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2. Importance of Understanding the Physical System in Selecting Separation of Variables Based Methods to Solve the Heat Conduction Partial Differential Equation.
- Author
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Florio, Laurie A.
- Subjects
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HEAT conduction , *DIFFERENTIAL equations , *HEAT equation , *RANDOM forest algorithms , *PHENOMENOLOGICAL theory (Physics) , *MATHEMATICS - Abstract
Separation of variables is a common method for producing an analytical based solution to partial differential equations. Despite the wide application of this method, often the physical phenomena described by the differential equations are not adequately involved in the discourse over the appropriate methods to solve a given problem, particularly in mathematics curricula. However, as mathematics is the tool to better understanding of the physical world, the meaning of the differential equation, boundary conditions, and initial conditions cannot be detached from the methods used to solve the differential equations. Failure to recognize the physical conditions being studied can lead to solution methods that are invalid or unphysical. This paper demonstrates how awareness of the physical nature of the system being investigated and its relationship to the mathematics can guide the selection of the relevant solution methods. To illustrate the importance of the comprehension of the physical meaning behind the mathematical equations and representations and the need to avoid rote application a solution technique, the logic behind the selection of the appropriate solution techniques for the one-dimensional transient heat conduction equation is considered under different imposed conditions which lead to different trends in system operation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. ON A SOLVABLE SYSTEM OF DIFFERENCE EQUATIONS OF SIXTH-ORDER.
- Author
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KARAKAYA, DILEK, YAZLIK, YASIN, and KARA, MERVE
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VARIATIONAL inequalities (Mathematics) , *MATHEMATICS , *DIFFERENTIAL equations , *REAL numbers , *ARITHMETIC - Abstract
In this paper, we study the following two-dimesional system of difference equations ... where the parameters a;b; c;d and the initial values x i; y i, i 2 f1;2;3;4;5;6g, are real numbers. We show that some subclasses of nonlinear two-dimensional system of difference equations are solvable in closed form. We also describe the forbidden set of solutions of the system of differ- ence equations. Some numerical examples are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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4. ADDITIVE MAPPINGS SATISFYING CERTAIN ALGEBRAIC EQUATIONS IN PRIME RINGS.
- Author
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MIR, HAJAR EL, MAMOUNI, ABDELLAH, and OUKHTITE, LAHCEN
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LAPLACIAN operator , *FRACTIONAL differential equations , *DIFFERENTIAL equations , *MATHEMATICS , *FIXED point theory - Abstract
In this paper we give a classification of endomorphisms and additive mappings of a prime ring satisfying certain algebraic identities. Moreover, we provide an example proving that the primeness hypothesis imposed in our theorems is not superfluous. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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5. AN EXISTENCE ETUDY FOR A TRIPLED SYSTEM WITH p-LAPLACIAN INVOLVING γ-CAPUTO DERIVATIVES.
- Author
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BEDDANI, HAMID, BEDDANI, MOUSTAFA, and DAHMANI, ZOUBIR
- Subjects
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LAPLACIAN operator , *FRACTIONAL differential equations , *DIFFERENTIAL equations , *MATHEMATICS , *FIXED point theory - Abstract
In this paper, we study the existence and uniqueness of solutions for a tripled system of fractional differential equations with nonlocal integro multi point boundary conditions by using the p-Laplacian operator and the γ-Caputo derivatives. The presented results are obtained by the two fixed point theorems of Banach and Krasnoselskii. An illustrative example is presented at the end to show the applicability of the obtained results. To the best of our knowledge, this is the first time where such problem is considered. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
6. A LOGICAL ALTERNATIVE FOR THE BURR PROBABILITY DISTRIBUTIONS.
- Author
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Ranjbaran, Abdolrasoul, Ranjbaran, Mohammad, Ranjbaran, Fatema, Falamaki, Masoud, Hashemi, Shamsedin, and Rousta, Ali Mohammad
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DATA analysis , *CUMULATIVE distribution function , *MATHEMATICS , *DIFFERENTIAL equations , *CURVES - Abstract
The analysis of real-world data in classical statistics is commenced by deriving a density function which requires a lengthy function selection and parameter estimation process. The process is supported by difficult integration to obtain the cumulative frequency function. In view of the difficulties in the existing methods, a super function called the Persian Probability Curve is proposed in this paper. This function is based on logical reasoning and mathematical concepts to overcome the shortcomings of the classical method. It is shown that the Persian probability is equivalent to that of the Burr in the governing differential equation. The validity of the work has been verified by comparing the obtained results with those of others. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
7. The simplest approach to solving the cubic nonlinear jerk oscillator with the non‐perturbative method.
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NONLINEAR oscillators , *NONLINEAR differential equations , *NONLINEAR equations , *DIFFERENTIAL equations , *MATHEMATICS , *DUFFING oscillators - Abstract
Mathematics and its applications try to make available a simple and accurate approach to handle nonlinear equations. The present paper investigates for the first time the application of the linearizing method to determine both the approximate frequency and displacement amplitude for the third‐order differential equations. The main merit of the linearized method is a collection of simplicity and accuracy for the solution of high‐order nonlinear conservative or non‐conservative oscillators. The current work sheds light on the approximate proposition to the damped third‐order nonlinear differential equations. The fast estimated solution is simulated to the accurate solution gained through the numerical methods. It is significant to conclude that the non‐perturbative method considered here is simpler and easier to apply than the other perturbation methods. This paper enriches some existing literature. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
8. Analytical and numerical treatment of the Volterra equation of the second kind.
- Author
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Jalel, Oday Hatem and Hussain, Huda Ismail
- Subjects
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VOLTERRA equations , *DIFFERENTIAL equations , *PROBABILITY theory , *MATHEMATICS - Abstract
This paper presents the study of numerical solutions to the linear integral Volterra equation of the second type because of this type of equations of great importance in several fields, including the science of population structures as well as in the study of physical problems and concepts such as elasticity. It is also used in mathematics in the study of probability theory, differential calculation and integration of equations, approximation problems, and boundary conditions. Two methods of numerical solution were used, which is the Adomian publication method, as this method leads to accurate calculations and approximate solutions of differential equations easily, as a solution to the Volterra equation of the second type was reached without the need for large calculations or any other transformations. The second method is the Taylor series, where we were able to find approximate solutions to the Volterra equation of the second type, as Taylor approximation is of great importance in numerical mathematics and the algorithms adopted to solve the equations. It was the use of the program Mathematica to draw the results that have been reached. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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9. Response Solutions in Degenerate Oscillators Under Degenerate Perturbations.
- Author
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Si, Wen and Yi, Yingfei
- Subjects
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NONLINEAR oscillators , *CANTOR sets , *LEBESGUE measure , *DIFFERENTIAL equations , *CONTINUOUS functions , *MATHEMATICS - Abstract
For a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi-periodically forced oscillators. These solutions are well-understood in quasi-periodically perturbed nonlinear oscillators either in the presence of large damping or in the non-degenerate cases with small or free damping. In this paper, we consider the existence of response solutions in quasi-periodically perturbed, second order differential equations, including nonlinear oscillators, of the form x ¨ + λ x l = ϵ f (ω t , x , x ˙) , x ∈ R , where λ is a constant, 0 < ϵ ≪ 1 is a small parameter, l > 1 is an integer, ω ∈ R d is a frequency vector, and f : T d × R 2 → R 1 is real analytic and non-degenerate in x up to a given order p ≥ 0 , i.e., [ f (· , 0 , 0) ] = [ ∂ f (· , 0 , 0) ∂ x ] = [ ∂ 2 f (· , 0 , 0) ∂ x 2 ] = ⋯ = [ ∂ p - 1 f (· , 0 , 0) ∂ x p - 1 ] = 0 and [ ∂ p f (· , 0 , 0) ∂ x p ] ≠ 0 , where [ ] denotes the average value of a continuous function on T d . In the case that λ = 0 and f is independent of x ˙ , the existence of response solutions was first shown by Gentile (Ergod Theory Dyn Syst 27:427–457, 2007) when p = 1 . This result was later generalized by Corsi and Gentile (Commun Math Phys 316:489–529, 2012; Ergod Theory Dyn Syst 35:1079–1140, 2015; Nonlinear Differ Equ Appl 24(1):article 3, 2017) to the case that p > 1 is odd. In the case λ ≠ 0 , the existence of response solutions is studied by the authors Si and Yi (Nonlinearity 33(11):6072–6099, 2020) when p = 0 . The present paper is devoted to the study of response solutions of the above quasi-periodically perturbed differential equations for the case λ ≠ 0 by allowing p > 0 . Under the conditions that 0 ≤ p < l / 2 and λ [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 when l - p is even, we obtain a general result which particularly implies the following: (1) If either l is odd and λ < 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 , then as ϵ sufficiently small response solutions exist for each ω satisfying a Brjuno-like non-resonant condition; (2) If either l is odd and λ > 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] < 0 , then there exists an ϵ ∗ > 0 sufficiently small and a Cantor set E ∈ (0 , ϵ ∗) with almost full Lebesgue measure such that response solutions exist for each ϵ ∈ E and ω satisfying a Diophantine condition. Similar results are also obtained in the case λ = ± ϵ which particularly concern the existence of large amplitude response solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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10. Interior and boundary regularity criteria for the 6D steady Navier-Stokes equations.
- Author
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Li, Shuai and Wang, Wendong
- Subjects
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HAUSDORFF measures , *HOLDER spaces , *DIFFERENTIAL equations , *MATHEMATICS - Abstract
It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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11. Inviscid limit for the full viscous MHD system with critical axisymmetric initial data.
- Author
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Maafa, Youssouf and Zerguine, Mohamed
- Subjects
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BESOV spaces , *DIFFERENTIAL equations , *INTEGRAL equations , *MATHEMATICS - Abstract
This paper establishes the global well-posedness issue for the full viscous MHD equations in the axisymmetric setting. Global solutions are obtained in critical Besov spaces uniformly to the viscosity when the resistivity is fixed in the spirit of [Abidi H, Hmidi T, Keraani S. On the global well-posedness for the axisymmetric Euler equations. Math. Ann. 2010;347:15–41.], [Hassainia Z. On the global well-posedness of the 3D axisymmetric resistive MHD equations. Ann. Henri Poincaré. 2022;23:2877-2917], [Hmidi T, Zerguine M. Inviscid limit axisymmetric Navier–Stokes system. Differential and Integral Equations. 2009;22(11–12):1223–1246.]. Furthermore, strong convergence in the resolution spaces with a rate of convergence is also studied. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
12. Generalized scale functions for spectrally negative Lévy processes.
- Author
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Contreras, Jesús and Rivero, Víctor
- Subjects
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LAPLACE transformation , *DIFFERENTIAL equations , *FUNCTIONALS , *MATHEMATICS , *BESSEL functions - Abstract
For a spectrally negative Lévy process, scale functions appear in the solution of twosided exit problems, and in particular in relation with the Laplace transform of the first time it exits a closed interval. In this paper, we consider the Laplace transform of more general functionals, which can depend simultaneously on the values of the process and its supremum up to the exit time. These quantities will be expressed in terms of generalized scale functions, which can be defined using excursion theory. In the case the functional does not depend on the supremum, these scale functions coincide with the ones found on the literature, see e.g. Li and Palmowski (2018) and therefore the results in this work are an extension of them. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
13. Designing and Teaching an Undergraduate Mathematical Modeling Course for Mathematics Majors and Minors.
- Author
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Rohde Poole, S. B.
- Subjects
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MATHEMATICAL models , *MINORS , *UNDERGRADUATES , *PREREQUISITES (Education) , *DIFFERENTIAL equations , *MATHEMATICS - Abstract
This paper is written to provide ideas and guide faculty who want to design a mathematical modeling course for undergraduate mathematics majors and minors. We discuss course goals, assignments, and projects that can be used to help students gain experience relevant for careers and mathematical modeling opportunities. The authors designed this course to build students' mathematical thought processes and toolbox, ability to analyze and evaluate mathematical models, mathematical modeling skills, and teamwork skills. The course described is intended as an upper division undergraduate course with a prerequisite of differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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14. The Best Proximity Points For Weak MT-cyclic Reich Type Contractions.
- Author
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Barootkoob, S., Lakzian, H., and Mitrović, Z. D.
- Subjects
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METRIC geometry , *DIFFERENTIAL equations , *INTEGRAL equations , *MATHEMATICS , *INTEGRALS - Abstract
In this paper, we introduce a weak MT-cyclic Reich type contractions and obtain the existence theorems for best proximity point for self-mappings defined on the complete metric spaces. Our results improve and generalize some results in literature. Also, we give some applications of our results to solving some classes of non-linear integral and differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
15. On Uniformly Rotating Binary Stars and Galaxies.
- Author
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Jang, Juhi and Seok, Jinmyoung
- Subjects
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BINARY stars , *EVOLUTION equations , *GALAXIES , *DIFFERENTIAL equations , *MATHEMATICS - Abstract
In this paper, we study the asymptotic profiles, uniqueness and orbital stability of McCann's uniformly rotating binary stars (Houston J Math 32(2):603–631, 2006) governed by the Euler–Poisson system. A new uniqueness result will be importantly used in stability analysis. Moreover, we apply our framework to the study of uniformly rotating binary galaxies of the Vlasov–Poisson system through Rein's reduction (Handbook of differential equations: evolutionary equations, vol III, pp 383–476, 2007). [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
16. Moments of generalized Cauchy random matrices and continuous-Hahn polynomials.
- Author
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Assiotis, Theodoros, Bedert, Benjamin, Gunes, Mustafa Alper, and Soor, Arun
- Subjects
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RANDOM matrices , *POLYNOMIALS , *DIFFERENTIAL equations , *MATHEMATICS , *DISTRIBUTION (Probability theory) , *INTEGERS - Abstract
In this paper we prove that, after an appropriate rescaling, the sum of moments of an N × N Hermitian matrix H sampled according to the generalized Cauchy (also known as Hua–Pickrell) ensemble with parameter s > 0 is a continuous-Hahn polynomial in the variable k. This completes the picture of the investigation that began in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) where analogous results were obtained for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Jacobi. Our strategy of proof is somewhat different from the one in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) due to the fact that the generalized Cauchy is the only classical ensemble which has a finite number of integer moments. Our arguments also apply, with straightforward modifications, to the other three cases studied in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) as well. We finally obtain a differential equation for the one-point density function of the eigenvalue distribution of this ensemble and establish the large N asymptotics of the moments. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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17. Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green's Function, in Nonstandard Analysis.
- Author
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Morita, Tohru and Sato, Ken-ichi
- Subjects
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FRACTIONAL differential equations , *NONSTANDARD mathematical analysis , *DIFFERENTIAL equations , *POLYNOMIALS , *GREEN'S functions , *MATHEMATICS - Abstract
Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green's function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
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