Article
Mathematics, Applied
Yan Cui, Xiaoyu Fu, Jiaxin Tian
Summary: In this paper, a fundamental inequality for a fourth order partial differential operator is established, and using this inequality, some Carleman estimates for the operator with suitable boundary conditions are proved. As an application, a resolvent estimate for the operator is obtained, which implies a log-type stabilization result for the plate equation.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Samy A. Harisa, Chokkalingam Ravichandran, Kottakkaran Sooppy Nisar, Nashat Faried, Ahmed Morsy
Summary: In this paper, the behavior of the neutral integro-differential equations of fractional order including the Caputo-Hadamard fractional derivative is analyzed using the topological degree method. The suggested fixed point technique is validated by specific numerical examples, demonstrating its wide applicability and high efficiency.
Article
Mathematics
Toshiyuki Tanisaki
Summary: The cohomology of the sheaf of twisted differential operators on the quantized flag manifold at a root of unity whose order is a prime power is described. It is shown that for the De Concini-Kac type quantized enveloping algebra, when the parameter q is specialized to a root of unity whose order is a prime power, the number of irreducible modules with a specified central character coincides with the dimension of the total cohomology group of the corresponding Springer fiber. This provides a weak version of a conjecture of Lusztig concerning non-restricted representations of the quantized enveloping algebra.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Daniel Barlet
Summary: This paper investigates the relationship between elementary symmetric functions and trace functions of complex variables. By using a specific family of differential operators, a left ideal killing all trace functions can be generated. As an application, a holonomic system is obtained, which is a quotient of a specific left ideal in the Weyl algebra.
MATHEMATISCHE ZEITSCHRIFT
(2022)
Review
Engineering, Mechanical
Vahid Reza Hosseini, Wennan Zou
Summary: This paper investigates the numerical solution of time-fractional convection diffusion equations (TF-CDEs), and develops a nonlocal model using the Peridynamic differential operator for discretization. The proposed scheme is analyzed for stability and error estimates, and numerical experiments are conducted to validate the theoretical analysis and demonstrate computational efficiency.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics
Jose R. dos Santos Filho, Mauricio Fronza da Silva
Summary: The authors have extended the conditions for global solvability of linear differential operators with real principal part defined on a n-dimensional manifold, under the assumption of the existence of a hyperbolic singular point, to the case of non-degenerate singular semi-hyperbolic points for n = 2.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Materials Science, Multidisciplinary
Jyoti Mishra
Summary: This paper investigates a system of partial differential equations that describes the behavior of the telegraph. By converting the time differential operators to nonlocal operators, nonlocal behaviors are incorporated into the mathematical formulation. Numerical solutions are presented using Newton's polynomial interpolation, and exact solutions are attempted to be derived through Laplace transform.
RESULTS IN PHYSICS
(2022)
Article
Engineering, Multidisciplinary
Shorish Omer Abdulla, Sadeq Taha Abdulazeez, Mahmut Modanli
Summary: In this research paper, the first investigation of the third-order fractional partial differential equation (FPDE) in the sense of the Caputo fractional derivative and the Atangana-Baleanu Caputo (ABC) fractional derivative is presented. The proposed problem is more general than the third-order linear time-varying systems model. A first-order finite difference scheme is developed for the converted problem and its stability inequality is demonstrated in Hilbert space. The explicit finite difference method (EFDM) is utilized to obtain approximate solutions for the third-order FPDE based on the Caputo and ABC fractional derivatives.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Physics, Mathematical
Muhammad Zafar, Jiequn Han, Xu-Hui Zhou, Heng Xiao
Summary: Partial differential equations (PDEs) play a crucial role in mathematical modeling but can be computationally expensive to solve. Neural operators offer a faster way to solve PDEs. This study compares the performance of two neural operators for transport PDEs and finds that the graph kernel network (GKN) operator performs slightly better than the vector cloud neural network (VCNN) operator, but at a higher computational cost.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Sunil Dutt Purohit, Dumitru Baleanu, Kamlesh Jangid
Summary: In this article, solutions of a generalised multiorder fractional partial differential equations involving the Caputo time-fractional derivative and the Riemann-Liouville space fractional derivatives are studied using the Laplace-Fourier transform technique. The proposed equations can be reduced to the Schrodinger equation, wave equation, and diffusion equation in a more general sense. Solutions of the equation proposed in the stochastic resetting theory in the context of Brownian motion are also found in a general regime.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Interdisciplinary Applications
Svetlin G. Georgiev, Khaled Zennir, Wiem Abedelmonem Salah Ben Khalifa, Amal Hassan Mohammed Yassin, Aymen Ghilen, Sulima Ahmed Mohammed Zubair, Najla Elzein Abukaswi Osman
Summary: This paper investigates a BVP for a class of impulsive fractional partial differential equations and proposes a new topological approach to prove the existence of at least one classical solution and at least two nonnegative classical solutions.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2022)
Article
Mathematics, Applied
A. Ghose-Choudhury, Sudip Garai
Summary: This article discusses the use of a comparison method to obtain exact solutions for nonlinear partial differential equations (PDEs) through their traveling wave reductions. The method, proposed by N. A. Kudryashov, is extended to include solutions expressed in terms of both the logistic function and the tanh$$ \tanh $$-class of functions. The article derives the standard set of second-order ordinary differential equations (ODEs) that have the logistic and tanh$$ \tanh $$ functions as solutions and also extends the analysis to third-order cases.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Guangchen Wang, Wencan Wang, Zhiguo Yan
Summary: This paper studies the optimal control problem of linear backward stochastic differential equations with quadratic cost functional under partial information. By utilizing the stochastic maximum principle and a decoupling technique, the problem is completely and explicitly solved. The optimal feedback representation and an explicit formula for the optimal cost are established through the derivation of three Riccati equations, a BSDE with filtering, and a SDE with filtering after obtaining a stochastic Hamiltonian system.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Resat Yilmazer, Karmina Ali
Summary: In this study, the discrete fractional nabla calculus operator is applied to investigate the k-hypergeometric differential equation in both homogeneous and nonhomogeneous states. Through classical transformations and parameter constraints, new exact fractional solutions are obtained for the guiding equation. The fractional nabla operator is beneficial for converting singular differential equations into fractional order equations, leading to the construction of several new exact fractional solutions for the given equation.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Hongyan Xu, Ling Xu, Hari Mohan Srivastava
Summary: This article deals with the description of the complete solutions to several Fermat type partial differential-difference equations (PDDEs), and provides some descriptions of the forms of transcendental entire solutions for these equations. The results are extensions and improvements of previous theorems, and are supported by a series of examples that demonstrate the accuracy of the conditions and forms of transcendental entire solutions.