1,843 results
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2. Special relativity and the Lorentz equations. Errors in Einstein’s 1905 paper.
- Author
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Beal, Alasdair N.
- Subjects
- *
TIME dilation , *LORENTZ theory , *LORENTZ transformations , *SPECIAL relativity (Physics) , *EQUATIONS , *CLOCKS & watches - Abstract
The explanation of Einstein’s special theory of relativity in his original 1905 paper is examined. His analysis is confusing, as terms x, y, z, t, etc., have different meanings at various points, and he presents equations based on different and inconsistent assumptions. Adding subscripts clarifies these issues but exposes errors in his reasoning. To calculate his transformation equations, he selects a combination of equations which gives results matching the Lorentz transformation but he ignores other possible valid solutions. Also his calculations contain serious errors. Therefore, he fails to prove that his theory leads to the Lorentz equations as a unique solution. Einstein’s analysis includes “moving” clocks which show “stationary” time t, so the idea that a moving clock should run slower than a stationary clock is incompatible with his theory. Also, his calculation of time dilation contains serious errors. As a result, he fails to provide a theoretical justification for his famous “clocks paradox.” [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. Mathematical Equation Identification and Solving System.
- Author
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Tambe, Riya, Bharadwaj, Rakhi, Vaze, Sanaya, Uttarwar, Varad, and Urane, Samarth
- Subjects
DEEP learning ,SYSTEM identification ,NATURAL language processing ,MANUAL labor ,EQUATIONS - Abstract
The rapid advancements in deep learning techniques have significantly impacted various fields, including mathematics and natural language processing. Mathematical equation identification and solving is a challenging task that has been studied by researchers for many years. Traditional approaches to this task typically rely on rule-based systems, hand-crafted features or manual work. However, these approaches are often limited in their ability to handle complex equations and there are also chances of error in it. The survey begins by discussing the challenges associated with mathematical equation identification and solving, such as the complexity of mathematical notation, variable representation, and equation structure. The paper provides an overview of the existing methodologies, algorithms, and frameworks employed in the development of systems capable of identifying and solving mathematical equations using deep learning techniques. [ABSTRACT FROM AUTHOR]
- Published
- 2024
4. Analytical solutions of the space--time fractional Kadomtsev--Petviashvili equation using the (G'/G)-expansion method.
- Author
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Hassaballa, Abaker, Salih, Mohyaldein, Mohamed Khamis, Gamal Saad, Gumma, Elzain, Adam, Ahmed M. A, and Satty, Ali
- Subjects
NONLINEAR evolution equations ,ANALYTICAL solutions ,ORDINARY differential equations ,NONLINEAR equations ,EQUATIONS ,NONLINEAR differential equations - Abstract
This paper focusses on the nonlinear fractional Kadomtsev-Petviashvili (FKP) equation in space-time, employing the conformable fractional derivative (CFD) approach. The main objective of this paper is to examine the application of the (G'/G)-expansion method in order to find analytical solutions to the FKP equation. The (G'/G)-expansion method is a powerful tool for constructing traveling wave solutions of nonlinear evolution equations. However, its application to the FKP equation remains relatively unexplored. By employing traveling wave transformation, the FKP equation was transformed into an ordinary differential equation (ODE) to acquire exact wave solutions. A range of exact analytical solutions for the FKP equation is obtained. Graphical illustrations are included to elucidate the physical characteristics of the acquired solutions. To demonstrate the impact of the fractional operator on results, the acquired solutions are exhibited for different values of the fractional order α, with a comparison to their corresponding exact solutions when taking the conventional scenario where α equals 1. The results indicate that the (G'/G)-expansion method serves as an efficient method and dependable in solving the nonlinear FKP equation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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5. Discussion of "A Reformed Empirical Equation for the Discharge Coefficient of Free-Flowing Type-A Piano Key Weirs".
- Author
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Yao, Li, Gu, Jianbei, He, Feifei, Li, Jinbo, and Shi, Sha
- Subjects
DISCHARGE coefficient ,WEIRS ,PIANO ,EQUATIONS - Abstract
This document discusses a research paper on the discharge coefficient of Type-A piano key weirs. The discussers appreciate the authors' efforts but suggest further analysis for one equation in the original paper. They conducted simulations and analysis, providing graphs and tables to support their findings. The document also includes a sensitivity analysis of input parameters affecting the discharge coefficient, showing the varying effects of different parameters. The study provides insights into estimating the discharge coefficient and was supported by funding from the Water Resources Department of Jiangxi Province and the Natural Science Foundation of Jiangxi Province in China. [Extracted from the article]
- Published
- 2024
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6. Initial value problems for fractional p-Laplacian equations with singularity.
- Author
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Hasanov, Mahir
- Subjects
MATHEMATICAL equivalence ,VOLTERRA equations ,MATHEMATICAL analysis ,EXISTENCE theorems ,EQUATIONS ,FRACTIONAL differential equations - Abstract
We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
7. Existence of solutions for a non-local equation on the Sierpiński gasket.
- Author
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Shokooh, Saeid
- Subjects
GASKETS ,EQUATIONS ,CLASSICAL solutions (Mathematics) - Abstract
This research paper introduces novel findings regarding a specific class of Kirchhoff problems. Building upon the theoretical groundwork established in a previous work by Ricceri [15], the paper centers its investigation on the Sierpiński gasket (V, |•|) situated in (R
n-1 ,- |•|), where n≥ 2. The intrinsic boundary of the Sierpiński gasket, denoted as Vo , comprises n corners. Under certain conditions, we prove that a Kirchhoff problem has two classical solutions in the space H0 ¹(V). [ABSTRACT FROM AUTHOR]- Published
- 2024
8. On the Controllability for the 1D-Heat Equation with Dirichlet Boundary condition, in the Presence of a Scale-Invariant Parameter.
- Author
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Benalia, Karim
- Subjects
HEAT equation ,EQUATIONS ,CONTROLLABILITY in systems engineering - Abstract
In this paper we study the controllability for the 1D-Heat equation with a Dirichlet boundary condition, in the presence of a scale-invariant parameter. First, we construct the scale-invariant solutions for the one-dimensional heat equation. Then we present our problem statement. We finally prove the Dirichlet boundary controllability. [ABSTRACT FROM AUTHOR]
- Published
- 2024
9. On the Cauchy Problem for a Two-component Peakon System With Cubic Nonlinearity.
- Author
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Wang, Ying and Zhu, Min
- Subjects
CAUCHY problem ,CONSERVATION laws (Physics) ,LEAD time (Supply chain management) ,VELOCITY ,EQUATIONS ,CONSERVATION laws (Mathematics) - Abstract
In this paper, we consider the Cauchy problem for a two-component peakon system with cubic nonlinearity, which is a natural multi-component extension of the single Camassa-Holm equation. We first establish the local well-posedness for the Cauchy problem of the system, and then derive a blow-up criteria and construct data that lead to the finite time blow-up by exploiting a special conservation law and by using the method of characteristics. It is worthwhile to point out that the classical approach to study the blow up phenomena heavily depends on the control of H 1 -norm of the velocity component. However, this two-component peakon system considered in this paper does not admit H 1 -norm conservation law. Our idea is to use the new conservation law H 1 = 1 6 ∫ R (u - u x) n d x = 1 6 ∫ R (v + v x) m d x (see Lemma 3.2) and the structure of the system to obtain the estimates on ‖ m ‖ L 1 and ‖ n ‖ L 1 (see Lemma 3.3). As a result, we can assert that the finite time blow-up can occur if and only if the slope of the transport velocity is unbounded below and derive a new blow-up result for strong solutions to the system. Finally, we also establish the persistence properties of the solutions to the integrable peakon system in weighted L ϕ p spaces for a large class of moderate weights. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. Underdetermined Equation Model Combined with Improved Krylov Subspace Basis for Solving Electromagnetic Scattering Problems.
- Author
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Cunjie Shen, Xinyuan Cao, Qi Qi, Yunuo Fan, Xiangxiang Liu, Xiaojing Kuang, Chenghua Fan, and Zhongxiang Zhang
- Subjects
KRYLOV subspace ,ELECTROMAGNETIC wave scattering ,CURRENT fluctuations ,MOMENTS method (Statistics) ,EQUATIONS ,COMPUTATIONAL complexity - Abstract
To accelerate the solution of electromagnetic scattering problems, compressive sensing (CS) has been introduced into the method of moments (MoM). Consequently, a computational model based on underdetermined equations has been proposed, which effectively reduces the computational complexity compared with the traditional MoM. However, while solving surface-integral formulations for three-dimensional targets by MoM, due to the severe oscillation of current signals, commonly used sparse bases become inapplicable, which renders the application of the underdetermined equation model quite challenging. To address this issue, this paper puts forward a scheme that employs Krylov subspace, which is constructed with low complexity by meticulously designing a group of non-orthogonal basis vectors, to replace the sparse transforms in the algorithmic framework. The principle of the method is elaborated in detail, and its effectiveness is validated through numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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11. Analyzing the physical behavior of optical fiber pulses using solitary wave solutions of the perturbed Chen–Lee–Liu equation.
- Author
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Khater, Mostafa M. A.
- Subjects
SOLITONS ,PULSE wave analysis ,SCHRODINGER equation ,TELECOMMUNICATION ,EQUATIONS ,OPTICAL communications - Abstract
This research paper investigates the behavior of optical fiber pulses by studying the solitary wave solutions of the perturbed Chen–Lee–Liu (CLL) equation. The perturbed CLL equation is derived from the perturbed Schrödinger equation. Two analytical approaches for obtaining the innovative soliton-wave solutions are presented in this paper, and their reliability is examined by using a well-established numerical method. The accuracy of pulse wave analysis in an optical cable is demonstrated through a series of graphical representations. Our study introduces scientific novelty, as evidenced by the comparative analysis of our data with those of previous research papers. This research contributes to enhancing the understanding of the kinetics and physical behavior of pulses in optical fibers, which holds implications for the advancement of optical communication technologies. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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12. A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics.
- Author
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Esen, Oğul, Sardón, Cristina, and Zajac, Marcin
- Subjects
HAMILTON'S equations ,EQUATIONS - Abstract
In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
13. On general Kirchhoff type equations with steep potential well and critical growth in R².
- Author
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Zhenluo Lou and Jian Zhang
- Subjects
POTENTIAL well ,EQUATIONS - Abstract
In this paper, we study the following Kirchhoff-type equation:... where M ∈ C(R
+ ,R+ ) is a general function, V ≥ 0 and its zero set may have several disjoint connected components, μ > 0 is a parameter, K is permitted to be unbounded above, and f has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhofftype equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case M ≡ 1. [ABSTRACT FROM AUTHOR]- Published
- 2024
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14. On an efficient numerical procedure for the Functionalized Cahn-Hilliard equation.
- Author
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Orizaga, Saulo, Ifeacho, Ogochukwu, and Owusu, Sampson
- Subjects
PARTIAL differential equations ,BENCHMARK problems (Computer science) ,SEPARATION (Technology) ,PHASE separation ,EQUATIONS - Abstract
The Functionalized Cahn Hilliard (FCH) equation was used to model micro-phase separation in mixtures of amphiphilic molecules in solvent. In this paper, we proposed a Tri-Harmonic Modified (THM) numerical approach for efficiently solving the FCH equation with symmetric double well potential by extending the ideas of the Bi-harmonic Modified (BHM) method. THM formulation allowed for the nonlinear terms in the FCH equation to be computed explicitly, leading to fast evaluations at every time step. We investigated the convergence properties of the new approach by using benchmark problems for phase-field models, and we directly compared the performance of the THM method with the recently developed scalar auxiliary variable (SAV) schemes for the FCH equation. The THM modified scheme was able to produce smaller errors than those obtained from the SAV formulation. In addition to this direct comparison with the SAV schemes, we tested the adaptability of our scheme by using an extrapolation technique which allows for errors to be reduced for longer simulation runs. We also investigated the adaptability of the THM method to other 6th order partial differential equations (PDEs) by considering a more complex form of the FCH equation with nonsymmetric double well potential. Finally, we also couple the THM scheme with a higher order timestepping method, (implicit-explicit) IMEX schemes, to demonstrate the robustness and adaptability of the new scheme. Numerical experiments are presented to investigate the performance of the new approach. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
15. Polynomial stability of transmission system for coupled Kirchhoff plates.
- Author
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Wang, Dingkun, Hao, Jianghao, and Zhang, Yajing
- Subjects
POLYNOMIALS ,ELASTICITY ,EXPONENTS ,MATHEMATICS ,EQUATIONS - Abstract
In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional damping (- Δ) 2 θ v t with θ ∈ [ 1 2 , 1 ] . By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional damping order. Specifically, when the inertia/elasticity ratios are not equal and θ ∈ [ 1 2 , 3 4 ] , the polynomial decay rate of the system is t - 1 / (10 - 4 θ) . When the inertia/elasticity ratios are not equal and θ ∈ [ 3 4 , 1 ] , the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional damping exponent 2 θ from [0, 1] to [1, 2]. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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16. Continuous data assimilation for the three-dimensional planetary geostrophic equations of large-scale ocean circulation.
- Author
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You, Bo
- Subjects
OCEAN circulation ,EQUATIONS ,SPATIAL resolution ,OCEAN - Abstract
The main objective of this paper is to consider a continuous data assimilation algorithm for the three-dimensional planetary geostrophic model in the case that the observable measurements, obtained continuously in time, may be contaminated by systematic errors. In this paper, we will provide some suitable conditions on the nudging parameter and the spatial resolution, which are sufficient to show that the approximation solution of the proposed continuous data assimilation algorithm converges to the unique exact unknown reference solution of the original system at an exponential rate, asymptotically in time, under the assumption that the observed data is free of error. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
17. Holder regularity of solutions and physical quantities for the ideal electron magnetohydrodynamic equations.
- Author
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Wang, Yanqing, Liu, Jitao, and He, Guoliang
- Subjects
PHYSICAL constants ,EULER equations ,ELECTRONS ,QUANTUM dots ,TRANSPORT equation ,EQUATIONS - Abstract
In this paper, we make the first attempt to figure out the differences on Hölder regularity in time of solutions and conserved physical quantities between the ideal electron magnetohydrodynamic equations concerning Hall term and the incompressible Euler equations involving convection term. It is shown that the regularity in time of magnetic field B is C_{t}^{\frac {\alpha }2} provided it belongs to L_{t}^{\infty } C_{x}^{\alpha } for any \alpha >0, its energy is C_{t}^{\frac {2\alpha }{2-\alpha }} as long as B belongs to L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0<\alpha <1 and its magnetic helicity is C_{t}^{\frac {2\alpha +1}{2-\alpha }} supposing B belongs to L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0<\alpha <\frac 12, which are quite different from the classical incompressible Euler equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. Estimates of Eigenvalues and Approximation Numbers for a Class of Degenerate Third-Order Partial Differential Operators.
- Author
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Muratbekov, Mussakan, Suleimbekova, Ainash, and Baizhumanov, Mukhtar
- Subjects
PARTIAL differential operators ,PARTIAL differential equations ,EIGENVALUES ,RECTANGLES ,EQUATIONS - Abstract
In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation numbers is proven. Here, we note that finding estimates of approximation numbers, as well as extremal subspaces, for a set of solutions to the equation is a task that is certainly important from both a theoretical and a practical point of view. The paper also obtained an upper bound for the eigenvalues. Note that, in this paper, estimates of eigenvalues and approximation numbers for the degenerate third-order partial differential operators are obtained for the first time. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. An Improved Porosity Calculation Algorithm for Particle Flow Code.
- Author
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Zhang, Siyu, Xin, Xiankang, Cui, Yongzheng, and Yu, Gaoming
- Subjects
GRANULAR flow ,FLUID flow ,POROSITY ,ALGORITHMS ,EQUATIONS - Abstract
The widely used discrete-element particle flow software PFC's (PFC 7.0 and previous versions) algorithm for calculating porosity is not sufficiently accurate. Because of this, when the particles are densely packed, the solution to the equation produces an algorithm exception for odd calculations of porosity, which results in the inability to calculate the results. This paper, based on a Darcy seepage model of fluid flow through a granular bed, analyzed the shortcomings of the two porosity calculation methods of PFC and the function analysis method. Combining this analysis with the theory of computer graphics, a new and efficient porosity calculation algorithm was proposed. The result showed that the new proposed porosity calculation algorithm calculated a more accurate and reasonable porosity field and made the iterative solution of the CFD equation more stable. This method makes porosity-related models of PFC more accurate. The algorithm can be not only used to calculate porosity, but also applied to other fields. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. Climatology of the terms and variables of transformed Eulerian-mean (TEM) equations from multiple reanalyses: MERRA-2, JRA-55, ERA-Interim, and CFSR.
- Author
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Fujiwara, Masatomo, Martineau, Patrick, Wright, Jonathon S., Abalos, Marta, Šácha, Petr, Kawatani, Yoshio, Davis, Sean M., Birner, Thomas, and Monge-Sanz, Beatriz M.
- Subjects
CLIMATOLOGY ,GRAVITY waves ,ENTHALPY ,EQUATIONS - Abstract
A 30-year (1980–2010) climatology of the major variables and terms of the transformed Eulerian-mean (TEM) momentum and thermodynamic equations is constructed by using four global atmospheric reanalyses: the Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2); the Japanese 55-year Reanalysis (JRA-55); the European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim); and the Climate Forecast System Reanalysis (CFSR). Both the reanalysis ensemble mean (REM) and the differences in each reanalysis from the REM are investigated in the latitude–pressure domain for December–January–February and for June–July–August. For the REM investigation, two residual vertical velocities (the original one and one evaluated from residual meridional velocity) and two mass streamfunctions (from meridional and vertical velocities) are compared. Longwave (LW) radiative heating and shortwave (SW) radiative heating are also shown and discussed. For the TEM equations, the residual terms are also calculated and investigated for their potential usefulness, as the residual term for the momentum equation should include the effects of parameterized processes such as gravity waves, while that for the thermodynamic equation should indicate the analysis increment. Inter-reanalysis differences are investigated for the mass streamfunction, LW and SW heating, the two major terms of the TEM momentum equation (the Coriolis term and the Eliassen–Palm flux divergence term), and the two major terms of the TEM thermodynamic equation (the vertical temperature advection term and the total diabatic heating term). The spread among reanalysis TEM momentum balance terms is around 10 % in Northern Hemisphere winter and up to 50 % in Southern Hemisphere winter. The largest uncertainties in the thermodynamic equation (about 50 %) are found in the vertical advection, for which the structure is inconsistent with the differences in heating. The results shown in this paper provide basic information on the degree of agreement among recent reanalyses in the stratosphere and upper troposphere in the TEM framework. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. Pogorelov estimates for semi-convex solutions of k-curvature equations.
- Author
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Chen, Xiaojuan, Tu, Qiang, and Xiang, Ni
- Subjects
PARTIAL differential equations ,EQUATIONS - Abstract
In this paper, we consider k-curvature equations \sigma _k(\kappa [M_u])=f(x,u,\nabla u) subject to (k+1)-convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J. 123 (2004), pp. 235–264]. By using the crucial concavity inequality for Hessian operator of Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23], we derive Pogorelov estimates of semi-convex admissible solutions for these k-curvature equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. Inverse problem of recovering a time-dependent nonlinearity appearing in third-order nonlinear acoustic equations.
- Author
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Fu, Song-Ren, Yao, Peng-Fei, and Yu, Yongyi
- Subjects
NONLINEAR equations ,GEOMETRICAL constructions ,THEORY of wave motion ,QUASILINEARIZATION ,EQUATIONS - Abstract
This paper is devoted to some inverse problems of recovering the nonlinearity for the Jordan–Moore–Gibson–Thompson equation, which is a third order nonlinear acoustic equation. This equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained with the small initial and boundary data. By the second order linearization to the nonlinear equation, and construction of complex geometric optics solutions for the linearized equation, the uniqueness of recovering the nonlinearity is derived. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. Simulation of coupled elasticity problem with pressure equation: hydroelastic equation.
- Author
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Hooshyarfarzin, Baharak, Abbaszadeh, Mostafa, and Dehghan, Mehdi
- Subjects
FINITE element method ,NATURAL gas ,ELASTICITY ,HYDRAULIC fracturing ,FLUID pressure ,EQUATIONS - Abstract
Purpose: The main aim of the current paper is to find a numerical plan for hydraulic fracturing problem with application in extracting natural gases and oil. Design/methodology/approach: First, time discretization is accomplished via Crank-Nicolson and semi-implicit techniques. At the second step, a high-order finite element method using quadratic triangular elements is proposed to derive the spatial discretization. The efficiency and time consuming of both obtained schemes will be investigated. In addition to the popular uniform mesh refinement strategy, an adaptive mesh refinement strategy will be employed to reduce computational costs. Findings: Numerical results show a good agreement between the two schemes as well as the efficiency of the employed techniques to capture acceptable patterns of the model. In central single-crack mode, the experimental results demonstrate that maximal values of displacements in x- and y- directions are 0.1 and 0.08, respectively. They occur around both ends of the line and sides directly next to the line where pressure takes impact. Moreover, the pressure of injected fluid almost gained its initial value, i.e. 3,000 inside and close to the notch. Further, the results for non-central single-crack mode and bifurcated crack mode are depicted. In central single-crack mode and square computational area with a uniform mesh, computational times corresponding to the numerical schemes based on the high order finite element method for spatial discretization and Crank-Nicolson as well as semi-implicit techniques for temporal discretizations are 207.19s and 97.47s, respectively, with 2,048 elements, final time T = 0.2 and time step size τ = 0.01. Also, the simulations effectively illustrate a further decrease in computational time when the method is equipped with an adaptive mesh refinement strategy. The computational cost is reduced to 4.23s when the governed model is solved with the numerical scheme based on the adaptive high order finite element method and semi-implicit technique for spatial and temporal discretizations, respectively. Similarly, in other samples, the reduction of computational cost has been shown. Originality/value: This is the first time that the high-order finite element method is employed to solve the model investigated in the current paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. Developing computational methods of heat flow using bioheat equation enhancing skin thermal modeling efficiency.
- Author
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Ostadhossein, Rouhollah and Hoseinzadeh, Siamak
- Subjects
FINITE difference method ,FINITE volume method ,THERMAL efficiency ,SPECIFIC heat capacity ,FINITE differences ,EQUATIONS ,THERMAL conductivity - Abstract
Purpose: The main objective of this paper is to investigate the response of human skin to an intense temperature drop at the surface. In addition, this paper aims to evaluate the efficiency of finite difference and finite volume methods in solving the highly nonlinear form of Pennes' bioheat equation. Design/methodology/approach: One-dimensional linear and nonlinear forms of Pennes' bioheat equation with uniform grids were used to study the behavior of human skin. The specific heat capacity, thermal conductivity and blood perfusion rate were assumed to be linear functions of temperature. The nonlinear form of the bioheat equation was solved using the Newton linearization method for the finite difference method and the Picard linearization method for the finite volume method. The algorithms were validated by comparing the results from both methods. Findings: The study demonstrated the capacity of both finite difference and finite volume methods to solve the one-dimensional and highly nonlinear form of the bioheat equation. The investigation of human skin's thermal behavior indicated that thermal conductivity and blood perfusion rate are the most effective properties in mitigating a surface temperature drop, while specific heat capacity has a lesser impact and can be considered constant. Originality/value: This paper modeled the transient heat distribution within human skin in a one-dimensional manner, using temperate-dependent physical properties. The nonlinear equation was solved with two numerical methods to ensure the validity of the results, despite the complexity of the formulation. The findings of this study can help in understanding the behavior of human skin under extreme temperature conditions, which can be beneficial in various fields, including medical and engineering. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. SOME QUENCHING PROBLEMS FOR ω-DIFFUSION EQUATIONS ON GRAPHS WITH A POTENTIAL AND A SINGULAR SOURCE.
- Author
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B., EDJA Kouamé, A. T., DIABATÉ Paterne, C., N'DRI Kouakou, and A., TOURÉ Kidjegbo
- Subjects
LAPLACIAN operator ,EQUATIONS ,HYPOTHESIS - Abstract
In this paper, we study the quenching phenomenon related to the ω-diffusion equation on graphs with a potential and a singular source u
t (x, t) = Δω u(x, t) + b(x)(1 - u(x, t))-p, where Δω is called the discrete weighted Laplacian operator. Under some appropriate hypotheses, we prove the existence and uniqueness of the local solution via Banach fixed point theorem. We also show that the solution of the problem quenches in a finite time and that the time-derivative blows up at the quenching time. Moreover, we estimate the quenching time and the quenching rate. Finally, we verify our results through some numerical examples. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
26. A new two-step iterative technique for efficiently solving absolute value equations.
- Author
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Gul, Nisar, Chen, Haibo, Iqbal, Javed, and Shah, Rasool
- Subjects
ABSOLUTE value ,EQUATIONS - Abstract
Purpose: This work presents a new two-step iterative technique for solving absolute value equations. The developed technique is valuable and effective for solving the absolute value equation. Various examples are given to demonstrate the accuracy and efficacy of the suggested technique. Design/methodology/approach: In this paper, we present a new two-step iterative technique for solving absolute value equations. This technique is very straightforward, and due to the simplicity of this approach, it can be used to solve large systems with great effectiveness. Moreover, under certain assumptions, we examine the convergence of the proposed method using various theorems. Numerical outcomes are conducted to present the feasibility of the proposed technique. Findings: This paper gives numerical experiments on how to solve a system of absolute value equations. Originality/value: Nowadays, two-step approaches are very popular for solving equations (1). For solving AVEs, Liu in Shams (2021), Ning and Zhou (2015) demonstrated two-step iterative approaches. Moosaei et al. (2015) introduced a novel approach that utilizes a generalized Newton's approach and Simpson's rule to solve AVEs. Zainali and Lotfi (2018) presented a two-step Newton technique for AVEs that converges linearly. Feng and Liu (2016) have proposed minimization approaches for AVEs and presented their convergence under specific circumstances. Khan et al. (2023), suggested a nonlinear CSCS-like technique and a Picard-CSCS approach. Based on the benefits and drawbacks of the previously mentioned methods, we will provide a two-step iterative approach to efficiently solve equation (1). The numerical results show that our proposed technique converges rapidly and provides a more accurate solution. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. Helmholtz decomposition with a scalar Poisson equation in elastic anisotropic media.
- Author
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Xin-Yu Fang, Gang Yao, Qing-Qing Zheng, Ping-Min Zhang, Di Wu, and Feng-Lin Niu
- Subjects
HELMHOLTZ equation ,ANISOTROPY ,POISSON'S equation ,SHEAR waves ,COMPUTATIONAL complexity ,SPATIAL filters ,EQUATIONS ,FOURIER transforms - Abstract
P- and S-wave separation plays an important role in elastic reverse-time migration. It can reduce the artifacts caused by crosstalk between different modes and improve image quality. In addition, P- and Swave separation can also be used to better understand and distinguish wave types in complex media. At present, the methods for separating wave modes in anisotropic media mainly include spatial nonstationary filtering, low-rank approximation, and vector Poisson equation. Most of these methods require multiple Fourier transforms or the calculation of large matrices, which require high computational costs for problems with large scale. In this paper, an efficient method is proposed to separate the wave mode for anisotropic media by using a scalar anisotropic Poisson operator in the spatial domain. For 2D problems, the computational complexity required by this method is 1/2 of the methods based on solving a vector Poisson equation. Therefore, compared with existing methods based on pseudo-Helmholtz decomposition operators, this method can significantly reduce the computational cost. Numerical examples also show that the P and S waves decomposed by this method not only have the correct amplitude and phase relative to the input wavefield but also can reduce the computational complexity significantly. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Fixed-Point Iteration Method for Uncertain Parameters in Dynamic Response of Systems with Viscoelastic Elements.
- Author
-
Łasecka-Plura, Magdalena
- Subjects
DYNAMICAL systems ,VISCOELASTIC materials ,LINEAR equations ,LINEAR systems ,EQUATIONS of motion ,EQUATIONS - Abstract
The paper presents a method for determining the dynamic response of systems containing viscoelastic damping elements with uncertain design parameters. A viscoelastic material is characterized using classical and fractional rheological models. The assumption is made that the lower and upper bounds of the uncertain parameters are known and represented as interval values, which are then subjected to interval arithmetic operations. The equations of motion are transformed into the frequency domain using Laplace transformation. To evaluate the uncertain dynamic response, the frequency response function is determined by transforming the equations of motion into a system of linear interval equations. Nevertheless, direct interval arithmetic often leads to significant overestimation. To address this issue, this paper employs the element-by-element technique along with a specific transformation to minimize redundancy. The system of interval equations obtained is solved iteratively using the fixed-point iteration method. As demonstrated in the examples, this method, which combines the iterative solving of interval equations with the proposed technique of equation formulation, enables a solution to be found rapidly and significantly reduces overestimation. Notably, this approach has been applied to systems containing viscoelastic elements for the first time. Additionally, the proposed notation accommodates both parallel and series configurations of damping elements and springs within rheological models. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. On Liouville-type theorems for k-Hessian equations with gradient terms.
- Author
-
Doerr, Cameron and Mohammed, Ahmed
- Subjects
NONLINEAR equations ,EQUATIONS - Abstract
In this paper, we investigate several Liouville-type theorems related to k-Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to S_k[D^2u]\ge h(u,|Du|) in \mathbb {R}^n. The results depend on some qualified growth conditions of h at infinity. A Liouville-type result to subsolutions of a prototype equation S_k[D^2u]=f(u)+g(u)\varpi (|Du|) is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to S_k[D^2u]=f(u)+g(u)|Du|^q for 0\le q
- Published
- 2024
- Full Text
- View/download PDF
30. Analytical and Numerical Investigation for the Inhomogeneous Pantograph Equation.
- Author
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Aldosari, Faten and Ebaid, Abdelhalim
- Subjects
PANTOGRAPH ,INITIAL value problems ,CATENARY ,EQUATIONS - Abstract
This paper investigates the inhomogeneous version of the pantograph equation. The current model includes the exponential function as the inhomogeneous part of the pantograph equation. The Maclaurin series expansion (MSE) is a well-known standard method for solving initial value problems; it may be easier than any other approaches. Moreover, the MSE can be used in a straightforward manner in contrast to the other analytical methods. Thus, the MSE is extended in this paper to treat the inhomogeneous pantograph equation. The solution is obtained in a closed series form with an explicit formula for the series coefficients and the convergence of the series is proved. Also, the analytic solutions of some models in the literature are recovered as special cases of the present work. The accuracy of the results is examined through several comparisons with the available exact solutions of some classes in the relevant literature. Finally, the residuals are calculated and then used to validate the accuracy of the present approximations for some classes which have no exact solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. Representations of Solutions of Time-Fractional Multi-Order Systems of Differential-Operator Equations.
- Author
-
Umarov, Sabir
- Subjects
SYSTEMS theory ,ORDINARY differential equations ,EXISTENCE theorems ,DIFFERENTIAL equations ,EQUATIONS - Abstract
This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. Attached Flows for Reaction–Diffusion Processes Described by a Generalized Dodd–Bullough–Mikhailov Equation.
- Author
-
Ionescu, Carmen and Petrisor, Iulian
- Subjects
REACTION-diffusion equations ,NONLINEAR differential equations ,INTEGRAL equations ,EQUATIONS ,ANALYTICAL solutions - Abstract
This paper uses the attached flow method for solving nonlinear second-order differential equations of the reaction–diffusion type. The key steps of the method consist of the following: (i) reducing the differentiability order by defining the first derivative of the variable as a new variable called the flow and (ii) a forced decomposition of the derivative-free term so that the flow appears explicitly in it. The resulting reduced equation is solved using specific balancing rules. Only step (i) would lead to an Abel-type equation with complicated integral solutions. Completed with (ii) and with the graduation procedure, the attached flow method used in the paper, without requiring such a great effort, allows for the obtaining of accurate analytical solutions. The method is applied here to a subclass of reaction–diffusion equations, the generalized Dodd–Bulough–Mikhailov equation, which includes a translation of the variable and nonlinearities up to order five. The equation is solved for each order of nonlinearity, and the solutions are discussed following the values of the parameters involved in the equation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. STUDY ON THE INTERACTION SOLUTION OF ZAKHAROV-KUZNETSOV EQUATION IN QUANTUM PLASMA.
- Author
-
Zhen ZHAO, Yue LIU, Yanni ZHANG, and Jing PANG
- Subjects
QUANTUM plasmas ,EQUATIONS ,ELECTRON plasma ,LOW temperatures - Abstract
The fundamental difference between quantum and traditional plasmas is the electron and ion composition, the former has a much higher density and extremely lower temperature, and it can be modelled by Zakharov-Kuznetsov (ZK) equation. In this paper, the Hirota bilinear method is used to study its solution properties. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. GENERALIZED VARIATIONAL PRINCIPLES FOR THE MODIFIED BENJAMIN-BONA-MAHONY EQUATION IN THE FRACTAL SPACE.
- Author
-
Xiao-Qun CAO, Si-Hang XIE, Hong-Ze LENG, Wen-Long TIAN, and Jia-Le YAO
- Subjects
VARIATIONAL principles ,CALCULUS of variations ,POROUS materials ,EQUATIONS ,FRACTALS ,FRACTAL dimensions - Abstract
Because variational principles are very important for some methods to get the numerical or exact solutions, it is very important to seek explicit variational formulations for the non-linear PDE. At first, this paper describes the modified Benjamin-Bona-Mahony equation in fractal porous media or with irregular boundaries. Then, by designing skillfully the trial-Lagrange functional, variational principles are successfully established for the modified Benjamin-Bona-Mahony equation in the fractal space, respectively. Furthermore, the obtained variational principles are proved correct by minimizing the functionals with the calculus of variations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. Application of q-Homotopy analysis method via fractional complex transformation for time fractional coupled Jaulent-Miodek equation.
- Author
-
Farid, Samreen, Nawaz, Rashid, Shah, Inayat Ali, and Bushnaq, Samia
- Subjects
EQUATIONS ,SENSES - Abstract
In this paper, the approximate solution of time fractional coupled Jaulent-Miodek equation has been found using q-Homotopy Analysis Method (q-HAM) with the help of Fractional Complex Transformation (FCT). Fractional derivative are described in the Caputo sense. Results obtained by (q-HAM) are compared with two dimensional Hermite Wavelet method. Results of the proposed method admit a remarkable accuracy over two dimensional Hermite Wavelet method. The accuracy of (q-HAM) increases as we increase the value of n as shown in the paper. All calculation are presented with the use of Mathematica. For n = 1 (q-HAM) reduces to Homotopy Analysis Method (HAM). [ABSTRACT FROM AUTHOR]
- Published
- 2024
36. Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities.
- Author
-
Cingolani, Silvia, Gallo, Marco, and Tanaka, Kazunaga
- Subjects
NONLINEAR equations ,EQUATIONS ,SCALAR field theory ,MOUNTAIN pass theorem - Abstract
In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (− Δ) s u + μ u = ( I α * F (u)) F ′ (u) in R N , (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, "Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities," Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, "Nonlinear scalar field equations II: existence of infinitely many solutions," Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C
1 -regularity. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
37. Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons' conjecture.
- Author
-
Guo, Yahong, Ma, Lingwei, and Zhang, Zhenqiu
- Subjects
PARABOLIC operators ,ELLIPTIC operators ,LOGICAL prediction ,EQUATIONS ,NONLINEAR equations ,PROBLEM solving - Abstract
In this paper, we consider the general dual fractional parabolic problem ∂ t α u (x , t) + L u (x , t) = f (t , u (x , t)) in R n × R. We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons' conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L and the Marchaud time derivative ∂ t α , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons' conjecture. It is worth noting that our results are novel even for a special case of L , the fractional Laplacian (−Δ)
s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
38. Non-Abelian Toda-type equations and matrix valued orthogonal polynomials.
- Author
-
Deaño, Alfredo, Morey, Lucía, and Román, Pablo
- Subjects
SYMMETRIC matrices ,EQUATIONS ,MATRICES (Mathematics) ,NONABELIAN groups ,ABELIAN functions ,LAX pair ,ORTHOGONAL polynomials ,POLYNOMIALS - Abstract
In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. Using a Mix of Finite Difference Methods and Fractional Differential Transformations to Solve Modified Black–Scholes Fractional Equations.
- Author
-
Sugandha, Agus, Rusyaman, Endang, Sukono, and Carnia, Ema
- Subjects
FINITE difference method ,BLACK-Scholes model ,OPTIONS (Finance) ,INVESTORS ,EQUATIONS - Abstract
This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. A new efficient two-step iterative method for solving absolute value equations.
- Author
-
Khan, Alamgir, Iqbal, Javed, and Shah, Rasool
- Subjects
ABSOLUTE value ,NONLINEAR equations ,FINITE difference method ,EQUATIONS - Abstract
Purpose: This study presents a two-step numerical iteration method specifically designed to solve absolute value equations. The proposed method is valuable and efficient for solving absolute value equations. Several numerical examples were taken to demonstrate the accuracy and efficiency of the proposed method. Design/methodology/approach: We present a two-step numerical iteration method for solving absolute value equations. Our two-step method consists of a predictor-corrector technique. The new method uses the generalized Newton method as the predictor step. The four-point open Newton-Cotes formula is considered the corrector step. The convergence of the proposed method is discussed in detail. This new method is highly effective for solving large systems due to its simplicity and effectiveness. We consider the beam equation, using the finite difference method to transform it into a system of absolute value equations, and then solve it using the proposed method. Findings: The paper provides empirical insights into how to solve a system of absolute value equations. Originality/value: This paper fulfills an identified need to study absolute value equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. The General Solution to a Classical Matrix Equation AXB = C over the Dual Split Quaternion Algebra.
- Author
-
Si, Kai-Wen and Wang, Qing-Wen
- Subjects
QUATERNIONS ,ALGEBRA ,EQUATIONS ,MATRICES (Mathematics) - Abstract
In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation A X B = C , and present the general solution expression when the solvability conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations A X = C and X B = C . Finally, we provide a numerical example to demonstrate the findings of this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. A Hamiltonian equation produces a variety of Painlevé integrable equations: solutions of distinct physical structures.
- Author
-
Wazwaz, Abdul-Majid
- Subjects
PAINLEVE equations ,SOCIAL impact ,EQUATIONS ,SOCIAL services ,SOLITONS - Abstract
Purpose: The purpose of this paper is to investigate a variety of Painlevé integrable equations derived from a Hamiltonian equation. Design/methodology/approach: The newly developed Painlevé integrable equations have been handled by using Hirota's direct method. The authors obtain multiple soliton solutions and other kinds of solutions for these six models. Findings: The developed Hamiltonian models exhibit complete integrability in analogy with the original equation. Research limitations/implications: The present study is to address these two main motivations: the study of the integrability features and solitons and other useful solutions for the developed equations. Practical implications: The work introduces six Painlevé-integrable equations developed from a Hamiltonian model. Social implications: The work presents useful algorithms for constructing new integrable equations and for handling these equations. Originality/value: The paper presents an original work with newly developed integrable equations and shows useful findings. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. Existence and multiplicity of solutions for fractional p1(x,⋅)&p2(x,⋅)-Laplacian Schrödinger-type equations with Robin boundary conditions.
- Author
-
Zhang, Zhenfeng, An, Tianqing, Bu, Weichun, and Li, Shuai
- Subjects
VARIATIONAL principles ,MULTIPLICITY (Mathematics) ,FRACTIONAL differential equations ,EQUATIONS ,SCHRODINGER equation ,FOUNTAINS - Abstract
In this paper, we study fractional p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland's variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in R N ∖ Ω ‾ for fractional order p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Exact solutions to the (3+1)-dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients.
- Author
-
Hao, Yating and Gao, Ben
- Subjects
NONLINEAR differential equations ,PARTIAL differential equations ,FLUID dynamics ,PLASMA physics ,EQUATIONS - Abstract
The (3 + 1) -dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients, which have very crucial applications in various areas, such as fluid dynamics, plasma physics and so on, are studied via unified and generalised unified method in this work. Solitary, soliton, elliptic, singular (periodic type) and non-singular (soliton-type) solutions of these two equations are extracted using the unified method. Otherwise, polynomial solutions in double-wave form and rational solutions in double-soliton form of the modified KdV equation with variable coefficients are acquired by exploiting the generalised unified method. The dynamical demeanours of these solutions help to comprehend the physical phenomena reflected by the equations, are depicted and analysed graphically for specific values of randomly undetermined parameters, which are diverse in each solution. The outcomes show that these two methods are quite trustworthy and effective to explore numerous solutions of nonlinear partial differential equations. We recognise that two approaches have never been utilised to study these two equations and work carried out in this paper is fresh and handy. Compared to previous methods, more comprehensive solutions can be obtained using them in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations.
- Author
-
Regmi, Samundra, Argyros, Ioannis K., and George, Santhosh
- Subjects
DIFFERENTIABLE dynamical systems ,EQUATIONS ,BANACH spaces ,ALGORITHMS - Abstract
In this study, we extended the applicability of a derivative-free algorithm to encompass the solution of operators that may be either differentiable or non-differentiable. Conditions weaker than the ones in earlier studies are employed for the convergence analysis. The earlier results considered assumptions up to the existence of the ninth order derivative of the main operator, even though there are no derivatives in the algorithm, and the Taylor series on the finite Euclidian space restricts the applicability of the algorithm. Moreover, the previous results could not be used for non-differentiable equations, although the algorithm could converge. The new local result used only conditions on the divided difference in the algorithm to show the convergence. Moreover, the more challenging semi-local convergence that had not previously been studied was considered using majorizing sequences. The paper included results on the upper bounds of the error estimates and domains where there was only one solution for the equation. The methodology of this paper is applicable to other algorithms using inverses and in the setting of a Banach space. Numerical examples further validate our approach. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Orbital stability of the sum of N peakons for the mCH-Novikov equation.
- Author
-
Wang, Jiajing, Deng, Tongjie, and Zhang, Kelei
- Subjects
CUBIC equations ,EQUATIONS ,ENERGY consumption ,SHALLOW-water equations - Abstract
This paper investigates a generalized Camassa–Holm equation with cubic nonlinearities (alias the mCH-Novikov equation), which is a generalization of some special equations. The mCH-Novikov equation possesses well-known peaked solitary waves that are called peakons. The peakons were proved orbital stable by Chen et al. in [Stability of peaked solitary waves for a class of cubic quasilinear shallow-water equations. Int Math Res Not. 2022;1–33]. We mainly prove the orbital stability of the multi-peakons in the mCH-Novikov equation. In this paper, it is proved that the sum of N fully decoupled peaks is orbitally stable in the energy space by using energy argument, combining the orbital stability of single peakons and local monotonicity of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations.
- Author
-
Zhang, Yue, Wang, Qing-Wen, and Xie, Lv-Ming
- Subjects
QUATERNIONS ,EQUATIONS ,LEAST squares ,MATRICES (Mathematics) ,COMMUTATIVE algebra - Abstract
This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Dual Quaternion Matrix Equation AXB = C with Applications.
- Author
-
Chen, Yan, Wang, Qing-Wen, and Xie, Lv-Ming
- Subjects
QUATERNIONS ,AUTOMATIC differentiation ,EQUATIONS ,COMPUTER graphics ,HERMITIAN forms ,MATRICES (Mathematics) ,QUATERNION functions - Abstract
Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation A X B = C has been extensively studied. However, there is currently limited information on matrix equation A X B = C regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation A X B = C , and present the expression for the general solution when it is solvable. As an application, we derive the ϕ -Hermitian solutions for dual quaternion matrix equation A X A ϕ = C , where the ϕ -Hermitian extends the concepts of Hermiticity and η -Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Scattering for a class of inhomogeneous generalized Hartree equations.
- Author
-
Saanouni, Tarek and Peng, Congming
- Subjects
EQUATIONS ,CONSERVATION laws (Mathematics) ,BLOWING up (Algebraic geometry) ,NONLINEAR equations - Abstract
This paper studies the asymptotic behavior of energy solutions to a non-linear generalized Hartree equation. Indeed, in the inter-critical regime, one revisits the scattering versus finite time blow-up of energy solutions with non-necessarily spherically symmetric datum. Here, one uses the new approach due to Dodson and Murphy. The novelty in this work is to express the scattering threshold in terms of some non-conserved quantities. The main result of this note seems to be stronger than the classical scattering versus finite time blow-up dichotomy given in terms of the conserved mass and energy by Holmer and Roudenko. Indeed, as an application, one investigates the scattering in three different regimes: under, at and beyond the ground state threshold. The main result given here, which can be seen as a criteria of scattering versus finite time blow-up of energy solutions, enables us to give a unified approach to deal with the above generalised Hartree equation in different regimes. This paper follows some new ideas presented recently in the classical Schrödinger equation with a local source term, by V.D. Dinh. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. Sturmian comparison theorem for hyperbolic equations on a rectangular prism.
- Author
-
Özbekler, Abdullah, İşler, Kübra Uslu, and Alzabut, Jehad
- Subjects
NONLINEAR equations ,PRISMS ,EQUATIONS ,LINEAR equations ,EIGENVALUES ,HYPERBOLIC differential equations - Abstract
In this paper, new Sturmian comparison results were obtained for linear and nonlinear hyperbolic equations on a rectangular prism. The results obtained for linear equations extended those given by Kreith [Sturmian theorems on hyperbolic equations, Proc. Amer. Math. Soc., 22 (1969), 277-281] in which the Sturmian comparison theorem for linear equations was obtained on a rectangular region in the plane. For the purpose of verification, an application was described using an eigenvalue problem. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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