Article
Mathematics, Applied
Juan Bory-Reyes, Marco Antonio Perez-de la Rosa
Summary: This study develops a general quaternionic structure for the local fractional Moisil-Teodorescu operator in Cantor-type cylindrical and spherical coordinate systems, and demonstrates its application in the Helmholtz equation with local fractional derivatives on Cantor sets through two examples.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Interdisciplinary Applications
Hossein Jafari, Hassan Kamil Jassim, Dumitru Baleanu, Yu-ming Chu
Summary: In this paper, the authors used the local fractional reduced differential transform method and the local fractional Laplace variational iteration method to obtain approximate solutions for coupled KdV equations, and compared the results obtained by both methods. The results indicate that these algorithms are suitable for linear and nonlinear problems in engineering and sciences.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2021)
Article
Mathematics, Applied
Menglibay Ruziev
Summary: In this paper, a nonlocal boundary value problem is investigated for a special type of equation. The equation consists of a fractional diffusion equation for y > 0 with the Riemann-Liouville fractional derivative, and a generalized equation of moisture transfer for y < 0. The unique solvability of the problem is proved.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2021)
Article
Mathematics, Applied
Obaid Algahtani, Sayed Saifullah, Amir Ali
Summary: This article investigates the fractional Drinfeld-Sokolov-Wilson system with fractal dimensions under the power-law kernel. The integral transform using the Adomian decomposition technique is applied to study the general series solution and the applications of the model with fractal-fractional dimensions. The numerical case with appropriate subsidiary conditions validates the model and provides a detailed numerical/physical interpretation. The results reveal the effects of minimizing the fractal dimension and reducing the fractional order on the solitary wave solution, as well as the behavior of the coupled system.
Article
Multidisciplinary Sciences
Safyan Mukhtar
Summary: In this study, the variational iteration transform method (VITM) and the Adomian decomposition (ADM) method were used to solve the second- and fourth-order fractional Boussinesq equations. Both methods are effective in approximating non-linear problems accurately and easily. Utilizing the fractional Atangana-Baleanu operator and ZZ transform, solutions for the equation were derived. Validation was done through two examples, and the results showed that both VITM and ADM methods are effective in obtaining accurate and reliable solutions for the time-fractional Boussinesq equation.
Article
Mathematics, Applied
Limin Zhang, Xianhua Tang, Sitong Chen
Summary: This paper studies the solutions of a fractional Kirchhoff equation in R-3, showing the existence of ground state solution and sign-changing solution using Moser iterative method and truncation technique.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Naveed Iqbal, Humaira Yasmin, Akbar Ali, Abdul Bariq, M. Mossa Al-Sawalha, Wael W. Mohammed
Summary: In this paper, the numerical solution of the Fornberg-Whitham equations was investigated using two powerful techniques: the modified decomposition technique and the modified variational iteration technique with fractional-order derivatives. The proposed approach was evaluated for accuracy through fractional order projected models, demonstrating its ease of implementation and effectiveness in analyzing complicated fractional-order linear and nonlinear differential equations.
JOURNAL OF FUNCTION SPACES
(2021)
Article
Mathematics, Applied
Ruchi Sharma, Pranay Goswami, Ravi Shanker Dubey, Fethi Bin Muhammad Belgacem
Summary: This paper introduces a new fractional derivative operator using the Lonezo-Hartely function (G-function). Based on this operator, a new fractional diffusion equation is proposed and analytically solved using Laplace transform. Some applications related to the operator are also discussed in corollaries.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Computer Science, Interdisciplinary Applications
Ghada H. Ibraheem, Mustafa Turkyilmazoglu
Summary: In this work, the optimal variational iteration method was used to solve fractional differential equations, with an introduction of a parameter to accelerate convergence. The results showed that the proposed method exhibited better convergence and accuracy compared to the standard variational iteration method.
JOURNAL OF COMPUTATIONAL SCIENCE
(2022)
Article
Mathematics, Applied
Songting Luo, Qing Huo Liu
Summary: Conventional finite difference and finite element methods for solving the high frequency Helmholtz equation often suffer from numerical dispersion errors, requiring refined meshes. Asymptotic methods like geometrical optics can avoid the pollution effect, but only provide locally valid approximations and fail to capture caustics. To efficiently obtain globally valid solutions without pollution effect, we use a fixed-point problem related to an exponential operator and apply unconditionally stable operator-splitting based pseudospectral schemes. Anderson acceleration is also incorporated to accelerate convergence. Numerical experiments in two and three dimensions demonstrate the effectiveness of the method.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics
Nehad Ali Shah, Ioannis Dassios, Essam R. El-Zahar, Jae Dong Chung, Somaye Taherifar
Summary: In this article, modified techniques such as the variational iteration transform and Shehu decomposition method are used to obtain an approximate analytical solution for the time-fractional Fornberg-Whitham equation. A comparison between the variational iteration transform method and the Shehu decomposition method is made, showing that both methods are effective, reliable, and straightforward. The solutions achieved are compared with exact results, demonstrating the efficiency and accuracy of these methods in solving a wide range of linear and non-linear problems across various scientific fields.
Article
Mathematics, Applied
Tian-Yi Li, Fang Chen, Hai-Wei Sun, Tao Sun
Summary: We propose two preconditioners based on the fast sine transformation for solving linear systems with ill-conditioned multilevel Toeplitz structure. These matrices are generated by discretizing the two-dimensional nonlocal Helmholtz equations with fractional Laplacian operators via the finite difference method. For complex wave numbers with nonnegative real parts, we give the spectral analysis of the preconditioned matrices. Numerical experiments also indicate that the proposed preconditioners outperform the existing preconditioners.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Interdisciplinary Applications
Dongping Li, Yankai Li, Fangqi Chen, Xiaozhou Feng
Summary: This paper investigates a new type of instantaneous and non-instantaneous impulsive boundary value problem, which involves the generalized ?-Caputo fractional derivative with a weight. By utilizing critical point theorems and some properties of ?-Caputo-type fractional integration and differentiation, the variational construction and multiplicity result of solutions are established.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Ramzi B. Albadarneh, Iqbal Batiha, A. K. Alomari, Nedal Tahat
Summary: This work introduces a new power series formula to approximate the Caputo fractional-order operator, which is successfully used for approximate solutions of linear and nonlinear fractional-order differential equations. Numerical examples are provided to validate the formula.
Article
Mathematics
Jing Chang, Jin Zhang, Ming Cai
Summary: In this paper, series solutions and approximate solutions of time-space fractional differential equations are obtained using two different analytical methods, namely the homotopy perturbation Sumudu transform method (HPSTM) and the variational iteration Laplace transform method (VILTM). It is found that the approximate solutions closely match the exact solutions. These solutions are significant for analyzing various phenomena and have not been reported in previous literature. The graphical presentations of the third approximate solutions for different values of order alpha are a notable feature of this work.
Article
Chemistry, Multidisciplinary
Biswajit Pandit, Mukesh Kumar Rawani, Amit Kumar Verma, Carlo Cattani
Summary: This paper analyzes the accuracy of the Haar wavelet approximation method on the singular boundary value problem and proposes a solution for higher-order nonlinear singular BVPs. Numerical verification shows that the estimated order of convergence agrees with the theoretical results.
JOURNAL OF MATHEMATICAL CHEMISTRY
(2023)
Article
Thermodynamics
Xiao-Jun Yang, Mahmoud Abdel-Aty, Lu-Lu Geng
Summary: In this article, a new odd entire function of order one is suggested, which is found to be a solution of the heat equation. A new conjecture is proposed that this function only has real zeros. The obtained result establishes a connection between the heat problem and number theory.
Article
Thermodynamics
Xiao-Jun Yang, Abdulrahman Ali Alsolami, Ahmed Refaie Ali
Summary: In this article, we explore the solution of the classical wave equation in one-dimensional space, which is an even entire function of order one. We propose a conjecture that this function only has purely real zeros in the entire complex plane. This conjecture reveals a new perspective on the connection between number theory and wave equation.
Article
Thermodynamics
Xiao-Jun Yang, Nasser Hassan Sweilam, Mustafa Bayram
Summary: This article discusses the use of entire functions as exact solutions for the Laplace and diffusion equations, considering them in the algebraic number field. The hypothesis is that these functions have purely real zeros throughout the entire complex plane, suggesting new connections between algebraic number theory and mathematical physics.
Article
Thermodynamics
Pei-Tao Qiu, Xiao-Jun Yang, Hai Pu
Summary: In this paper, a slurry seepage dynamics model is developed and the critical conditions for instability of the model are discussed. The effects of power index, effective mobility, and non-Darcian flow factors on the seepage velocity are analyzed. The results show that the boundary between the stability and instability zones in the 2-D logarithmic parameter space is a straight line, and the slope of the line decreases with increasing power index.
Article
Thermodynamics
Xiao-Jun Yang, Tasawar Hayat
Summary: This article discusses a first-order entire function constructed by the Fourier sine integral, which is the solution to the one-dimensional diffusion equation in the theory of modular form.
Article
Thermodynamics
Xiao-Jun Yang, Mahmoud Abdel-Aty, Tasawar Hayat
Summary: This article discusses the entire functions associated with cusp forms' L-functions, where the entire function defined by the Fourier cosine transform serves as the solution for the diffusion equation in one dimension. Additionally, three conjectures are proposed regarding the zeros of three entire functions with order one, using the theory of entire functions.
Article
Thermodynamics
Hossein Jafari, Muslim Yusif Zair, Hassan Kamil Jassim
Summary: In this study, the fractional Laplace variational iteration method (FLVIM) is applied to explore solutions of the fractional Navier-Stokes equation. Using the theory of fixed points and Banach spaces, the uniqueness and convergence of the general fractional differential equation solutions obtained by the proposed method are investigated. Error analysis of the fractional Laplace variational iteration method solution is also conducted, demonstrating the validity and reliability of this method for solving fractional Navier-Stokes equations, with obtained solutions matching previously established ones.
Article
Mathematics
Mohsen Alimohammady, Carlo Cattani, Morteza Koozehgar Kalleji
Summary: This paper discusses the existence of multiple solutions to a degenerate weighted elliptic equation, where the number of solutions depends on the degeneracy term, specifically the number of subdomains of omega\a(-1)(0) whose boundary is formed by submanifolds with 1-codimension. Using the Ljusternik-Schnirelman principle, the corresponding eigenvalue problem is studied and the nonnegative eigenvalue sequence of the main problem is obtained. The regularity of the solutions on C-1,C-alpha (omega) is also discussed.
QUAESTIONES MATHEMATICAE
(2023)
Article
Mathematics
Hassan Kamil Jassim, Mohammed Abdulshareef Hussein
Summary: Recently, researchers have developed a new method called the Hussein-Jassim (HJ) method for solving nonlinear fractional ordinary differential equations, which have wide-ranging applications in various scientific fields. The HJ method is based on a power series of fractional order and provides approximate solutions for these equations. The results of this study demonstrate a high degree of consistency between the approximate solutions obtained using the HJ method and the exact solutions.
Article
Mathematics, Applied
Ayman Shehata, Ghazi S. Khammash, Carlo Cattani
Summary: This paper investigates the classical and fractional properties of the (r)R(s) matrix function using the Hilfer fractional operator. The theory focuses on special matrix functions, including gamma, beta, and Gauss hypergeometric matrix functions. Additionally, this paper establishes the connection between the (r)R(s) matrix function and other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials.
Article
Mathematics, Applied
Weam G. Alharbi, Abdullah F. Shater, Abdelhalim Ebaid, Carlo Cattani, Mounirah Areshi, Mohammed M. Jalal, Mohammed K. Alharbi
Summary: This study investigates a generalized model of the COVID-19 pandemic by introducing two different definitions in fractional calculus. The solutions of both models are derived and applied to predict the behavior of infected cases in eight European countries. The validity of the results reported in the literature is also discussed.
Article
Mathematics, Applied
Ozge Ozalp Guller, Carlo Cattani, Ecem Acar, Sevilay Kirci Serenbay
Summary: In this study, we propose a new type of nonlinear Bernstein-Chlodowsky operators based on q-integers. We firstly define the nonlinear q-Bernstein-Chlodowsky operators of max-product kind. Then, we provide an error estimation for the q-Bernstein Chlodowsky operators of max-product kind using a suitable generalizition of the Shisha-Mond Theorem. Furthermore, we present upper estimates of the approximation error for some subclasses of functions.
Article
Thermodynamics
Xiao-Jun Yang, Abdulrahman Ali Alsolami, Sohail Nadeem
Summary: This article presents a new anomalous relaxation model using the Sonine kernel and the general fractional derivative proposed in [Yang et al., General fractional derivatives with applications in viscoelasticity, Academic Press, New York, USA, 2020]. The mathematical model's solution is obtained by employing Laplace transform. A detailed comparison between classical and anomalous relaxation models is discussed. This result offers a mathematical tool to simulate the anomalous relaxation behavior of complex materials.
Article
Physics, Multidisciplinary
Jagdev Singh, Hassan Kamil Jassim, Devendra Kumar, Ved Prakash Dubey
Summary: This paper presents the use of the local fractional natural decomposition method (LFNDM) for solving a local fractional Poisson equation. Numerical examples with computer simulations are provided to demonstrate the effectiveness and convenience of LFNDM.
COMMUNICATIONS IN THEORETICAL PHYSICS
(2023)
