Article
Mathematics
Bahrom Y. Irgashev
Summary: This paper constructs similarity solutions for a model equation with multiple characteristics of an arbitrary integer order, showing that the structure of these solutions depends on the simplicity of the derivative orders with respect to the variables x and y. Various frequent cases are considered, where these solutions are expressed linearly as fundamental solutions of well-known equations.
GEORGIAN MATHEMATICAL JOURNAL
(2022)
Article
Mathematics, Applied
Ariel E. Barton, Michael J. Duffy Jr
Summary: We establish the Caccioppoli inequality, a reverse Holder inequality inspired by Meyers' classic estimate, and construct the fundamental solution for linear elliptic differential equations of order 2m with certain lower order terms.
ADVANCED NONLINEAR STUDIES
(2023)
Article
Mathematics
Zharasbek Baishemirov, Abdumauvlen Berdyshev, Ainur Ryskan
Summary: The solvability issues of the boundary value problem with mixed conditions for a four-dimensional second-order Gellerstedt equation are studied in this paper. The uniqueness of the solution is proven using an energy integral's method, and the solution is constructed in explicit form using Green's function method, which involves the hypergeometric Lauricella's function.
Article
Engineering, Multidisciplinary
S. Tudu, S. P. Mondal, A. Ahmadian, A. K. Mahmood, S. Salahshour, M. Ferrara
Summary: In this article, a new representation of type-2 fuzzy numbers called generalized type-2 fuzzy numbers was proposed for solving boundary value problems. Theorems were developed and examples were provided to understand the nature of fuzzy solutions in various cases.
ALEXANDRIA ENGINEERING JOURNAL
(2021)
Article
Mathematics, Applied
Waleed M. Abd-Elhameed, Youssri H. Youssri
Summary: This paper presents novel formulae for the high-order derivatives of Chebyshev polynomials of the fifth-kind, involving terminating F-4(3)(1) hypergeometric functions. Utilizing the derived formulas, a spectral tau algorithm is implemented for numerically solving the convection-diffusion equation, with investigation into the convergence and error analysis of the algorithm.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Zharasbek Baishemirov, Abdumauvlen Berdyshev, Anvar Hasanov
Summary: In this paper, four fundamental solutions are constructed for an elliptic equation with two lines of different order of degeneration in the first quarter of the plane (x,y). These solutions are expressed by Appell's hypergeometric functions F2 of two variables. Using the decomposition formula of the Appell's functions, it is proven that these fundamental solutions have a logarithmic singularity as r approaches 0.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Qi-Rui Li, Dongrui Wan, Xu-Jia Wang
Summary: The Christoffel problem concerning convex solutions on the unit sphere is equivalent to the Laplace equation, and necessary and sufficient conditions can be found using the Green function of the Laplacian. By expressing the problem on the entire space, simpler conditions for solvability can be obtained. The L-p extension of the Christoffel problem is also studied with sufficient conditions provided for the case p > 2.
SCIENCE CHINA-MATHEMATICS
(2021)
Article
Mathematics, Applied
Ali Muhib, Hammad Alotaibi, Omar Bazighifan, Kamsing Nonlaopon
Summary: This paper explores the oscillation of solutions for a class of second-order neutral functional differential equations by proposing new criteria to ensure that all obtained solutions are oscillatory. The results obtained can be used to develop and provide theoretical support for the oscillation study for this class of equations, and an illustrated example is given to demonstrate the effectiveness of the new criteria.
Article
Mathematics
Flavio Crisanti, Clemente Cesarano, Artur Ishkhanyan
Summary: The Grad-Shafranov plasma equilibrium equation was originally solved analytically in toroidal geometry, and it has been compared with the limiting case of the standard toroidal geometry in terms of the Fock functions. In this paper, the complete analytical solution of the Grad-Shafranov equation is obtained in terms of the general Heun functions by using the prolate elliptical coordinate system.
Article
Mathematics
Antonella Lupica, Clemente Cesarano, Flavio Crisanti, Artur Ishkhanyan
Summary: The study presents solutions of the three-dimensional Laplace equation in prolate elliptic geometry using linear combinations of generalized hypergeometric functions, which are compared to solutions obtained in standard toroidal geometry.
Article
Physics, Multidisciplinary
V. H. Badalov, S. Badalov
Summary: In this paper, the authors present the bound state solution of the modified radial Klein-Gordon equation for a generalised tanh-shaped hyperbolic potential using the Nikiforov-Uvarov method. The energy eigenvalues and radial wave functions are expressed in terms of Jacobi polynomials for arbitrary l states. It is shown that the energy eigenvalues strongly depend on the potential parameters for quantum states. The results for the generalised tanh-shaped hyperbolic potential and its energy eigenvalues are found to be in good agreement with previous findings, and the calculated rovibrational energies for H-2, HCl, and O-2 match experimental results. Overall, this potential model is shown to be a viable candidate for describing various quantum systems simultaneously.
COMMUNICATIONS IN THEORETICAL PHYSICS
(2023)
Article
Mathematics, Applied
Muajebah Hidan, Mohamed Akel, Hala Abd-Elmageed, Mohamed Abdalla
Summary: In this work, an extension of the k-Wright ((k, tau)-Gauss) hypergeometric matrix function is defined and certain properties of this function are obtained. Furthermore, this function is presented as a solution for fractional kinetic equations.
Review
Mathematics
Natanael Karjanto
Summary: Exact analytical expressions for the spatial Fourier spectrum of the fundamental bright soliton solution for the 1+1-dimensional nonlinear Schrodinger equation are derived. It is shown that the Fourier transform for the hyperbolic secant shape is shape-preserving and can be represented by a convergent infinite series. The fundamental bright soliton has important applications in nonlinear fiber optics and optical telecommunication systems.
Article
Mathematics, Applied
Wei Shi, Zhaqilao
Summary: In this paper, the generalized Darboux transformation (GDT) method and the Taylor expansion are used to construct solutions for the Kundu equation, including the multiple breather solution on a periodic background and the mixed solutions of multiple soliton and multiple breather solution. An odd-fold Darboux transformation is obtained starting from the Lax pair under the framework of Kaup-Newell (KN) system. The odd-fold GDT is then expressed in the form of determinants using the Taylor expansion. Various solutions are obtained by selecting different parameters in the determinants. Modulation instability (MI) for the Kundu equation is also studied.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Multidisciplinary Sciences
Vahid Reza Hosseini, Farzaneh Yousefi, W. -N. Zou
Summary: This study introduces a novel meshless technique for solving diffusion problems within cell biology, computer graphics, image processing, and fluid flow. It presents a variable-order time fractional diffusion equation and uses a meshfree method based on singular boundary method and dual reciprocity method on three-dimensional arbitrary geometry for numerical solutions. Results confirm the stability and convergent of the proposed method on high-dimensional domains, demonstrating its reliability and accuracy on complex geometries.
JOURNAL OF ADVANCED RESEARCH
(2021)
