaku State University, Baku, Azerbaijan.Journal Article20230620A class of quasilinear second-order parabolic equations with discontinuous coefficients is considered in this work. The analog of Harnack inequality is proved for the non-negative solutions of these equations.https://cmde.tabrizu.ac.ir/article_17801_bdc7d6684b508beffd426e9eae17d2f3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901Gradient estimates for a nonlinear equation under the almost Ricci soliton condition7107181774610.22034/cmde.2024.56129.2344ENSakinehHajiaghasiDepartment of Pure Mathematics, Faculty of Science,
Imam Khomeini International University,
Qazvin, Iran.ShahroudAzamiDepartment of pure Mathematics, Faculty of Sciences
Imam Khomeini International University,
Qazvin, Iran.Journal Article20230410In this paper, we study the gradient estimate for the positive solutions of the equation $\Delta u+au(\log u)^{p}+bu=f$ on an almost Ricci soliton $(M^{n},g,X,\lambda)$. In a special case, when $X=\nabla h$ for a smooth function $h$, we derive a gradient estimate for an almost gradient Ricci soliton.https://cmde.tabrizu.ac.ir/article_17746_61d8a49d26af1e89be7c358d2ecda0c9.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901An efficient computational method based on exponential B-splines for a class of fractional sub-diffusion equations7197401774710.22034/cmde.2024.57372.2398ENAnshimaSinghDepartment of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, India.0000-0001-6317-3011SunilKumarDepartment of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, India.0000-0001-9991-1012HiginioRamosScientific Computing Group, Universidad de Salamanca, Plaza de la Merced, Salamancay, 37008, Spain.0000-0003-2791-6230Journal Article20230704The primary objective of this research is to develop and analyze a robust computational method based on exponential B splines for solving fractional sub-diffusion equations. The fractional operator includes the Mittag-Leffler function of one parameter in the form of a kernel that is non-local and non-singular in nature. The current approach is based on an effective finite difference method for discretizing in time, and the exponential B-spline functions for discretizing in space. The proposed scheme is proven to be unconditionally stable and convergent. Also, the unique solvability of the method is established. Numerical simulations conducted for multiple test examples validate the agreement between the obtained theoretical results and the corresponding numerical outcomes.https://cmde.tabrizu.ac.ir/article_17747_71f4d91173b85bcd4d83a5ff87b558bd.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901Existence and uniqueness of positive solutions for a Hadamard fractional integral boundary value problem7417481773610.22034/cmde.2023.51601.2150ENAsgharAhmadkhanluDepartment of Mathematics, Faculty of Basic Science, Azarbaijan Shahid Madani University, Km 35 Tabriz-Maragheh Road, Tabriz, Iran.ShabnamJamshidzadehDepartment of Mathematics, Faculty of Basic Science, Azarbaijan Shahid Madani University, Km 35 Tabriz-Maragheh Road, Tabriz, Iran.Journal Article20220515The main aim of this paper is to study a kind of boundary value problem with an integral boundary condition including Hadamard-type fractional differential equations. To do this, upper and lower solutions are used to guarantee their existence, and Schauder’s fixed point theorem is used to prove the uniqueness of the positive solutions to this problem. An illustrated example is presented to explain the theorems that have been proved.https://cmde.tabrizu.ac.ir/article_17736_2ab12d48bfa61389523d3a11241fded5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901Solitary waves with two new nonlocal boussinesq types equations using a couple of integration schemes7497621776710.22034/cmde.2024.57112.2390ENIslamSamirDepartment of Physics and Mathematics Engineering, Faculty of Engineering,
Ain Shams University-11517, Cairo, Egypt.Ahmed H.ArnousAin Shams University-11517, Cairo, Egypt.Ahmed M.ElsherbenyDepartment of Physics and Mathematics Engineering, Faculty of Engineering,
Ain Shams University-11517, Cairo, Egypt.MohammadMirzazadehDepartment of Engineering Sciences, Faculty of Technology and Engineering, East
of Guilan, University of Guilan, Rudsar-Vajargah, Iran.Mir SajjadHashemiDepartment of Computer Engineering, Biruni University, 34010 Istanbul, Turkey.MostafaEslamiDepartment of Applied Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran.0000-0001-6168-7916Journal Article20230614The Boussinesq equation and its related types are able to provide a significant explanation for a variety of different physical processes that are relevant to plasma physics, ocean engineering, and fluid flow. Within the framework of shallow water waves, the aim of this research is to find solutions for solitary waves using newly developed nonlocal models of Boussinesq’s equations. The extraction of bright and dark solitary wave solutions along with bright–dark hybrid solitary wave solutions is accomplished through the implementation of two integration algorithms. The general projective Riccati equations method and the enhanced Kudryashov technique are the ones that have been implemented as techniques. The enhanced Kudryashov method combines the benefits of both the original Kudryashov method and the newly developed Kudryashov method, which may generate bright, dark, and singular solitons. The Projective Riccati structure is determined by two functions that provide distinct types of hybrid solitons. The solutions get increasingly diverse as these functions are combined. The techniques that were applied are straightforward and efficient enough to provide an approximation of the solutions discovered in the research. Furthermore, these techniques can be utilized to solve various kinds of nonlinear partial differential equations in mathematical physics and engineering. In addition, plots of the selected solutions in three dimensions, two dimensions, and contour form are provided.https://cmde.tabrizu.ac.ir/article_17767_9b570a142b07d83c5343d81ac87a06e9.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901An innovative computational approach for fuzzy space-time fractional Telegraph equation via the new iterative transform method7637791773710.22034/cmde.2024.52246.2185ENKishor AshokKshirsagarDepartment of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.0000-0001-8123-8608Vasant RNikamDepartment of Mathematics, Loknete Vyankatrao Hiray Arts, Science and Commerce College, Nashik, Maharashtra, India.Shrikisan BGaikwadDepartment of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.Shivaji AshokTarateDepartment of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.0000-0001-7942-5682Journal Article20220624In this paper, the Fuzzy Sumudu Transform Iterative method (FSTIM) was applied to find the exact fuzzy solution of the fuzzy space-time fractional telegraph equations using the Fuzzy Caputo Fractional Derivative operator. The Telegraph partial differential equation is a hyperbolic equation representing the reaction-diffusion process in various fields. It has applications in engineering, biology, and physics. The FSTIM provides a reliable and efficient approach for obtaining approximate solutions to these complex equations improving accuracy and allowing for fine-tuning and optimization for better approximation results. The work introduces a fuzzy logic-based approach to Sumudu transform iterative methods, offering flexibility and adaptability in handling complex equations. This innovative methodology considers uncertainty and imprecision, providing comprehensive and accurate solutions, and advancing numerical methods. Solving the fuzzy space-time fractional telegraph equation used a fusion of the Fuzzy Sumudu transform and iterative approach. Solution of fuzzy fractional telegraph equation finding analytically and interpreting its results graphically. Throughout the article, whenever we draw graphs, we use Mathematica Software. We successfully employed FSTIM, which is elegant and fast to convergence.https://cmde.tabrizu.ac.ir/article_17737_1e84092f025ee6ef85e8645381d636ef.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901A numerical approach for solving the Fractal ordinary differential equations7807901773410.22034/cmde.2023.55868.2331ENNooshinPashmakianDepartment of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran.AliFarajzadehDepartment of Mathematics, Razi University, Kermanshah, Iran.NordinParandinDepartment of Mathematics, Kermanshah Branch, Islamic
Azad University, Kermanshah, Iran.NasrinKaramikabirDepartment of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran.Journal Article20230316In this paper, fractal differential equations are solved numerically. Here, the typical fractal equation is considered as follows:<br />$$\frac{du(t)}{dt^{\alpha}}=f\left\{ t,u(t)\right\},~~~\alpha>0,$$<br /> $f$ can be a nonlinear function and the main goal is to get $u(t)$. The continuous and discrete modes of this method have differences, so the subject must be carefully studied. How to solve fractal equations in their discrete form will be another goal of this research and also its generalization to higher dimensions than other aspects of this research.https://cmde.tabrizu.ac.ir/article_17734_553432829c9da6375ee08c7c2c077198.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901Eigenvalue intervals of parameters for iterative systems of nonlinear Hadamard fractional boundary value problems7918071778810.22034/cmde.2024.57505.2407ENBodduMuralee Bala KrushnaDepartment of Mathematics, MVGR College of Engineering(Autonomous), Vizianagaram, 535005, India.KhuddushMahammadDepartment of Mathematics, Chegg India Pvt. Ltd., Visakhapatnam, 530002, Andhra Pradesh, India.0000-0002-1236-8334KapulaRajendra PrasadDepartment of Applied Mathematics, College of Science and Technology, Andhra University, Visakhapatnam, 530003, India.Journal Article20230712This study uses a classic fixed point theorem of cone type in a Banach space to identify the eigenvalue intervals of parameters for which an iterative system of a Hadamard fractional boundary value problem has at least one positive solution. To the best of our knowledge, no attempt has been made to obtain such results for Hadamard-type problems in the literature. We provided an example to illustrate the feasibility of our findings in order to show how effective they are. https://cmde.tabrizu.ac.ir/article_17788_a250b1e014f35788f25262bf8132d0f6.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901Higher-order multi-step Runge-Kutta-Nystr\"om methods with frequency-dependent coefficients for second-order initial value problem $u^{\prime \prime}=f(x,u,u^{\prime})$8088261780010.22034/cmde.2024.57684.2418ENAthraa AbdulsalamJasim1- Department of Mathematics and Computer Applications, College of Sciences,
Al-Nahrain University, Jadriya, Baghdad, Iraq.\\
2- Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.NorazakSenu1- Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.\\
2- Department of Mathematics and Statistics, Universiti Putra Malaysia, 43400
Serdang, Selangor, Malaysia.ZanariahAbdul Majid1- Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.\\
2- Department of Mathematics and Statistics, Universiti Putra Malaysia, 43400
Serdang, Selangor, Malaysia.Nik Mohd AsriNik Long1- Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.\\
2- Department of Mathematics and Statistics, Universiti Putra Malaysia, 43400
Serdang, Selangor, Malaysia.Journal Article20230724In this study, for the numerical solution of general second-order ordinary differential equations (ODEs) that exhibit oscillatory or periodic behavior, fifth- and sixth-order explicit multi-step Runge-Kutta-Nystr¨om (MSGRKN) methods, respectively, are constructed. The parameters of the proposed methods rely on the frequency ω of each problem whose solution is a linear combination of functions {e(iωx), e(−iωx)} or {cos(ωx), sin(ωx)}. The study also includes an analysis of the linear stability of the suggested methods. The numerical results indicate the efficiency of the proposed methods in solving such problems compared to methods with similar characteristics in the literature.https://cmde.tabrizu.ac.ir/article_17800_32aa02f32cbf31a75668ec8846a38e50.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901A Chebyshev pseudo-spectral based approach for solving Troesch’s problem with convergence analysis8278411780510.22034/cmde.2024.56313.2355ENMehrdadGhaznaviFaculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.0000-0002-6274-7076Mohammad HadiNoori SkandariFaculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.Journal Article20230424In this article, the Chebyshev pseudo-spectral (CPS) method is presented for solving Troesch’s problem, which is a singular, highly sensitive, and nonlinear boundary problem and occurs in the consideration of the confinement of a plasma column by radiation pressure. Here, a continuous time optimization (CTO) problem corresponding to Troesch’s problem is first proposed. Then, the Chebyshev pseudo-spectral method is used to convert the CTO problem to a discrete-time optimization problem its optimal solution can be found by nonlinear programming methods. The feasibility and convergence of the generated approximate solutions are analyzed. The proposed method is used to solve various kinds of Troesch’s equations. The obtained results have been compared with approximate solutions resulting from well known numerical methods. It can be confirmed that the numerical solutions resulting from this method are completely acceptable and accurate, compared with other techniques. https://cmde.tabrizu.ac.ir/article_17805_cb5e503560e55c66e9c2cb5e7e7d1f83.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901Generalization of Katugampola fractional kinetic equation involving incomplete H-function8428561775610.22034/cmde.2024.57294.2395ENNishant-Department of Mathematics, Malaviya National Institute of Technology Jaipur, India.SanjayBhatterDepartment of Mathematics, Malaviya National Institute of Technology Jaipur, India.Sunil DuttPurohitDepartment of HEAS (Mathematics), Rajasthan Technical University, India. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon.0000-0002-1098-5961Journal Article20230627In this article, Katugampola fractional kinetic equation (KE) has been expressed in terms of polynomials along with incomplete H-function, incomplete Meijer’s G-function, incomplete Fox-Wright function, and incomplete generalized hypergeometric function, weighing the novel significance of the fractional KE that appear in a variety of scientific and engineering scenarios. τ-Laplace transform is used to solve the Kathugampola fractional KE. The obtained solutions have been presented with some real values and the simulation was done via MATLAB. Furthermore, the numerical and graphical interpretations are also mentioned to illustrate the main results. Each of the obtained conclusions is of a general nature and is capable of generating the solutions to several fractional KE.https://cmde.tabrizu.ac.ir/article_17756_079b9b23718981a1820c36278e3a7a06.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398212420240901The use of the Sinc-collocation method for solving steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether8578651774510.22034/cmde.2024.55413.2304ENFatemehZabihiDepartment of Applied Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, 87317-51167, Iran.Journal Article20230214In this paper, the Sinc-collocation method is applied to solve a system of coupled nonlinear differential equations that report the chemical reaction of carbon dioxide CO2 and phenyl glycidyl ether in solution. The model has Dirichlet and Neumann boundary conditions. The given scheme has transformed this problem into some algebraic equations. The approach is quite simple to handle and the new numerical solutions are compared with some known solutions, which shows that the new technique is accurate and efficient.https://cmde.tabrizu.ac.ir/article_17745_dde25d86bcccad62a79a463e02261b1d.pdf