Journal articles: 'Euler Equation'

1

Lettau, Martin, and Sydney C. Ludvigson. "Euler equation errors." Review of Economic Dynamics 12, no. 2 (April 2009): 255–83. http://dx.doi.org/10.1016/j.red.2008.11.004.

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Bodaghi, Abasalt, Hossein Moshtagh, and Amir Mousivand. "Characterization and Stability of Multi-Euler-Lagrange Quadratic Functional Equations." Journal of Function Spaces 2022 (October 10, 2022): 1–9. http://dx.doi.org/10.1155/2022/3021457.

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The aim of the current article is to characterize and to prove the stability of multi-Euler-Lagrange quadratic mappings. In other words, it reduces a system of equations defining the multi-Euler-Lagrange quadratic mappings to an equation, say, the multi-Euler-Lagrange quadratic functional equation. Moreover, some results corresponding to known stability (Hyers, Rassias, and Gӑvruta) outcomes regarding the multi-Euler-Lagrange quadratic functional equation are presented in quasi- β -normed and Banach spaces by using the fixed point methods. Lastly, an example for the nonstable multi-Euler-Lagrange quadratic functional equation is indicated.

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Marian, Daniela, Sorina Anamaria Ciplea, and Nicolaie Lungu. "Hyers–Ulam Stability of Order k for Euler Equation and Euler–Poisson Equation in the Calculus of Variations." Mathematics 10, no. 15 (July 22, 2022): 2556. http://dx.doi.org/10.3390/math10152556.

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In this paper, we define and study Hyers–Ulam stability of order 1 for Euler’s equation and Hyers–Ulam stability of order m−1 for the Euler–Poisson equation in the calculus of variations in two special cases, when these equations have the form y″(x)=f(x) and y(m)(x)=f(x), respectively. We prove some estimations for Jyx−Jy0x, where y is an approximate solution and y0 is an exact solution of the corresponding Euler and Euler-Poisson equations, respectively. We also give two examples.

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Karthigai Selvam, S. "The Characteristic Equation of the Euler-Cauchy Differential Equation and its Related Solution Using MATLAB." Asian Journal of Science and Applied Technology 10, no. 1 (May 15, 2021): 1–4. http://dx.doi.org/10.51983/ajsat-2021.10.1.2792.

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The behavior of nature is usually modelled with Differential Equations in various forms. Depending on the constrains and the accuracy of a model, the connected equations may be more or less complicated. For simple models we may use Non Homogeneous Equations but in general, we have to deal with Homogeneous ones since from a physicists point of view nature seems to be Homogeneous. In many applications of sciences, for solving many of them, often appear equations of type nth order Linear differential equations, where the number of them is Euler-Cauchy differential equations. i.e. Euler-Cauchy differential equations often appear in analysis of computer algorithms, notably in analysis of quick sort and search trees; a number of physics and engineering applications. In this paper, the researcher aims to present the solutions of a homogeneous Euler-Cauchy differential equation from the roots of the characteristics equation related with this differential equation using MATLAB. It is hoped that this work can contribute to minimize the lag in teaching and learning of this important Ordinary Differential Equation.

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GUHA, PARTHA. "EULER–POINCARÉ FLOWS ON THE LOOP BOTT–VIRASORO GROUP AND SPACE OF TENSOR DENSITIES AND (2 + 1)-DIMENSIONAL INTEGRABLE SYSTEMS." Reviews in Mathematical Physics 22, no. 05 (June 2010): 485–505. http://dx.doi.org/10.1142/s0129055x10003989.

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Following the work of Ovsienko and Roger ([54]), we study loop Virasoro algebra. Using this algebra, we formulate the Euler–Poincaré flows on the coadjoint orbit of loop Virasoro algebra. We show that the Calogero–Bogoyavlenskii–Schiff equation and various other (2 + 1)-dimensional Korteweg–deVries (KdV) type systems follow from this construction. Using the right invariant H1 inner product on the Lie algebra of loop Bott–Virasoro group, we formulate the Euler–Poincaré framework of the (2 + 1)-dimensional of the Camassa–Holm equation. This equation appears to be the Camassa–Holm analogue of the Calogero–Bogoyavlenskii–Schiff type (2 + 1)-dimensional KdV equation. We also derive the (2 + 1)-dimensional generalization of the Hunter–Saxton equation. Finally, we give an Euler–Poincaré formulation of one-parameter family of (1 + 1)-dimensional partial differential equations, known as the b-field equations. Later, we extend our construction to algebra of loop tensor densities to study the Euler–Poincaré framework of the (2 + 1)-dimensional extension of b-field equations.

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Melliani, S., L. Chadli, and A. Harir. "Fuzzy Euler differential equation." SOP Transactions on Applied Mathematics 2, no. 1 (January 31, 2015): 1–12. http://dx.doi.org/10.15764/am.2015.01001.

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Hasan, Amna, Hakeem A. Othman, and Sami H. Altoum. "q-Euler Lagrange Equation." American Journal of Applied Sciences 16, no. 9 (September 1, 2019): 283–88. http://dx.doi.org/10.3844/ajassp.2019.283.288.

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8

CIPRA, BARRY A. "Solutions to Euler Equation." Science 239, no. 4839 (January 29, 1988): 464. http://dx.doi.org/10.1126/science.239.4839.464.

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9

CHEN, JUN. "SUBSONIC FLOWS FOR THE FULL EULER EQUATIONS IN HALF PLANE." Journal of Hyperbolic Differential Equations 06, no. 02 (June 2009): 207–28. http://dx.doi.org/10.1142/s0219891609001873.

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We study the subsonic flows governed by full Euler equations in the half plane bounded below by a piecewise smooth curve asymptotically approaching the x1-axis. Nonconstant conditions in the far field are prescribed to ensure the real Euler flows. The Euler system is reduced to a single elliptic equation for the stream function. The existence, uniqueness, and asymptotic behaviors of the solutions for the reduced equation are established by the Schauder fixed point argument and some delicate estimates. The existence of subsonic flows for the original Euler system is proved based on the results for the reduced equation, and their asymptotic behaviors in the far field are also obtained.

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10

Liu, Mingshuo, Huanhe Dong, Yong Fang, and Yong Zhang. "Lie Symmetry Analysis of Burgers Equation and the Euler Equation on a Time Scale." Symmetry 12, no. 1 (December 19, 2019): 10. http://dx.doi.org/10.3390/sym12010010.

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As a powerful tool that can be used to solve both continuous and discrete equations, the Lie symmetry analysis of dynamical systems on a time scale is investigated. Applying the method to the Burgers equation and Euler equation, we get the symmetry of the equation and single parameter groups on a time scale. Some group invariant solutions in explicit form for the traffic flow model simulated by a Burgers equation and Euler equation with a Coriolis force on a time scale are studied.

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11

Kulenović, M. R. S. "Oscillation of the Euler differential equation with delay." Czechoslovak Mathematical Journal 45, no. 1 (1995): 1–6. http://dx.doi.org/10.21136/cmj.1995.128506.

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12

Guha, Partha. "Nonholonomic deformation of coupled and supersymmetric KdV equations and Euler–Poincaré–Suslov method." Reviews in Mathematical Physics 27, no. 04 (May 2015): 1550011. http://dx.doi.org/10.1142/s0129055x15500117.

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Recently, Kupershmidt [38] presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by Karasu-Kalkani et al. [31]. In this paper, we demonstrate that Kupershmidt's method can be interpreted as an infinite-dimensional analogue of the Euler–Poincaré–Suslov (EPS) formulation. In a finite-dimensional case, we modify Kupershmidt's deformation of the Euler top equation to obtain the standard EPS construction on SO(3). We extend Kupershmidt's infinite-dimensional construction to construct a nonholonomic deformation of a wide class of coupled KdV equations, where all these equations follow from the Euler–Poincaré–Suslov flows of the right invariant L2 metric on the semidirect product group [Formula: see text], where Diff (S1) is the group of orientation preserving diffeomorphisms on a circle. We generalize our construction to the two-component Camassa–Holm equation. We also give a derivation of a nonholonomic deformation of the N = 1 supersymmetric KdV equation, dubbed as sKdV6 equation and this method can be interpreted as an infinite-dimensional supersymmetric analogue of the Euler–Poincaré–Suslov (EPS) method.

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13

Khalili Golmankhaneh, Alireza, and Carlo Cattani. "Fractal Logistic Equation." Fractal and Fractional 3, no. 3 (July 11, 2019): 41. http://dx.doi.org/10.3390/fractalfract3030041.

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In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics.

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14

Tang, Xiuli, Xiuqing Wang, and Ganshan Yang. "Stability and Unstability of the Standing Wave to Euler Equations." Advances in Applied Mathematics and Mechanics 9, no. 4 (January 18, 2017): 818–38. http://dx.doi.org/10.4208/aamm.2016.m1425.

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AbstractIn this paper, we first discuss the well-posedness of linearizing equations, and then study the stability and unstability of the 3-D compressible Euler Equation, by analysing the existence of saddle point. In addition, we give the existence of local solutions of the compressible Euler equation.

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Dao, Cong-Binh, and Viet-Chinh Mai. "Applications of common energy variability methods for establishing the equilibrium equation in mechanics." IOP Conference Series: Materials Science and Engineering 1289, no. 1 (August 1, 2023): 012067. http://dx.doi.org/10.1088/1757-899x/1289/1/012067.

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Abstract The principles of the energy variation method are commonly utilized in mechanics. Energy is a scalar variable, so these are more convenient and simple to establish the equilibrium equations compared to vector-based approaches (i.e. using forces and displacements). The present article applied the theorem of the energy variation method in order to set the equilibrium equations for various complicated problems. Four examples of applying the energy variation method include the differential equation of the Euler – Bernoulli beam based on the energy method, the system of Equilibrium Equations of the Euler–Bernoulli beam with the theorem of Least Work, the principle of maximum work to establish the equation of motions for the Euler–Bernoulli beam and the equation of motion for the Euler–Bernoulli beam by the virtual work theorem, have been implemented. The results obtained from this study open up further research directions on the application of the energy variation method in mechanics as well as in the analysis theory of beam bridges.

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SUKADANA, I. GEDE PUTU MIKI, I. NYOMAN WIDANA, and KETUT JAYANEGARA. "SOLUSI DARI PERSAMAAN CAUCHY–EULER NONHOMOGEN KASUS LOGARITMIK." E-Jurnal Matematika 9, no. 2 (May 26, 2020): 125. http://dx.doi.org/10.24843/mtk.2020.v09.i02.p289.

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Ordinary differential equation is one form of differential equations that are often found in everyday life. One form of ordinary differential equations which has non–constant coefficients is the Cauchy–Euler differential equation. In the nonhomogeneous Cauchy–Euler differential equations, the undetermined coefficient and the parameter variation were the most method that often used to find the particular solution. This paper aimed to show a new solution that was shorter than the previous methods for nonhomogeneous Cauchy–Euler differential equations with the right side was a logarithmic form. The new solution had been proven to produce the same solution as the ordinary solution sought using the undetermined coefficient method.

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17

Anderson, Douglas R., and Masakazu Onitsuka. "Hyers–Ulam Stability for Quantum Equations of Euler Type." Discrete Dynamics in Nature and Society 2020 (May 18, 2020): 1–10. http://dx.doi.org/10.1155/2020/5626481.

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Many applications using discrete dynamics employ either q-difference equations or h-difference equations. In this work, we introduce and study the Hyers–Ulam stability (HUS) of a quantum (q-difference) equation of Euler type. In particular, we show a direct connection between quantum equations of Euler type and h-difference equations of constant step size h with constant coefficients and an arbitrary integer order. For equation orders greater than two, the h-difference results extend first-order and second-order results found in the literature, and the Euler-type q-difference results are completely novel for any order. In many cases, the best HUS constant is found.

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18

Filipovic, Mirjana, and Miomir Vukobratovic. "Expansion of source equation of elastic line." Robotica 26, no. 6 (November 2008): 739–51. http://dx.doi.org/10.1017/s0263574708004347.

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SUMMARYThe paper is concerned with the relationship between the equation of elastic line motion, the “Euler-Bernoulli approach” (EBA), and equation of motion at the point of elastic line tip, the “Lumped-mass approach” (LMA). The Euler–Bernoulli equations (which have for a long time been used in the literature) should be expanded according to the requirements of the motion complexity of elastic robotic systems. The Euler–Bernoulli equation (based on the known laws of dynamics) should be supplemented with all the forces that are participating in the formation of the bending moment of the considered mode. This yields the difference in the structure of Euler–Bernoulli equations for each mode. The stiffness matrix is a full matrix. Mathematical model of the actuators also comprises coupling between elasticity forces. Particular integral of Daniel Bernoulli should be supplemented with the stationary character of elastic deformation of any point of the considered mode, caused by the present forces. General form of the elastic line is a direct outcome of the system motion dynamics, and can not be described by one scalar equation but by three equations for position and three equations for orientation of every point on that elastic line. Simulation results are shown for a selected robotic example involving the simultaneous presence of elasticity of the joint and of the link (two modes), as well the environment force dynamics.

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19

Liu, Hailiang, and Ferdinand Thein. "On the invariant region for compressible Euler equations with a general equation of state." Communications on Pure & Applied Analysis 20, no. 7-8 (2021): 2751. http://dx.doi.org/10.3934/cpaa.2021084.

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<p style="text-indent:20px;">The state space for solutions of the compressible Euler equations with a general equation of state is examined. An arbitrary equation of state is allowed, subject only to the physical requirements of thermodynamics. An invariant region of the resulting Euler system is identified and the convexity property of this region is justified by using only very minimal thermodynamical assumptions. Finally, we show how an invariant-region-preserving (IRP) limiter can be constructed for use in high order finite-volume type schemes to solve the compressible Euler equations with a general constitutive relation.</p>

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20

Duyunova, Anna, Valentin Lychagin, and Sergey Tychkov. "Quotients of Euler Equations on Space Curves." Symmetry 13, no. 2 (January 25, 2021): 186. http://dx.doi.org/10.3390/sym13020186.

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Quotients of partial differential equations are discussed. The quotient equation for the Euler system describing a one-dimensional gas flow on a space curve is found. An example of using the quotient to solve the Euler system is given. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations (ODEs). Possible solutions of the ODE system are discussed.

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21

Zang, Aibin. "Uniform time of existence of the smooth solution for 3D Euler-α equations with periodic boundary conditions." Mathematical Models and Methods in Applied Sciences 28, no. 10 (September 2018): 1881–97. http://dx.doi.org/10.1142/s0218202518500458.

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After reformulating the incompressible Euler-[Formula: see text] equations in 3D periodic domain, one obtains that there exists a unique classical solution of Euler-[Formula: see text] equations in uniform time interval independent of [Formula: see text]. It is shown that the solutions of the Euler-[Formula: see text] converge to the corresponding solutions of Euler equation in [Formula: see text] in space, uniformly in time. It also follows that the [Formula: see text] [Formula: see text] solutions of Euler-[Formula: see text] equations exist in any fixed sub-interval of the maximum existing interval for the Euler equations provided that initial velocity is regular enough and [Formula: see text] is sufficiently small.

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22

Shilin, A. P. "A hypersingular integro-differential equation of the Euler type." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 56, no. 1 (April 6, 2020): 17–29. http://dx.doi.org/10.29235/1561-2430-2020-56-1-17-29.

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In this paper, we study an integro-differential equation on a closed curve located on the complex plane. The integrals included in the equation are understood as a finite part by Hadamard. The coefficients of the equation have a particular structure. The analytical continuation method is applied. The equation is reduced to a boundary value linear conjugation problem for analytic functions and linear Euler differential equations in the domains of the complex plane. Solutions of the Euler equations, which are unambiguous analytical functions, are sought. The conditions of solvability of the initial equation are given explicitly. The solution of the initial equation obtained under these conditions is also given explicitly. Examples are considered.

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Dooner, David B., and Michael W. Griffis. "On Spatial Euler–Savary Equations for Envelopes." Journal of Mechanical Design 129, no. 8 (July 13, 2006): 865–75. http://dx.doi.org/10.1115/1.2735339.

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Presented are three equations that are believed to be original and new to the kinematics community. These three equations are extensions of the planar Euler–Savary relations (for envelopes) to spatial relations. All three spatial forms parallel the existing well established planar Euler–Savary equations. The genesis of this work is rooted in a system of cylindroidal coordinates specifically developed to parameterize the kinematic geometry of generalized spatial gearing and consequently a brief discussion of such coordinates is provided. Hyperboloids of osculation are introduced by considering an instantaneously equivalent gear pair. These analog equations establish a relation between the kinematic geometry of hyperboloids of osculation in mesh (viz., second-order approximation to the axode motion) to the relative curvature of conjugate surfaces in direct contact (gear teeth). Planar Euler–Savary equations are presented first along with a discussion on the terms in each equation. This presentation provides the basis for the proposed spatial Euler–Savary analog equations. A lot of effort has been directed to establishing generalized spatial Euler–Savary equations resulting in many different expressions depending on the interpretation of the planar Euler–Savary equation. This work deals with the interpretation where contacting surfaces are taken as the spatial analog to the contacting planar curves.

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24

Ossicini, Andrea. "On the Nature of Some Euler’s Double Equations Equivalent to Fermat’s Last Theorem." Mathematics 10, no. 23 (November 27, 2022): 4471. http://dx.doi.org/10.3390/math10234471.

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In this work, I provide a new rephrasing of Fermat’s Last Theorem, based on an earlier work by Euler on the ternary quadratic forms. Effectively, Fermat’s Last Theorem can be derived from an appropriate use of the concordant forms of Euler and from an equivalent ternary quadratic homogeneous Diophantine equation able to accommodate a solution of Fermat’s extraordinary equation. Following a similar and almost identical approach to that of A. Wiles, I tried to translate the link between Euler’s double equations (concordant/discordant forms) and Fermat’s Last Theorem into a possible reformulation of the Fermat Theorem. More precisely, through the aid of a Diophantine equation of second degree, homogeneous and ternary, solved not directly, but as a consequence of the resolution of the double Euler equations that originated it, I was able to obtain the following result: the intersection of the infinite solutions of Euler’s double equations gives rise to an empty set and this only by exploiting a well-known Legendre Theorem, which concerns the properties of all the Diophantine equations of the second degree, homogeneous and ternary. The impossibility of solving the second degree Diophantine equation thus obtained is possible using well-known techniques at the end of 18th century (see Euler, Lagrange and Legendre) and perhaps present in Fermat’s brilliant mind.

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25

Fabian, Andrew, and Hieu D. Nguyen. "Paradoxical Euler: Integrating by Differentiating." Mathematical Gazette 97, no. 538 (March 2013): 61–74. http://dx.doi.org/10.1017/s002555720000543x.

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Every student of calculus learns that one typically solves a differential equation by integrating it. However, as Euler showed in his 1758 paper (E236),Exposition de quelques paradoxes dans le calcul intégral(Explanation of certain paradoxes in integral calculus) [1], there are differential equations that can be solved by actually differentiating them again. This initially seems paradoxical or, as Euler describes it in the introduction of his paper:Here I intend to explain a paradox in integral calculus that will seem rather strange: this is that we sometimes encounter differential equations in which it would seem very difficult to find the integrals by the rules of integral calculus yet are still easily found. not by the method of integration. but rather in differentiating the proposed equation again; so in these cases, a repeated differentiation leads us to the sought integral.

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Martynov, V. K. "Euler Elastics and its Application." Izvestiya MGTU MAMI 2, no. 2 (January 20, 2008): 147–52. http://dx.doi.org/10.17816/2074-0530-69678.

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In engineering practice the solutions of nonlinear differential equations assume ever greater importance. Euler equation of elastic is the particular solution of such kind. The paper examines approximate solution and its applications. The comparison with previously obtained results is made.

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ELMACI, DENİZ, NURCAN BAYKUŞ SAVAŞANERİL, FADİME DAL, and MEHMET SEZER. "EULER AND TAYLOR POLYNOMIALS METHOD FOR SOLVING VOLTERRA TYPE INTEGRO DIFFERENTIAL EQUATIONS WITH NONLINEAR TERMS." Journal of Science and Arts 21, no. 2 (June 30, 2021): 395–406. http://dx.doi.org/10.46939/j.sci.arts-21.2-a07.

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In this study, the first order nonlinear Volterra type integro-differential equations are used in order to identify approximate solutions concerning Euler polynomials of a matrix method based on collocation points. This method converts the mentioned nonlinear integro-differential equation into the matrix equation with the utilization of Euler polynomials along with collocation points. The matrix equation is a system of nonlinear algebraic equations with the unknown Euler coefficients. Additionally, this approach provides analytic solutions, if the exact solutions are polynomials. Furthermore, some illustrative examples are presented with the aid of an error estimation by using the Mean-Value Theorem and residual functions. The obtained results show that the developed method is efficient and simple enough to be applied. And also, convergence of the solutions of the problems were examined. In order to obtain the matrix equations and solutions for the selected problems, code was developed in MATLAB.

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You, Cuilian, Yan Cheng, and Hongyan Ma. "Stability of Euler Methods for Fuzzy Differential Equation." Symmetry 14, no. 6 (June 20, 2022): 1279. http://dx.doi.org/10.3390/sym14061279.

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The Liu process is a fuzzy process whose membership function is a symmetric function on an expected value. The object of this paper was a fuzzy differential equation driven by Liu process. Since the existing fuzzy Euler solving methods (explicit Euler scheme, semi-implicit Euler scheme, and implicit Euler scheme) have the same convergence, to compare them, we presented four stabilities, i.e., asymptotical stability, mean square stability, exponential stability, and A stability. By choosing special fuzzy differential equation as a test equation, we deduced that mean square stability is equivalent to exponential stability. Furthermore, an explicit fuzzy Euler scheme and semi-implicit fuzzy Euler scheme showed asymptotical stability and mean square stability, while an explicit fuzzy Euler scheme failed to meet A stability but that an implicit fuzzy Euler scheme is A stable, and whether semi-implicit fuzzy Euler scheme is A stable depends on the values of α and λ.

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Yilmazer, R., and O. Ozturk. "N-Fractional Calculus Operator Method to the Euler Equation." Issues of Analysis 25, no. 2 (December 2018): 144–52. http://dx.doi.org/10.15393/j3.art.2018.5730.

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Skałba, Mariusz. "On Euler-von Mangoldt's equation." Colloquium Mathematicum 69, no. 1 (1996): 143–45. http://dx.doi.org/10.4064/cm-69-1-143-145.

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Papanicolaou, Vassilis G. "The periodic Euler-Bernoulli equation." Transactions of the American Mathematical Society 355, no. 09 (May 29, 2003): 3727–59. http://dx.doi.org/10.1090/s0002-9947-03-03315-4.

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Alshaikey, Salha, Narjess T. Khalifa, Hakeem A. Othman, and Hafedh Rguigui. "Quantum generalized Euler heat equation." St. Petersburg Polytechnical University Journal: Physics and Mathematics 3, no. 4 (December 2017): 365–76. http://dx.doi.org/10.1016/j.spjpm.2017.10.006.

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Hsu, Cheng-Hsiung, Song-Sun Lin, and Tetu Makino. "On the relativistic Euler equation." Methods and Applications of Analysis 8, no. 1 (2001): 159–208. http://dx.doi.org/10.4310/maa.2001.v8.n1.a7.

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Fatima, Aeeman, Fazal M. Mahomed, and Chaudry Masood Khalique. "Noether symmetries and exact solutions of an Euler–Bernoulli beam model." International Journal of Modern Physics B 30, no. 28n29 (November 10, 2016): 1640011. http://dx.doi.org/10.1142/s0217979216400117.

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In this paper, a Noether symmetry analysis is carried out for an Euler–Bernoulli beam equation via the standard Lagrangian of its reduced scalar second-order equation which arises from the standard Lagrangian of the fourth-order beam equation via its Noether integrals. The Noether symmetries corresponding to the reduced equation is shown to be the inherited Noether symmetries of the standard Lagrangian of the beam equation. The corresponding Noether integrals of the reduced Euler–Lagrange equations are deduced which remarkably allows for three families of new exact solutions of the static beam equation. These are shown to contain all the previous solutions obtained from the standard Lie analysis and more.

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QU, KUN, CHANG SHU, and JINSHENG CAI. "DEVELOPING LBM-BASED FLUX SOLVER AND ITS APPLICATIONS IN MULTI-DIMENSION SIMULATIONS." International Journal of Modern Physics: Conference Series 19 (January 2012): 90–99. http://dx.doi.org/10.1142/s2010194512008628.

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In this paper, a new flux solver was developed based on a lattice Boltzmann model. Different from solving discrete velocity Boltzmann equation and lattice Boltzmann equation, Euler/Navier-Stokes (NS) equations were solved in this approach, and the flux at the interface was evaluated with a compressible lattice Boltzmann model. This method combined lattice Boltzmann method with finite volume method to solve Euler/NS equations. The proposed approach was validated by some simulations of one-dimensional and multi-dimensional problems.

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Li, Guanfeng, Yong Wang, and Gejun Bao. "Variational Integrals of a Class of Nonhomogeneous𝒜-Harmonic Equations." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/697974.

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We introduce a class of variational integrals whose Euler equations are nonhomogeneous𝒜-harmonic equations. We investigate the relationship between the minimization problem and the Euler equation and give a simple proof of the existence of some nonhomogeneous𝒜-harmonic equations by applying direct methods of the calculus of variations. Besides, we establish some interesting results on variational integrals.

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Alan, Sule, Kadir Atalay, and Thomas F. Crossley. "EULER EQUATION ESTIMATION ON MICRO DATA." Macroeconomic Dynamics 23, no. 8 (June 25, 2018): 3267–92. http://dx.doi.org/10.1017/s1365100518000032.

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Consumption Euler equations are important tools in empirical macroeconomics. When estimated on micro data, they are typically linearized, so standard IV or GMM methods can be employed to deal with the measurement error that is endemic to survey data. However, linearization, in turn, may induce serious approximation bias. We numerically solve and simulate six different life-cycle models, and then use the simulated data as the basis for a series of Monte Carlo experiments in which we evaluate the performance of linearized Euler equation estimation. We sample from the simulated data in ways that mimic realistic data structures. The linearized Euler equation leads to biased estimates of the EIS, but that bias is modest when there is a sufficient time dimension to the data, and sufficient variation in interest rates. However, a sufficient time dimension can only realistically be achieved with a synthetic cohort. Estimates from synthetic cohorts of sufficient length, while often exhibiting small mean bias, are quite imprecise. We also show that in all data structures, estimates are less precise in impatient models.

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ESCHER, JOACHIM, and MARCUS WUNSCH. "RESTRICTIONS ON THE GEOMETRY OF THE PERIODIC VORTICITY EQUATION." Communications in Contemporary Mathematics 14, no. 03 (June 2012): 1250016. http://dx.doi.org/10.1142/s0219199712500162.

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We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group DIFF∞(𝕊1) of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, cf. [A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. 162 (2005) 1377–1389], the axisymmetric Euler flow in ℝd (see [H. Okamoto and J. Zhu, Some similarity solutions of the Navier–Stokes equations and related topics, Taiwanese J. Math. 4 (2000) 65–103]), and De Gregorio's vorticity model equation as introduced in [S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys. 59 (1990) 1251–1263].

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Verma, Mahendra K. "Boltzmann equation and hydrodynamic equations: their equilibrium and non-equilibrium behaviour." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2175 (June 22, 2020): 20190470. http://dx.doi.org/10.1098/rsta.2019.0470.

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This short article summarizes the key features of equilibrium and non-equilibrium aspects of Boltzmann and hydrodynamic equations. Under equilibrium, the Boltzmann equation generates uncorrelated random velocity that corresponds to k 2 energy spectrum for the Euler equation. The latter spectrum is produced using initial configuration with many Fourier modes of equal amplitudes but with random phases. However, for a large-scale vortex as an initial condition, earlier simulations exhibit a combination of k −5/3 (in the inertial range) and k 2 (for large wavenumbers) spectra, with the range of k 2 spectrum increasing with time. These simulations demonstrate an approach to equilibrium or thermalization of Euler turbulence. In addition, they also show how initial velocity field plays an important role in determining the behaviour of the Euler equation. In non-equilibrium scenario, both Boltzmann and Navier–Stokes equations produce similar flow behaviour, for example, Kolmogorov’s k −5/3 spectrum in the inertial range. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

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Iantovics, Laszlo Barna, and Florin Felix Nichita. "On the Colored and the Set-Theoretical Yang–Baxter Equations." Axioms 10, no. 3 (July 2, 2021): 146. http://dx.doi.org/10.3390/axioms10030146.

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This paper is related to several articles published in AXIOMS, SCI, etc. The main concepts of the current paper are the colored Yang–Baxter equation and the set-theoretical Yang–Baxter equation. The Euler formula, colagebra structures, and means play an important role in our study. We show that some new solutions for a certain system of equations lead to colored Yang–Baxter operators, which are related to an Euler formula for matrices, and the set-theoretical solutions to the Yang–Baxter equation are related to means. A new coalgebra is obtained and studied.

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Adji, Abdourahman Haman, and Shankishvili Lamara Dmitrievna. "On Classical and Distributional Solutions of a Higher Order Singular Linear Differential Equation in the Space K’." International Journal of Pure Mathematics 10 (April 11, 2023): 1–7. http://dx.doi.org/10.46300/91019.2023.10.1.

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In this research work, we aim to find and describe all the classical solutions of the homogeneous linear singular differential equation of order l in the space of K' distributions. Recall that in our previous research, the results of which have been published in some journals, we had undertaken similar studies in the case of a singular differential equation of the Euler type of second order, when the conditions were carried out. That said, our intentions in this article are therefore to generalize the results obtained and recently published, focusing our research on the situation of the homogeneous singular linear differential equation of order l of Euler type. In this orientation, we base ourselves on the classical theory of ordinary linear differential equations and look for the particular solution to the equation considered in the form of the distribution with a parameter to be determined, which we replace in the latter. Depending on the nature of the roots of the characteristic polynomial of the homogeneous equation we identify, case by case, all the solutions indicated in the sense of distributions in the space K'. In this same work, we return to the non-homogeneous equation of order l of the same Euler type, whose second member consists only of the derivative of order s of the Dirac-delta distribution studied in our previous work, to fully describe all the solutions of the latter in the sense of distributions in the space K'. We finalize this work by making an important remark emphasizing the interest in undertaking research of the same objective of finding a general solution, by studying the singular differential equations of the same higher-order l with the particularity of being of Euler types on the left and Euler on the right in the space of distributions K’.

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Gavrilyuk, Sergey, and Keh-Ming Shyue. "Hyperbolic approximation of the BBM equation." Nonlinearity 35, no. 3 (February 18, 2022): 1447–67. http://dx.doi.org/10.1088/1361-6544/ac4c49.

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Abstract It is well known that the Benjamin–Bona–Mahony (BBM) equation can be seen as the Euler–Lagrange equation for a Lagrangian expressed in terms of the solution potential. We approximate the Lagrangian by a two-parameter family of Lagrangians depending on three potentials. The corresponding Euler–Lagrange equations can be then written as a hyperbolic system of conservations laws. The hyperbolic BBM system has two genuinely nonlinear eigenfields and one linear degenerate eigenfield. Moreover, it can be written in terms of Riemann invariants. Such an approach conserves the variational structure of the BBM equation and does not introduce the dissipation into the governing equations as it usually happens for the classical relaxation methods. The state-of-the-art numerical methods for hyperbolic conservation laws such as the Godunov-type methods are used for solving the ‘hyperbolized’ dispersive equations. We find good agreement between the corresponding solutions for the BBM equation and for its hyperbolic counterpart.

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Fiori, Simone. "Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors." Mathematics 7, no. 10 (October 10, 2019): 935. http://dx.doi.org/10.3390/math7100935.

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The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler–Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge–Kutta integration method tailored to Lie groups.

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Moradpour, H., Y. Heydarzade, C. Corda, A. H. Ziaie, and S. Ghaffari. "Black hole solutions and Euler equation in Rastall and generalized Rastall theories of gravity." Modern Physics Letters A 34, no. 37 (December 6, 2019): 1950304. http://dx.doi.org/10.1142/s0217732319503048.

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Focusing on the special case of generalized Rastall theory, as a subclass of the non-minimal curvature-matter coupling theories in which the field equations are mathematically similar to the Einstein field equations in the presence of cosmological constant, we find two classes of black hole (BH) solutions including (i) conformally flat solutions and (ii) non-singular BHs. Accepting the mass function definition and by using the entropy contents of the solutions along with thermodynamic definitions of temperature and pressure, we study the validity of Euler equation on the corresponding horizons. Our results show that the thermodynamic pressure, meeting the Euler equation, is not always equal to the pressure components appeared in the gravitational field equations and satisfies the first law of thermodynamics, a result which in fact depends on the presumed energy definition. The requirements of having solutions with equal thermodynamic and Hawking temperatures are also studied. Additionally, we study the conformally flat BHs in the Rastall framework. The consequences of employing generalized Misner–Sharp mass in studying the validity of the Euler equation are also addressed.

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Hutahaean, Syawaluddin. "Wavelength and Wave Period Relationship with Wave Amplitude: A Velocity Potential Formulation." International Journal of Advanced Engineering Research and Science 9, no. 8 (2022): 387–93. http://dx.doi.org/10.22161/ijaers.98.44.

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In this study, the equation that expresses the explicit relationship between the wave number and wave amplitude, as well as wave period and wave amplitude are established. The wave number and the wave period are calculated solely using the input wave amplitude. The equation is formulated with the velocity potential of the solution to Laplace’s equation to the hydrodynamic conservation equations, such as the momentum equilibrium equation, Euler Equation for conservation of momentum, and by working on the kinematic bottom and free surface boundary condition.In this study, the equation that expresses the explicit relationship between the wave number and wave amplitude, as well as wave period and wave amplitude are established. The wave number and the wave period are calculated solely using the input wave amplitude. The equation is formulated with the velocity potential of the solution to Laplace’s equation to the hydrodynamic conservation equations, such as the momentum equilibrium equation, Euler Equation for conservation of momentum, and by working on the kinematic bottom and free surface boundary condition.

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Lu, Chang-Na, Sheng-Xiang Chang, Luo-Yan Xie, and Zong-Guo Zhang. "Generation and solutions to the time-space fractional coupled Navier-Stokes equations." Thermal Science 24, no. 6 Part B (2020): 3899–905. http://dx.doi.org/10.2298/tsci2006899l.

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In this paper, a Lagrangian of the coupled Navier-Stokes equations is proposed based on the semi-inverse method. The fractional derivatives in the sense of Riemann-Liouville definition are used to replace the classical derivatives in the Lagrangian. Then the fractional Euler-Lagrange equation can be derived with the help of the fractional variational principles. The Agrawal?s method is devot?ed to lead to the time-space fractional coupled Navier-Stokes equations from the above Euler-Lagrange equation. The solution of the time-space fractional coupled Navier-Stokes equations is obtained by means of RPS algorithm. The numerical results are presented by using exact solutions.

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Baculíková, Blanka, and Jozef Džurina. "Oscillation of the third order Euler differential equation with delay." Mathematica Bohemica 139, no. 4 (2014): 649–55. http://dx.doi.org/10.21136/mb.2014.144141.

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Wang, Jinhuan, Yicheng Pang, and Yu Zhang. "Limits of Solutions to the Isentropic Euler Equations for van der Waals Gas." International Journal of Nonlinear Sciences and Numerical Simulation 20, no. 3-4 (May 26, 2019): 461–73. http://dx.doi.org/10.1515/ijnsns-2018-0263.

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AbstractIn this paper, we consider limit behaviors of Riemann solutions to the isentropic Euler equations for a non-ideal gas (i.e. van der Waals gas) as the pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for van der Waals gas is solved. Then it is proved that, as the pressure vanishes, any Riemann solution containing two shock waves to the isentropic Euler equation for van der Waals gas converges to the delta shock solution to the transport equations and any Riemann solution containing two rarefaction waves tends to the vacuum state solution to the transport equations. Finally, some numerical simulations completely coinciding with the theoretical analysis are demonstrated.

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Geng, Shifeng, Feimin Huang, Guanghui Jin, and Xiaochun Wu. "The time asymptotic expansion for the compressible Euler equations with time-dependent damping." Nonlinearity 36, no. 10 (August 29, 2023): 5075–96. http://dx.doi.org/10.1088/1361-6544/acecf6.

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Abstract For the compressible Euler equations with time-dependent damping − 1 ( 1 + t ) λ ρ u , we propose a time asymptotic expansion around the self-similar solution of the generalized porous media equation and rigorously justify this expansion as λ ∈ ( 1 7 , 1 ) . Furthermore, the above expansion is the best asymptotic profile of the solution to the compressible Euler equations with time-dependent damping.

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Janev, Marko, Teodor Atanackovic, and Stevan Pilipovic. "Noether’s theorem for Herglotz type variational problems utilizing complex fractional derivatives." Theoretical and Applied Mechanics 48, no. 2 (2021): 127–42. http://dx.doi.org/10.2298/tam210913011j.

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This is a review article which elaborates the results presented in [1], where the variational principle of Herglotz type with a Lagrangian that depends on fractional derivatives of both real and complex orders is formulated and the invariance of this principle under the action of a local group of symmetries is determined. The conservation law for the corresponding fractional Euler Lagrange equation is obtained and a sequence of approximations of a fractional Euler?Lagrange equation by systems of integer order equations established and analyzed.

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Journal articles on the topic 'Euler Equation'

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