Article
Mathematics, Applied
Anshima Singh, Sunil Kumar, Jesus Vigo-Aguiar
Summary: This article introduces two new approximations (CPL2-1 sigma and CPL-2 formulas) for the Caputo-Prabhakar fractional derivative and proves their error bounds. These approximations are then applied in the numerical treatment of a reaction-diffusion problem with variable coefficients, and the stability and convergence of the numerical schemes are thoroughly analyzed using the discrete energy method. The numerical results demonstrate the feasibility and superiority of the proposed schemes.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Computer Science, Interdisciplinary Applications
Deeksha Singh, Farheen Sultana, Rajesh K. Pandey
Summary: This work aims to numerically approximate the Caputo-Prabhakar derivative and use it to solve the time-fractional advection-diffusion equation defined in the Caputo-Prabhakar sense. Two schemes are proposed to approximate the time-fractional derivative using different interpolation functions, and their convergence order and analytical error bounds are discussed. Numerical examples validate the feasibility and stability of these schemes.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2022)
Article
Mathematics, Applied
Anatoly A. Alikhanov, Chengming Huang
Summary: This paper focuses on constructing L2 type difference analog of the Caputo fractional derivative, studying its fundamental features, and using it to generate difference schemes with different orders in space and time for time fractional diffusion equations. The stability and convergence of the schemes are proven, and numerical computations support the obtained results.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Eric Ngondiep
Summary: This paper proposes a high-order numerical method for solving the multidimensional convection-diffusion-reaction equation with time-fractional derivative. The stability and error estimates of the method are analyzed, showing its unconditional stability and temporal accuracy of order O(tau(2+alpha)), where tau is the time step and 0 < alpha < 1. Numerical experiments confirm the theory and demonstrate the convergence accuracy of the proposed scheme to be O(tau(2+alpha) + h(4)), with h representing the space step.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Sadia Arshad, Mubashara Wali, Jianfei Huang, Sadia Khalid, Nosheen Akbar
Summary: In this article, a finite difference scheme and a fourth-order approximation are examined for solving a time-fractional diffusion equation with fourth-order derivative in space, subject to homogeneous and non-homogeneous boundary conditions. The proposed numerical scheme is shown to be unconditionally stable and have a convergence accuracy of order O(tau(2) + h(4)), as demonstrated through theoretical analysis and numerical examples.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2022)
Article
Mathematics
Saadoune Brahimi, Ahcene Merad, Adem Kilicman
Summary: This paper investigates a Caputo time fractional advection-diffusion equation with nonhomogeneous integral-type boundary conditions and proves the existence and uniqueness of the solution using the a priori estimate method. The approximate solution is established by a combination of the finite difference method and numerical integration, and numerical tests are conducted to demonstrate the usefulness of the obtained results.
Article
Mathematics, Applied
Alessandra Jannelli
Summary: In this paper, an adaptive procedure is proposed for solving time-fractional advection-diffusion-reaction equations involving the Caputo derivative, focusing on adaptivity in the time direction by defining a step size selection function based on the local behavior of the solution. The new approach is easy to implement with low computational cost, and test problems confirm the accuracy and efficiency of the step-size selection procedure for solving fractional partial differential equations.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Engineering, Multidisciplinary
Qammar Rubbab, Mubbashar Nazeer, Fayyaz Ahmad, Yu-Ming Chu, M. Ijaz Khan, Seifedine Kadry
Summary: The study focuses on the unsteady fractional advection-diffusion equation in cylindrical geometry with time-exponential concentration on a cylindrical surface. The Caputo-Fabrizio time-fractional derivative is used for modeling, and analytical solutions for solute concentration are obtained through integral transformations. Numerical schemes and comparison are presented to investigate the impact of memory and drift velocity on solute concentration.
ALEXANDRIA ENGINEERING JOURNAL
(2021)
Article
Mathematics, Applied
Reetika Chawla, Komal Deswal, Devendra Kumar, Dumitru Baleanu
Summary: This paper presents a new approach to investigate the time-fractional advection-dispersion equation extensively used in studying transport processes. A new modified fractional derivative operator, based on Atangana-Baleanu's definition of a derivative in the Caputo sense, is proposed, involving singular and non-local kernels. A numerical approximation of this new operator is provided and applied to an advection-dispersion equation. Through Fourier analysis, it is proved that the proposed scheme is unconditionally stable. Numerical examples validate the theoretical results and demonstrate the proficiency of the numerical scheme.
Article
Mathematics, Applied
Mubashara Wali, Sadia Arshad, Sayed M. Eldin, Imran Siddique
Summary: In this study, the approximate solutions for time-space fractional linear and nonlinear diffusion equations are obtained. A finite difference approach is used to solve both linear and nonlinear fractional order diffusion problems. The Riesz fractional derivative in space is approximated using a centered difference scheme. The stability and convergence of the proposed scheme are analyzed, and the results show that the recommended method converges at a rate of O(delta t2 + h2) for mesh size h and time steps delta t. The application of the model is also examined through graphic results and numerical examples.
Article
Computer Science, Interdisciplinary Applications
Komal Taneja, Komal Deswal, Devendra Kumar
Summary: This paper presents a compact finite difference scheme for solving time-fractional diffusion-advection-reaction equations. The scheme utilizes temporal graded and harmonic meshes to recover the accuracy lost by traditional numerical methods near the initial time. The stability and convergence of the proposed numerical technique are analyzed, and three numerical experiments confirm its proficiency and effectiveness.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Golsa Sayyar, Seyed Mohammad Hosseini, Farinaz Mostajeran
Summary: This paper introduces a high-order approach for solving time-space fractional diffusion equations, demonstrating its stability and convergence through numerical examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics
Elsayed Mahmoud, Temirkhan S. Aleroev
Summary: This article presents the analytical and numerical solution of a one-dimensional space-time fractional advection diffusion equation. The analytical solution is carried out using the separation of variables method, and the numerical solution is based on constructing the Crank-Nicolson finite difference scheme. The convergence and unconditional stability of the solution are investigated.
Article
Mathematics, Applied
Adivi Sri Venkata Ravi Kanth, Neetu Garg
Summary: This paper focuses on numerically solving the variable coefficient multiterm time fractional advection-diffusion equation using exponential B-splines. The temporal part is discretized using the Crank-Nicolson method and spatial part by exponential B-splines. The results show unconditional stability, convergence rates, and superiority over other methods in both time and space directions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Engineering, Multidisciplinary
Mir Sajjad Hashemi, Mohammad Mirzazadeh, Mustafa Bayram, Sayed M. El Din
Summary: This paper introduces a numerical technique for solving the Cauchy non-homogeneous time-fractional diffusion-wave equation. The method utilizes shifted Chebyshev polynomials to collocate the problem in the time dimension, and introduces certain functions to homogenize the resulting system of ordinary differential equations, producing an approximate solution to the problem. Through extensive experimentation, the effectiveness and accuracy of the proposed technique are demonstrated, highlighting its efficiency and precision in approximating solutions. The convergence of the method is also investigated, providing theoretical insights into its performance. Overall, this paper contributes a valuable numerical tool for addressing the Cauchy non-homogeneous time-fractional diffusion-wave equation, offering enhanced efficiency and accuracy compared to existing methods.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Automation & Control Systems
Yanan Qiu, Xiaogeng Liang, Zhiyong Dai, Jianxiong Cao, YangQuan Chen
IET CONTROL THEORY AND APPLICATIONS
(2015)
Article
Mathematics, Applied
Jianxiong Cao, Changpin Li, YangQuan Chen
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2015)
Article
Mathematics, Applied
Jianxiong Cao, Yanan Qiu
APPLIED MATHEMATICS LETTERS
(2016)
Article
Mathematics, Applied
Jianxiong Cao, Changpin Li, YangQuan Chen
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2015)
Article
Physics, Multidisciplinary
Jianxiong Cao, Changpin Li
CENTRAL EUROPEAN JOURNAL OF PHYSICS
(2013)
Article
Mathematics, Applied
Qihong Shi, Suqin Chang, Xiaobing Zhang, Jianxiong Cao
ADVANCES IN DIFFERENCE EQUATIONS
(2018)
Article
Mathematics, Applied
Jianxiong Cao, Guojie Song, Jie Wang, Qihong Shi, Sujing Sun
APPLIED MATHEMATICS LETTERS
(2019)
Article
Mathematics, Applied
Yong-Liang Zhao, Pei-Yong Zhu, Xian-Ming Gu, Xi-Le Zhao, Jianxiong Cao
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Ke Li, Jianxiong Cao, Jin-Man He
Article
Mathematics, Applied
Yihong Wang, Jianxiong Cao, Junliang Fu
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2020)
Article
Mathematics, Applied
Jianxiong Cao, Yanan Qiu, Guojie Song
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2017)
Proceedings Paper
Automation & Control Systems
Jianxiong Cao, YangQuan Chen, Changpin Li
2015 AMERICAN CONTROL CONFERENCE (ACC)
(2015)
Proceedings Paper
Computer Science, Theory & Methods
Jianxiong Cao, Changpin Li, YangQuan Chen
2014 IEEE/ASME 10TH INTERNATIONAL CONFERENCE ON MECHATRONIC AND EMBEDDED SYSTEMS AND APPLICATIONS (MESA 2014)
(2014)
Article
Mathematics, Applied
M. S. Bruzon, T. M. Garrido, R. de la Rosa
Summary: We study a family of generalized Zakharov-Kuznetsov modified equal width equations in (2+1)-dimensions involving an arbitrary function and three parameters. By using the Lie group theory, we classify the Lie point symmetries of these equations and obtain exact solutions. We also show that this family of equations admits local low-order multipliers and derive all local low-order conservation laws through the multiplier approach.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Dohee Jung, Changbum Chun
Summary: The paper presents a general approach to enhance the Pade iterations for computing the matrix sign function by selecting an arbitrary three-point family of methods based on weight functions. The approach leads to a multi-parameter family of iterations and allows for the discovery of new methods. Convergence and stability analysis as well as numerical experiments confirm the improved performance of the new methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Abhishek Yadav, Amit Setia, M. Thamban Nair
Summary: In this paper, we propose a Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations using Chebyshev polynomials. We prove the well-posedness of the system and derive a theoretical error bound and convergence order. The numerical examples validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fernando Chacon-Gomez, M. Eugenia Cornejo, Jesus Medina, Eloisa Ramirez-Poussa
Summary: The use of decision rules allows for reliable extraction of information and inference of conclusions from relational databases, but the concepts of decision algorithms need to be extended in fuzzy environments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper considers the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay. To solve this problem numerically, a higher-order difference method is constructed and the error estimate for its solution is obtained. Based on the method of energy estimates, the fully discrete scheme is shown to be convergent of order four in space and of order two in time. The given numerical results illustrate the convergence and effectiveness of the numerical method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Tong-tong Shang, Guo-ji Tang, Wen-sheng Jia
Summary: The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound and upper bound of the solutions in the sense of the infinity norm of the TLCP are obtained when the problem has a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fabio Difonzo, Pawel Przybylowicz, Yue Wu
Summary: This paper focuses on the existence, uniqueness, and approximation of solutions of delay differential equations (DDEs) with Caratheodory type right-hand side functions. It presents the construction of the randomized Euler scheme for DDEs and investigates its error. Furthermore, the paper reports the results of numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Priyanka Roy, Geetanjali Panda, Dong Qiu
Summary: In this article, a gradient based descent line search scheme is proposed for solving interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational perspectives. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by the proposed scheme.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Zhongqian Wang, Changqing Ye, Eric T. Chung
Summary: In this paper, the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions for elasticity equations in high contrast media is developed. The method offers advantages such as independence of target region's contrast from precision and significant impact of oversampling domain sizes on numerical accuracy. Furthermore, this is the first proof of convergence of CEM-GMsFEM with mixed boundary conditions for elasticity equations. Numerical experiments demonstrate the method's performance.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Samaneh Soradi-Zeid, Maryam Alipour
Summary: The Laguerre polynomials are a new set of basic functions used to solve a specific class of optimal control problems specified by integro-differential equations, namely IOCP. The corresponding operational matrices of derivatives are calculated to extend the solution of the problem in terms of Laguerre polynomials. By considering the basis functions and using the collocation method, the IOCP is simplified into solving a system of nonlinear algebraic equations. The proposed method has been proven to have an error bound and convergence analysis for the approximate optimal value of the performance index. Finally, examples are provided to demonstrate the validity and applicability of this technique.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Almudena P. Marquez, Maria L. Gandarias, Stephen C. Anco
Summary: A generalization of the KP equation involving higher-order dispersion is studied. The Lie point symmetries and conservation laws of the equation are obtained using Noether's theorem and the introduction of a potential. Sech-type line wave solutions are found and their features, including dark solitary waves on varying backgrounds, are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Susanne Saminger-Platz, Anna Kolesarova, Adam Seliga, Radko Mesiar, Erich Peter Klement
Summary: In this article, we study real functions defined on the unit square satisfying basic properties and explore the conditions for generating bivariate copulas using parameterized transformations and other constructions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Lulu Tian, Nattaporn Chuenjarern, Hui Guo, Yang Yang
Summary: In this paper, a new local discontinuous Galerkin (LDG) algorithm is proposed to solve the incompressible Euler equation in two dimensions on overlapping meshes. The algorithm solves the vorticity, velocity field, and potential function on different meshes. The method employs overlapping meshes to ensure continuity of velocity along the interfaces of the primitive meshes, allowing for the application of upwind fluxes. The article introduces two sufficient conditions to maintain the maximum principle of vorticity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Mingyu He, Wenyuan Liao
Summary: This paper presents a numerical method for solving reaction-diffusion equations in spatially heterogeneous domains, which are commonly used to model biological applications. The method utilizes a fourth-order compact alternative directional implicit scheme based on Pade approximation-based operator splitting techniques. Stability analysis shows that the method is unconditionally stable, and numerical examples demonstrate its high efficiency and high order accuracy in both space and time.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)