Journal articles: 'Third degree equations'

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Dawidowicz, Antoni Leon. "How to solve third degree equations without moving to complex numbers." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 12 (December 31, 2020): 123–31. http://dx.doi.org/10.24917/20809751.12.6.

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During the Renaissance, the theory of algebraic equations developed in Europe. It is about finding a solution to the equation of the formanxn + . . . + a1x + a0 = 0,represented by coefficients subject to algebraic operations and roots of any degree. In the 16th century, algorithms for the third and fourth-degree equations appeared. Only in the nineteenth century, a similar algorithm for thehigher degree was proved impossible. In (Cardano, 1545) described an algorithm for solving third-degree equations. In the current version of this algorithm, one has to take roots of complex numbers that even Cardano didnot know.This work proposes an algorithm for solving third-degree algebraic equations using only algebraic operations on real numbers and elementary functions taught at High School.

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Baica, Malvina. "Diophantine equations and identities." International Journal of Mathematics and Mathematical Sciences 8, no. 4 (1985): 755–77. http://dx.doi.org/10.1155/s0161171285000849.

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The general diophantine equations of the second and third degree are far from being totally solved. The equations considered in this paper are i) x2−my2=±1 ii) x3+my3+m2z3−3mxyz=1iii) Some fifth degree diopantine equationsInfinitely many solutions of each of these equations will be stated explicitly, using the results from the ACF discussed before.It is known that the solutions of Pell's equation are well exploited. We include it here because we shall use a common method to solve these three above mentioned equations and the method becomes very simple in Pell's equations case.Some new third and fifth degree combinatorial identities are derived from units in algebraic number fields.

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Mačys, Juvencijus, and Jurgis Sušinskas. "Classification of cubic equations." Lietuvos matematikos rinkinys, no. 59 (December 20, 2018): 46–53. http://dx.doi.org/10.15388/lmr.b.2018.7.

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Rather unexpectably all real equations of the fourth degree are solvable by real means. So we canclassify all real equations of the third and fourth degree. In this article we classify real cubics. Thereal quartics will be classified in another article.

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Doytsher, Yerahmiel, and John K. Hall. "Interpolation of DTM using bi-directional third-degree parabolic equations, with FORTRAN subroutines." Computers & Geosciences 23, no. 9 (November 1997): 1013–20. http://dx.doi.org/10.1016/s0098-3004(97)00061-7.

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Migda, Janusz, Małgorzata Migda, and Magdalena Nockowska-Rosiak. "Asymptotic properties of solutions of third order difference equations." Applicable Analysis and Discrete Mathematics 14, no. 1 (2020): 1–19. http://dx.doi.org/10.2298/aadm180826006m.

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We consider the difference equation of the form ?(rn?(pn?xn)) = anf (x?(n)) + bn. We present sufficient conditions under which, for a given solution y of the equation ?(rn?(pn?yn)) = 0, there exists a solution x of the nonlinear equation with the asymptotic behavior xn = yn + zn, where z is a sequence convergent to zero. Our approach allows us to control the degree of approximation, i.e., the rate of convergence of the sequence We examine two types of approximation: harmonic approximation when zn = o(ns), s ? 0, and geometric approximation when zn = o(?n), ? ? (0, 1).

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Gordoa, Pilar R., and Andrew Pickering. "New Bäcklund transformations for the third and fourth Painlevé equations to equations of second order and higher degree." Physics Letters A 282, no. 3 (April 2001): 152–56. http://dx.doi.org/10.1016/s0375-9601(01)00178-5.

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Houzel, Christian. "Sharaf al-Dīn al-Ṭūsī et le polygone de Newton." Arabic Sciences and Philosophy 5, no. 2 (September 1995): 239–62. http://dx.doi.org/10.1017/s0957423900002046.

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The Treatise on Equations of Sharaf al-Dīn al-Ṭūsī (2nd half of the 12th century) is in the tradition of ‛Umar al-Khayyām (d. 1131). However, it has two special features. First, it contains a full discussion of the existence of a solution for third-degree equations, which al-Ṭūsī establishes by proving that the conic curves that represent this solution effectively intersect – a proof based on an intuitive notion of connexity. Secondly, al-Ṭūsī develops algorithms for the numerical resolution of these third-degree equations. The first stage of one of these algorithms follows a procedure which is akin to the so-called method of Newton's polygon.

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Ammar, Boukhemis, and Zerouki Ebtissem. "Classical2-orthogonal polynomials and differential equations." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–32. http://dx.doi.org/10.1155/ijmms/2006/12640.

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We construct the linear differential equations of third order satisfied by the classical2-orthogonal polynomials. We show that these differential equations have the following form:R4,n(x)Pn+3(3)(x)+R3,n(x)P″n+3(x)+R2,n(x)P′n+3(x)+R1,n(x)Pn+3(x)=0, where the coefficients{Rk,n(x)}k=1,4are polynomials whose degrees are, respectively, less than or equal to4,3,2, and1. We also show that the coefficientR4,n(x)can be written asR4,n(x)=F1,n(x)S3(x), whereS3(x)is a polynomial of degree less than or equal to3with coefficients independent ofnanddeg⁡(F1,n(x))≤1. We derive these equations in some cases and we also quote some classical2-orthogonal polynomials, which were the subject of a deep study.

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Ульянова, И., and I. Ulyanova. "Introduction of a New Variable Asa Universal Technique of the Solution of Rational Equations Above the Second Degree." Profession-Oriented School 6, no. 2 (May 22, 2018): 31–37. http://dx.doi.org/10.12737/article_5ae471f61bbbd2.24029294.

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The article reveals the role of equations at the basic and profi le levels of teaching students mathematics at school. Various ways of solving rational equations of the third and fourth degree, based on the technique of the introduction of a new variable, are demonstrated. This technique is universal in mathematics. It fi nds its application both in algebra, in the solution of inequalities, in equations and their systems, and in geometry, where the additionally constructed geometric fi gure — a segment, an angle, a circle, a triangle — appears as a new variable. The methods of solving rational equations, shown by the author in this article, will be especially useful and interesting for students of profi le classes, as well as for applicants and students of higher educational institutions and mathematics teachers.

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LLIBRE, JAUME, ANA CRISTINA MEREU, and MARCO ANTONIO TEIXEIRA. "Limit cycles of the generalized polynomial Liénard differential equations." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 2 (November 12, 2009): 363–83. http://dx.doi.org/10.1017/s0305004109990193.

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AbstractWe apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ + f(x)ẋ + g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m − 1)/2] limit cycles, where [·] denotes the integer part function.

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Hatch, C. T., and A. P. Pisano. "Modeling, Simulation, and Modal Analysis of a Hydraulic Valve Lifter With Oil Compressibility Effects." Journal of Mechanical Design 113, no. 1 (March 1, 1991): 46–54. http://dx.doi.org/10.1115/1.2912750.

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A two-degree-of-freedom (2-DOF), analytical model of a hydraulic valve lifter is derived. Special features of the model include the effects of bulk oil compressibility, multimode behavior due to plunger check valve modeling, and provision for the inclusion of third and fourth body displacements to aid in the use of the model in extended, multi-DOF systems. It is shown that motion of the lifter plunger and body must satisfy a coupled system of third-order, nonlinear differential equations of motion. It is also shown that the special cases of zero oil compressibility and/or 1-DOF motion of lifter plunger can be obtained from the general third-order equations. For the case of zero oil compressibility, using Newtonian fluid assumptions, the equations of motion are shown to reduce to a system of second-order, linear differential equations. The differential equations are numerically integrated in five scenarios designed to test various aspects of the model. A modal analysis of the 2-DOF, compressible model with an external contact spring is performed and is shown to be in excellent agreement with simulation results.

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LIN, XIAOJIE, BENSHENG ZHAO, and ZENGJI DU. "A third-order multi-point boundary value problem at resonance with one three dimensional kernel space." Carpathian Journal of Mathematics 30, no. 1 (2014): 93–100. http://dx.doi.org/10.37193/cjm.2014.01.13.

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This paper deals with a third order nonlinear differential equations with multi-point boundary conditions. By using the coincidence degree theory, we establish some existence results of the problem at resonance under some appropriate conditions. The emphasis here is that the dimension of the linear operator is equal to three. We also give an example to demonstrate our results.

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Mazrooei-Sebdani, Reza. "Homogeneous rational difference equations of degree 1: convergence, monotone and oscillatory solutions for second- and third-order cases." Journal of Difference Equations and Applications 18, no. 12 (December 2012): 1979–2018. http://dx.doi.org/10.1080/10236198.2011.606177.

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Yakuto, K. L. "POSITIVE INTEGER SOLUTION OF THE MATRIX EQUATION Xn = A FOR THIRD-ORDER MATRICES IN THE CASE OF POSITIVE INTEGERS n." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 54, no. 2 (July 1, 2018): 164–78. http://dx.doi.org/10.29235/1561-2430-2018-54-2-164-178.

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The problem of the positive integer solution of the equation Xn = A for different-order matrices is important to solve a large range of problems related to the modeling of economic and social processes. The need to solve similar problems also arises in areas such as management theory, dynamic programming technique for solving some differential equations. In this connection, it is interesting to question the existence of positive and positive integer solutions of the nonlinear equations of the form Xn = A for different-order matrices in the case of the positive integer n. The purpose of this work is to explore the possibility of using analytical methods to obtain positive integer solutions of nonlinear matrix equations of the form Xn = A where A, X are the third-order matrices, n is the positive integer. Elements of the original matrix A are integer and positive numbers. The present study found that when the root of the nth degree of the third-order matrix will have zero diagonal elements and nonzero and positive off-diagonal elements, the root of the nth degree of the third-order matrix will have two zero diagonal elements and nonzero positive off-diagonal elements. It was shown that to solve the problem of finding positive integer solutions of the matrix equation for third-order matrices in the case of the positive integer n, the analytical techniques can be used. The article presents the formulas that allow one to find the roots of positive integer matrices for n = 3,…,5. However, the methodology described in the article can be adopted to find the natural roots of the third-order matrices for large n.

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HAN, M., Y. LIN, and P. YU. "A STUDY ON THE EXISTENCE OF LIMIT CYCLES OF A PLANAR SYSTEM WITH THIRD-DEGREE POLYNOMIALS." International Journal of Bifurcation and Chaos 14, no. 01 (January 2004): 41–60. http://dx.doi.org/10.1142/s0218127404009247.

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The focus of the paper is mainly on the existence of limit cycles of a planar system with third-degree polynomial functions. A previously developed perturbation technique for computing normal forms of differential equations is employed to calculate the focus values of the system near equilibrium points. Detailed studies have been provided for a number of cases with certain restrictions on system parameters, giving rise to a complete classification for the local dynamical behavior of the system. In particular, a sufficient condition is established for the existence of k small amplitude limit cycles in the neighborhood of a high degenerate critical point. The condition is then used to show that the system can have eight and ten small amplitude (local) limit cycles for a set of particular parameter values.

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ZHANG, W., and X. Y. GUO. "PERIODIC AND CHAOTIC OSCILLATIONS OF A COMPOSITE LAMINATED PLATE USING THE THIRD-ORDER SHEAR DEFORMATION PLATE THEORY." International Journal of Bifurcation and Chaos 22, no. 05 (May 2012): 1250103. http://dx.doi.org/10.1142/s0218127412501039.

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An analysis on nonlinear oscillations and chaotic dynamics is presented for a simply-supported symmetric cross-ply composite laminated rectangular thin plate with parametric and forcing excitations in the case of 1:3:3 internal resonance. Based on Reddy's third-order shear deformation plate theory and the von Karman-type equations, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate can be established via the Hamilton's principle. Such partial differential equations are further discretized by the Galerkin method to form a three-degree-of-freedom coupled nonlinear system including the cubic nonlinear terms. The method of multiple scales is then employed to derive a set of averaged equations. Through the stability analysis, the steady-state solutions of the averaged equations are provided. An illustrative case of 1:3:3 internal resonance and fundamental parametric resonance, 1/3 subharmonic resonance is considered. Numerical simulation is applied to investigate the intrinsically nonlinear behavior of the composite laminated rectangular thin plate. With certain external load excitations, the simulation results demonstrate that the nonlinear dynamical system of the composite laminated plate exhibits different kinds of periodic and chaotic motions.

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Hernandez-Gonzalez, Miguel, and Michael V. Basin. "State estimation for stochastic polynomial systems with switching in the state equation." Transactions of the Institute of Measurement and Control 40, no. 9 (November 21, 2017): 2732–39. http://dx.doi.org/10.1177/0142331217737134.

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The problem of designing a mean-square filter has been studied for stochastic polynomial systems, where the state equation switches between two different nonlinear functions, over linear observations. A switching signal depends on a random variable modelled as a Bernoulli distributed sequence that takes the quantities of zero and one. The differential equations for the state estimate and the error covariance matrix are obtained in a closed form by expressing the conditional expectation of polynomial terms as functions of the estimate and covariance matrix. Finite-dimensional filtering equations are obtained for a particular case of a third-degree polynomial system. Numerical simulations are carried out in two cases of switching between different linear and second degree polynomial functions.

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Omole, Ezekiel, A. A. Aigbiremhon, and Abosede Funke Familua. "A THREE-STEP INTERPOLATION TECHNIQUE WITH PERTURBATION TERM FOR DIRECT SOLUTION OF THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS." FUDMA JOURNAL OF SCIENCES 5, no. 2 (July 7, 2021): 365–76. http://dx.doi.org/10.33003/fjs-2021-0502-556.

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In this paper, we developed a new three-step method for numerical solution of third order ordinary differential equations. Interpolation and collocation methods were used by choosing interpolation points at steps points using power series, while collocation points at step points, using a combination of powers series and perturbation terms gotten from the Legendre polynomials, giving rise to a polynomial of degree and equations. All the analysis on the method derived shows that it is zero-stable, convergent and the region of stability is absolutely stable. Numerical examples were provided to test the performance of the method. Results obtained when compared with existing methods in the literature, shows that the method is accurate and efficient

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Souza, Fábio Nicácio. "A SOLUÇÃO DAS QUADRÁTICAS E CÚBICAS NA HISTÓRIA." Ciência e Natura 37 (August 7, 2015): 555. http://dx.doi.org/10.5902/2179460x14595.

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http://dx.doi.org/10.5902/2179460X14595For a long time the search to solve equations pushed the brilliant minds in the human History. Arithmetical, geometric and algebraic mechanisms ere developed bringing advance to the mathematics knowledge. A lot of these resources were "simplified" and nowadays they're handled in the classroom overruling among them the algebraic manipulation. This article, gotten from chapter 1 of Math Master dissertation in the PROFMAT Program from UFRPE and named A Geometrical Approach to the cubic Equations shows as from the history of algebra a geometrical manner, not usual anymore, to find the solution to second and third degree equations.

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Mazrooei-Sebdani, Reza. "Non-autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case." Mathematical Methods in the Applied Sciences 37, no. 4 (June 19, 2013): 518–23. http://dx.doi.org/10.1002/mma.2810.

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Mata Rodríguez, Carlos M. "OPUSCULO SOBRE LA SOLUCIÓN DE LA ECUACIÓN CÚBICA." Revista Ingeniería, Matemáticas y Ciencias de la Información 8, no. 15 (January 31, 2021): 67–74. http://dx.doi.org/10.21017/rimci.2021.v8.n15.a93.

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Алероев, Темирхан Султанович, and Магомедюсуф Владимирович Гасанов. "A necessary and sufficient condition for the existence of a mobile singular points for a third-order nonlinear differential equation." Вестник Чувашского государственного педагогического университета им. И.Я. Яковлева. Серия: Механика предельного состояния, no. 1(47) (June 30, 2021): 49–55. http://dx.doi.org/10.37972/chgpu.2021.1.47.004.

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Рассматривается нелинейное уравнение третьего порядка с полиномом второй степени в правой части. Отличительной чертой этого класса уравнений является наличие подвижных особенностей, что делает эти уравнения неразрешимыми в квадратурах. В работе получены интервальные критерии существование подвижных особых точек. Представленная теория является подспорьем для написания различных алгоритмов в различных программных комплексах для нахождения подвижных особых точек. A nonlinear third-order equation with second degree polynomial on the right. The hallmark of this class equations is the presence of movable singularities, which makes these equations undecidable in quadratures. The work obtained interval criteria the existence of movable singular points. The theory presented is help for writing various algorithms in various software complexes for finding movable singular points.

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Dong, Lu, Yu Xin Hao, Jian Hua Wang, and Li Yang. "Nonlinear Vibration of Functionally Graded Material Cylindrical Shell Based on Reddy’s Third-Order Plates and Shells Theory." Advanced Materials Research 625 (December 2012): 18–24. http://dx.doi.org/10.4028/www.scientific.net/amr.625.18.

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In this paper, an analysis on nonlinear dynamics of a simply supported functionally graded material (FGM) cylindrical shell subjected to the different excitation in thermal environment. Material properties of cylindrical shell are assumed to be temperature-dependent. Based on the Reddy’s third-order plates and shells theory[1], the nonlinear governing partial differential equations of motion for the FGM cylindrical shell are derived by using Hamilton’s principle. Galerkin’s method is utilized to transform the partial differential equations into a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined parametric and external excitation. The effects played by different excitation and system initial conditions on the nonlinear vibration of the cylindrical shell are studied. In addition, the Runge–Kutta method is used to find out the nonlinear dynamic responses of the FGM cylindrical shell.

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Smith, Blair J., and Donald R. Scott. "Level of Structural Aggregation and Predictive Accuracy of Milk Supply Response Estimates." Northeastern Journal of Agricultural and Resource Economics 15, no. 1 (April 1986): 32–36. http://dx.doi.org/10.1017/s0899367x00001306.

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Milk supply response was estimated for Pennsylvania using three different levels of structural aggregation. The base level involved the estimation of milk production in a single equation. Under the second method, production was the product of two equations: milk per cow and number of milk cows. The third method factored production into three equations: milk per cow, number of dairy farms, and number of cows per farm. As expected, the greater the degree of disaggregation the more was learned about the structural aspects of milk production. At the same time, predictive accuracy generally decreased, but the differences among models was slight.

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Green, Thomas A., Carl J. Schreck, Nathan S. Johnson, and Sonya Stevens Heath. "Education Backgrounds of TV Weathercasters." Bulletin of the American Meteorological Society 100, no. 4 (April 1, 2019): 581–88. http://dx.doi.org/10.1175/bams-d-17-0047.1.

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Abstract In the early days of television, most weathercasters lacked formal training in meteorology and instead relied on forecasts from other sources. Over the decades, degreed meteorologists became more common. A third category has recently emerged: people with certificates in broadcast meteorology from Mississippi State University (MSU). This certification and the related broadcast meteorology degrees from MSU provide weathercasters with an understanding of meteorology without advanced calculus or differential equations. This study makes no judgment on how a weathercaster’s education background might affect their on-air presentations but notes these courses are required by most guidelines for meteorological degrees, as well as the American Meteorological Society's Certified Broadcast Meteorologist (CBM) program. This study conducts a unique survey of television meteorologists using the education history listed on their station's website or LinkedIn. The backgrounds of 421 meteorologists were examined with the equivalent of a 94% response rate. Overall, 21% had a broadcast meteorology degree or certification from MSU, 64% had a traditional meteorology degree from MSU or another institution, 2% minored in meteorology or had military training, and 12% listed no or a partial education background in the field. Another way of viewing the data is that the MSU broadcast program alone has nearly as many graduates as the four largest traditional programs combined in our sample. These results were further broken down for various subsets of weathercasters, resulting in statistically significant variations by market size, region, ownership group, and gender.

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Орлов, Виктор Николаевич, and Рио-Рита Вадимовна Разакова. "An approximate solution of the one class third order onlinear differential equation in the analyticity domain." Вестник Чувашского государственного педагогического университета им. И.Я. Яковлева. Серия: Механика предельного состояния, no. 1(43) (March 27, 2020): 45–54. http://dx.doi.org/10.37972/chgpu.2020.43.1.005.

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В работе рассмотрен класс нелинейных дифференциальных уравнений третьего порядка с полиномиальной правой частью шестой степени. Доказана теорема существования и единственности решения в области аналитичности. Построено аналитическое приближенное решение. Предложен вариант оптимизации априорных оценок с помощью апостериорных. Проведен численный эксперимент. There is a class of third-order nonlinear differential equations with polynomial right part of the sixth degree considered in the paper. The existence and uniqueness theorem of a solution in the domain of analyticity is proved by authors. There is an analytical approximate solution which was constructed by V. Orlov and R. Razakova. A variant of optimization of a priori estimates using posterior ones is proposed by authors. A numerical experiment is carried out too.

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Messias, Marcelo, and Rafael Paulino Silva. "Determination of Nonchaotic Behavior for Some Classes of Polynomial Jerk Equations." International Journal of Bifurcation and Chaos 30, no. 08 (June 30, 2020): 2050117. http://dx.doi.org/10.1142/s0218127420501175.

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In this work, by using an algebraic criterion presented by us in an earlier paper, we determine the conditions on the parameters in order to guarantee the nonchaotic behavior for some classes of nonlinear third-order ordinary differential equations of the form [Formula: see text] called jerk equations, where [Formula: see text] is a polynomial of degree [Formula: see text]. This kind of equation is often used in literature to study chaotic dynamics, due to its simple form and because it appears as mathematical model in several applied problems. Hence, it is an important matter to determine when it is chaotic and also nonchaotic. The results stated here, which are proved using the mentioned algebraic criterion, corroborate and extend some results already presented in literature, providing simpler proofs for the nonchaotic behavior of certain jerk equations. The algebraic criterion proved by us is quite general and can be used to study nonchaotic behavior of other types of ordinary differential equations.

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ANDREEV, P. A. "FIRST PRINCIPLES DERIVATION OF NLS EQUATION FOR BEC WITH CUBIC AND QUINTIC NONLINEARITIES AT NONZERO TEMPERATURE: DISPERSION OF LINEAR WAVES." International Journal of Modern Physics B 27, no. 06 (February 5, 2013): 1350017. http://dx.doi.org/10.1142/s0217979213500173.

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We present a derivation of the quantum hydrodynamic (QHD) equations for neutral bosons. We consider the short-range interaction between particles. This interaction consist of a binary interaction U( ri, rj) and a three-particle interaction (TPI) U( ri, rj, rk) and the last one does not include binary interaction between particles. From QHD equations for Bose–Einstein condensate we derive a nonlinear Schrödinger equation. This equation was derived for zero temperature and contains the nonlinearities of the third and the fifth degree. Explicit form of the constant of the TPI is obtained. First of all, developed method we used for studying of dispersion of the linear waves. Dispersion characteristics of the linear waves are compared for different particular cases. We make comparison of the two-particle interaction in the third order by the interaction radius (TOIR) and TPI at the zero temperature. We consider influence of the temperature on the dispersion of the elementary excitations. For this aim we derive a system of the QHD equations at nonzero temperature. Obtained system of equation is an analog of the well-known two-fluid hydrodynamics. Moreover, it is generalization of the two-fluid hydrodynamics equations due to TPI. Explicit expressions of the velocities for the first and the second sound via the concentration of superfluid and noncondensate components is calculated.

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Eremin, A. V., and K. V. Gubareva. "Analytical solution to the problem of heat transfer using boundary conditions of the third kind." Vestnik IGEU, no. 6 (2019): 67–74. http://dx.doi.org/10.17588/2072-2672.2019.6.067-074.

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Non-stationary heat transmission within solid bodies is described using parabolic and hyperbolic equations. Currently, numerical methods for studying the processes of heat and mass transfer in the flows of liquids and gases have disseminated. Modern programs allow the automatic construction of computational grids, solutions to the systems of equations and offer a wide range of tools for analysis. Approximate analytical solutions have significant advantages compared to numerical ones. In particular, the solutions obtained in an analytical form allow performing parametric analysis of the system under study, configuration and programming of measurement devices, etc. Based on the joint use of additional desired function and additional boundary conditions in the integral method of heat balance, a method of mathematical modeling for the heat transfer process in a plate under symmetric boundary conditions of the third kind has been developed. Using the heat flux density as a new desired function, the method for solving heat conduction problems with boundary conditions of the third kind has been proposed. Finding a solution to the partial differential equation with respect to the temperature function presents integrating an ordinary differential equation with respect to the heat flux density on the surface of the studied zone. It has been shown that isotherms appear on the surface of the plate with a certain initial velocity which depends on the heat transfer intensity. The calculation results have been compared to the exact solution. The presented method can be used in determining the heat flux density of buildings and heating devices, finding heat losses during convective heat transfer and designing heat transfer equipment. The results can be applied to increase the validity and reliability of the calculation of actual losses and balance of thermal energy. The method reliability, validity and a high degree of approximation with about 3% inaccuracy have been demonstrated. The accuracy of the solution depends on the number of approximations performed and is determined by the degree of the approximating polynomial.

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Chronowski, Antoni. "Creative mathematical mental activity of students while solving problems requiring use of Vieta's formulas II." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 12 (December 24, 2020): 109–21. http://dx.doi.org/10.24917/20809751.12.5.

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This article is a continuation of the article (Chronowski, Powazka, 2016). In this article the next tasks regarding the use of Vieta’s formulas for third degree polynomials (equations) with real coefficients are considered. The Olympic tasks in this article are an inspiration to pupils for creative mathematical activity. In this text, I limited myself to comments, hintsand didactic suggestions regarding directly presented tasks and their solutions. The most of the general didactic considerations included in our article (Chronowski, Powazka, 2016) are also valid in this paper.

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Toffoli, A., M. Benoit, M. Onorato, and E. M. Bitner-Gregersen. "The effect of third-order nonlinearity on statistical properties of random directional waves in finite depth." Nonlinear Processes in Geophysics 16, no. 1 (February 24, 2009): 131–39. http://dx.doi.org/10.5194/npg-16-131-2009.

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Abstract. It is well established that third-order nonlinearity produces a strong deviation from Gaussian statistics in water of infinite depth, provided the wave field is long crested, narrow banded and sufficiently steep. A reduction of third-order effects is however expected when the wave energy is distributed on a wide range of directions. In water of arbitrary depth, on the other hand, third-order effects tend to be suppressed by finite depth effects if waves are long crested. Numerical simulations of the truncated potential Euler equations are here used to address the combined effect of directionality and finite depth on the statistical properties of surface gravity waves; only relative water depth kh greater than 0.8 are here considered. Results show that random directional wave fields in intermediate water depths, kh=O(1), weakly deviate from Gaussian statistics independently of the degree of directional spreading of the wave energy.

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Bian, Xiao Li, and Shuang Bao Li. "Nonlinear Oscillations of a Composite Laminated Plate with Parametrically and Externally Excitations." Advanced Materials Research 699 (May 2013): 641–44. http://dx.doi.org/10.4028/www.scientific.net/amr.699.641.

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Nonlinear oscillations of a simply-supported symmetric cross-ply composite laminated rectangular thin plate are investigated in this paper. The rectangular thin plate is subjected to the transversal and in-plane excitations. Based on the Reddy’s third-order shear deformation plate theory and the stress-strain relationship of the composite laminated plate, a two-degree-of-freedom non-autonomous nonlinear system governing equations of motions for the composite laminated rectangular thin plate is derived by using the Galerkin’s method. Numerical simulations illustrate that there exist complex nonlinear oscillations for composite laminated rectangular thin plate.

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Egorov, A. D. "Approximate formulas for the evaluation of the mathematical expectation of functionals from the solution to the linear Skorohod equation." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 57, no. 2 (July 16, 2021): 198–205. http://dx.doi.org/10.29235/1561-2430-2021-57-2-198-205.

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This paper is devoted to the construction of approximate formulas for calculating the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod stochastic differential equation with a random initial condition. To calculate the mathematical expectations of nonlinear functionals from random processes, functional analogs of quadrature formulas have been developed, based on the requirement of their accuracy for functional polynomials of a given degree. Most often, formulas are constructed that are exact for polynomials of the third degree [1–9], which are used to obtain an initial approximation and in combination with approximations of the original random process. In the latter case, they are usually also exact for polynomials of a given degree and are called compound formulas. However, in the case of processes specified in the form of compound functions from other random processes the constructed functional quadrature formulas, as a rule, have great computational complexity and cannot be used for computer implementation. This is exactly what happens in the case of functionals from the solutions of stochastic equations. In [1, 2], the approaches to solving this problem were considered for some types of Ito equations in martingales. The solution of the problem is simplified in the cases when the solution of the stochastic equation is found in explicit form: the corresponding approximations were obtained in the cases of the linear equations of Ito, Ito – Levy and Skorohod in [3–11]. In [7, 8, 11], functional quadrature formulas were constructed that are exact for the approximations of the expansions of the solutions in terms of orthonormal functional polynomials and in terms of multiple stochastic integrals. This work is devoted to the approximate calculation of the mathematical expectations of nonlinear functionals from the solution of the linear Skorokhod equation with a leading Wiener process and a random initial condition. A new approach to the construction of quadrature formulas, exact for functional polynomials of the third degree, based on the use of multiple Stieltjes integrals over functions of bounded variation in the sense of Hardy – Krause, is proposed. A composite approximate formula is also constructed, which is exact for second-order functional polynomials, converging to the exact expectation value, based on a combination of the obtained quadrature formula and an approximation of the leading Wiener process. The test examples illustrating the application of the obtained formulas are considered.

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Zhumanazarova, Assiya, and Young Im Cho. "Uniform Approximation to the Solution of a Singularly Perturbed Boundary Value Problem with an Initial Jump." Mathematics 8, no. 12 (December 14, 2020): 2216. http://dx.doi.org/10.3390/math8122216.

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In this study, a third-order linear integro-differential equation with a small parameter at two higher derivatives was considered. An asymptotic expansion of the solution to the boundary value problem for the considered equation is constructed by considering the phenomenon of an initial jump of the second degree zeroth order on the left end of a given segment. The asymptotics of the solution has been sought in the form of a sum of the regular part and the part of the boundary layer. The terms of the regular part are defined as solutions of integro-differential boundary value problems, in which the equations and boundary conditions contain additional terms, called the initial jumps of the integral terms and solutions. Boundary layer terms are defined as solutions of third-order differential equations with initial conditions. A theorem on the existence, uniqueness, and asymptotic representation of a solution is presented along with an asymptotic estimate of the remainder term of the asymptotics. The purpose of this study is to construct a uniform asymptotic approximation to the solution to the original boundary value problem over the entire considered segment.

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Ma, W. S., and W. Zhang. "Resonant Chaotic Dynamics of a Symmetric Cross-Ply Composite Laminated Plate Under Transverse and In-Plane Excitations." International Journal of Bifurcation and Chaos 30, no. 07 (June 15, 2020): 2050106. http://dx.doi.org/10.1142/s0218127420501060.

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The resonant chaotic dynamics of a symmetric cross-ply composite laminated plate are studied using the exponential dichotomies and an averaging procedure for the first time. The partial differential governing equations of motion for the symmetric cross-ply composite laminated plate are derived by using Reddy’s third-order shear deformation plate theory and von Karman type equation. The partial differential governing equations of motion are discretized into two-degree-of-freedom nonlinear systems including the quadratic and cubic nonlinear terms by using Galerkin method. There exists a fixed point of saddle-focus in the linear part for two-degree-of-freedom nonlinear system. The Melnikov method containing the terms of the nonhyperbolic mode is developed to investigate the resonant chaotic motions of the symmetric cross-ply composite laminated plate. The obtained results indicate that the nonhyperbolic mode of the symmetric cross-ply composite laminated plate does not affect the critical conditions in the occurrence of chaotic motions in the resonant case. When the resonant chaotic motion occurs, we can draw a conclusion that the resonant chaotic motions of the hyperbolic subsystem are shadowed for the full nonlinear system of the symmetric cross-ply composite laminated plate.

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Hao, Y. X., W. Zhang, and X. L. Ji. "Nonlinear Dynamic Response of Functionally Graded Rectangular Plates under Different Internal Resonances." Mathematical Problems in Engineering 2010 (2010): 1–12. http://dx.doi.org/10.1155/2010/738648.

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The nonlinear dynamic response of functionally graded rectangular plates under combined transverse and in-plane excitations is investigated under the conditions of 1 : 1, 1 : 2 and 1 : 3 internal resonance. The material properties are assumed to be temperature-dependent and vary along the thickness direction. The thermal effect due to one-dimensional temperature gradient is included in the analysis. The governing equations of motion for FGM rectangular plates are derived by using Reddy's third-order plate theory and Hamilton's principle. Galerkin's approach is utilized to reduce the governing differential equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms, which are then solved numerically by using 4th-order Runge-Kutta algorithm. The effects of in-plane excitations on the internal resonance relationship and nonlinear dynamic response of FGM plates are studied.

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Kim, Hyun Soo, Wooseok Ryu, Shi-baek Park, and Yong Je Choi. "3-Degree-of-freedom electromagnetic vibration energy harvester with serially connected leaf hinge joints." Journal of Intelligent Material Systems and Structures 30, no. 2 (November 7, 2018): 308–22. http://dx.doi.org/10.1177/1045389x18806397.

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This article presents a new design method of a planar 3-degree-of-freedom serial manipulator-type electromagnetic vibration energy harvester in which any desired ratio of power peaks and three target resonant frequencies can be specified arbitrarily. The design of the harvester aims to achieve minimum difference between the power peaks generated at target frequencies. The geometrical positions of three normal modes are first determined and the corresponding stiffness matrix of the harvester is found. Second, the stiffness matrix can be synthesized by three serially connected torsional springs. Third, the leaf hinge joints corresponding to torsional springs are designed using the newly developed design equations. Finally, the array and the locations of the magnets are found using the sequential quadratic programming (SQP) algorithm. The experiments are conducted to verify the design method. Three resonant frequencies are measured at 23.4, 29.2, and 34.8 Hz comparing to the target frequencies of 25, 30, and 35 Hz. The peak powers of 1.28, 0.89, and 1.32 mW are obtained across the optimal load resistor of 1.01 kΩ under the condition of the constant acceleration of 1.5 m/s2.

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Kriauzienė, Rima, Andrej Bugajev, and Raimondas Čiegis. "A THREE-LEVEL PARALLELISATION SCHEME AND APPLICATION TO THE NELDER-MEAD ALGORITHM." Mathematical Modelling and Analysis 25, no. 4 (October 13, 2020): 584–607. http://dx.doi.org/10.3846/mma.2020.12139.

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We consider a three-level parallelisation scheme. The second and third levels define a classical two-level parallelisation scheme and some load balancing algorithm is used to distribute tasks among processes. It is well-known that for many applications the efficiency of parallel algorithms of these two levels starts to drop down after some critical parallelisation degree is reached. This weakness of the twolevel template is addressed by introduction of one additional parallelisation level. s an alternative to the basic solver some new or modified algorithms are considered on this level. The idea of the proposed methodology is to increase the parallelisation degree by using possibly less efficient algorithms in comparison with the basic solver. As an example we investigate two modified Nelder-Mead methods. For the selected application, a Schro¨dinger equation is solved numerically on the second level, and on the third level the parallel Wang’s algorithm is used to solve systems of linear equations with tridiagonal matrices. A greedy workload balancing heuristic is proposed, which is oriented to the case of a large number of available processors. The complexity estimates of the computational tasks are model-based, i.e. they use empirical computational data.

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PIELORZ, AMALIA, and DANUTA SADO. "ON REGULAR AND IRREGULAR NONLINEAR VIBRATIONS IN TORSIONAL DISCRETE-CONTINUOUS SYSTEMS." International Journal of Bifurcation and Chaos 21, no. 10 (October 2011): 3073–82. http://dx.doi.org/10.1142/s0218127411030386.

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The paper deals with regular and irregular nonlinear vibrations of discrete-continuous systems torsionally deformed. The systems consist of an arbitrary number of shafts connected by rigid bodies. In the systems, a local nonlinearity having a soft type characteristic is introduced. This nonlinearity is described by the polynomial of the third degree. General governing equations using the wave approach are derived for a multimass system. Detailed numerical considerations are presented for a two-mass system and a three-mass system. The possibility of occurrence of irregular vibrations is discussed on the basis of the Poincaré maps and bifurcation diagrams.

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Achour, Bachir. "Design of a Pressurized Rectangular-Shaped Conduit Using the Rough Model Method (Part 1)." Applied Mechanics and Materials 641-642 (September 2014): 261–66. http://dx.doi.org/10.4028/www.scientific.net/amm.641-642.261.

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The rough model method is successfully used to design a pressurized rectangular shaped conduit characterized by two linear dimensions. In this study, the focus is on the calculation of the horizontal linear dimension of the conduit. In a first step, the method is applied to a referential rough model in order to establish the relationships that govern its hydraulic characteristics. The obtained equations are of the third degree and are easily solved by trigonometric and hyperbolic functions. In a second step, these equations are used to easily deduce the linear dimension sought by introducing a non-dimensional correction factor. Practical example is taken to enable the hydraulic engineer to better understanding the advocated method and to observe the facility with which design of such a geometric profile can be performed. The calculation uses a strict minimum of data measurable in practice, in particular the absolute roughness.

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Wang, Lai, and Xi Tong Dong. "Influence of Earthquake Directions on Wind Turbine Tower under Seismic Action." Advanced Materials Research 243-249 (May 2011): 3883–88. http://dx.doi.org/10.4028/www.scientific.net/amr.243-249.3883.

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Influence of earthquake directions on wind turbine tower under seismic action are numerically investigated in this paper. First, equations of motion and an integrated finite element model of a wind turbine system consisting of a rotor, a nacelle and a tower shaft are established. Second, the finite element modal analysis is discussed. Third, relationships between upper displacements in x, y directions and bottom bending stress when the angle is 0, 30, 45, 60, 90 degree respectively between earthquake directions and concentrated eccentric mass direction (x direction) are analyzed by adjusted Taft seismic wave .The results show that: seismic responses of a wind turbine tower are remarkable and seismic action may be the dominant factor in the design of wind turbine towers that located at a seismically active zone. Under different earthquake directions structure’s dynamic responses are different, 90 degree with regard to x direction is the most unfavorable direction. Both maximum upper displacements in x, y directions and bottom bending stress appear at 90 degree direction with regard to x direction.

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Khusainov, D. Ya, J. Diblík, Z. Svoboda, and Z. Šmarda. "Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone." Abstract and Applied Analysis 2011 (2011): 1–23. http://dx.doi.org/10.1155/2011/154916.

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The present investigation deals with global instability of a generaln-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaev's method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.

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Doha, E. H., and Y. H. Youssri. "On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain." Nonlinear Engineering 8, no. 1 (January 28, 2019): 318–27. http://dx.doi.org/10.1515/nleng-2018-0073.

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Abstract Herein, three important theorems were stated and proved. The first relates the modified generalized Laguerre expansion coefficients of the derivatives of a function in terms of its original expansion coefficients; and an explicit expression for the derivatives of modified generalized Laguerre polynomials of any degree and for any order as a linear combination of modified generalized Laguerre polynomials themselves is also deduced. The second theorem gives new modified generalized Laguerre coefficients of the moments of one single modified generalized Laguerre polynomials of any degree. Finally, the third theorem expresses explicitly the modified generalized Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its modified generalized Laguerre coefficients. Some spectral applications of these theorems for solving ordinary differential equations with varying coefficients and some specific applied differential problems, by reducing them to recurrence relations in their expansion coefficients of the solution are considered.

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MILED, MAROUANE BEN. "MESURER LE CONTINU, DANS LA TRADITION ARABE DES LIVRES V ET X DES ÉLÉMENTS." Arabic Sciences and Philosophy 18, no. 1 (March 2008): 1–18. http://dx.doi.org/10.1017/s0957423908000453.

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In order to find positive solutions for third-degree equations, which he did not know how to solve for roots, ‘Umar al-Khayyām proceeds by the intersections of conic sections. The representation of an algebraic equation by a geometrical curve is made possible by the choices of units of measure for lengths, surfaces, and volumes. These units allow a numerical quantity to be associated with a geometrical magnitude. Is there a trace of this unit in the mathematicians to whom al-Khayyām refers directly in his Algebra? How does this unit enable the measurement of quantities and rational and irrational relations? We find answers to these questions in the commentaries to Books V and X of the Elements.

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Sarkar, Tanmay. "A Priori Error Analysis of a Discontinuous Galerkin Scheme for the Magnetic Induction Equation." Computational Methods in Applied Mathematics 20, no. 1 (January 1, 2020): 121–40. http://dx.doi.org/10.1515/cmam-2018-0032.

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AbstractWe perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. In order to obtain the quasi-optimal convergence incorporating second-order Runge–Kutta schemes for time discretization, we need a strengthened {4/3}-CFL condition ({\Delta t\sim h^{4/3}}). To overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge–Kutta scheme for time discretization. We demonstrate the error estimates in {L^{2}}-sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree {l\geq 1} under the standard CFL condition.

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Zheng, Bin, and Qinghua Feng. "A New Approach for Solving Fractional Partial Differential Equations in the Sense of the Modified Riemann-Liouville Derivative." Mathematical Problems in Engineering 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/307371.

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Based on a fractional complex transformation, certain fractional partial differential equation in the sense of the modified Riemann-Liouville derivative is converted into another ordinary differential equation of integer order, and the exact solutions of the latter are assumed to be expressed in a polynomial in Jacobi elliptic functions including the Jacobi sine function, the Jacobi cosine function, and the Jacobi elliptic function of the third kind. The degree of the polynomial can be determined by the homogeneous balance principle. With the aid of mathematical software, a series of exact solutions for the fractional partial differential equation can be found. For demonstrating the validity of this approach, we apply it to solve the space fractional KdV equation and the space-time fractional Fokas equation. As a result, some Jacobi elliptic functions solutions for the two equations are obtained.

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RING, GERARD J. F. "The hyperbolic theory of light scattering, tensile strength, and density in paper." November 2011 11, no. 11 (December 1, 2011): 9–18. http://dx.doi.org/10.32964/tj10.11.9.

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The hyperbolic theory of light scattering, tensile strength, and density in paper describes a sheet of paper as a matrix of cellulose with a characteristic material strength and open and closed pores dispersed through the cellulose, forming solid foam. This paper presents two principle hyperbolic equations. The first describes the conservation of tensile strength and the second describes the conservation of mass as functions of light scattering. Three additional equations are derived. The first relates tensile strength to sheet density, the second relates light scattering to total pore volume, and the third relates tensile strength to total pore volume. The conservation of tensile strength equation varies with processing for a given cellulose pulp. Variable refining levels at constant levels of wet-pressing produce separate curves for each wet-pressing level. Correspondingly, variable wet-pressing levels produce separate curves for each level of constant refining. Hydrolyzing pulps or cutting fibers to shorten the average degree of cellulose polymerization shifts the curves to lower tensile strengths. The conservation of mass equation results in a single curve, regardless of processing or reduction in the degree of cellulose polymerization. The concept of relative bonded area measurement using light scattering as developed by Ingmanson and Thode is shown to be invalid for paper by this hyperbolic theory. The validity of relative bonded area by any technique is also questioned.

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Hao, W. L., W. Zhang, and M. H. Yao. "Multipulse Chaotic Dynamics of Six-Dimensional Nonautonomous Nonlinear System for a Honeycomb Sandwich Plate." International Journal of Bifurcation and Chaos 24, no. 11 (November 2014): 1450138. http://dx.doi.org/10.1142/s0218127414501387.

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This paper studies the global bifurcations and multipulse chaotic dynamics of a four-edge simply supported honeycomb sandwich rectangular plate under combined in-plane and transverse excitations. Based on the von Karman type equation for the geometric nonlinearity and Reddy's third-order shear deformation theory, the governing equations of motion are derived for the four-edge simply supported honeycomb sandwich rectangular plate. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional nonautonomous nonlinear system is simplified to a three-order standard form by using the normal form method. The extended Melnikov method is improved to investigate the six-dimensional nonautonomous nonlinear dynamical system in a mixed coordinate. The global bifurcations and multipulse chaotic dynamics of the four-edge simply supported honeycomb sandwich rectangular plate are studied by using the improved extended Melnikov method. The multipulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.

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Santos, María Jesús, Alejandro Medina, José Miguel Mateos Roco, and Araceli Queiruga-Dios. "Compartmental Learning versus Joint Learning in Engineering Education." Mathematics 9, no. 6 (March 20, 2021): 662. http://dx.doi.org/10.3390/math9060662.

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Sophomore students from the Chemical Engineering undergraduate Degree at the University of Salamanca are involved in a Mathematics course during the third semester and in an Engineering Thermodynamics course during the fourth one. When they participate in the latter they are already familiar with mathematical software and mathematical concepts about numerical methods, including non-linear equations, interpolation or differential equations. We have focused this study on the way engineering students learn Mathematics and Engineering Thermodynamics. As students use to learn each matter separately and do not associate Mathematics and Physics, they separate each matter into different and independent compartments. We have proposed an experience to increase the interrelationship between different subjects, to promote transversal skills, and to make the subjects closer to real work. The satisfactory results of the experience are exposed in this work. Moreover, we have analyzed the results obtained in both courses during the academic year 2018–2019. We found that there is a relation between both courses and student’s final marks do not depend on the course.

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Filip, Ovidiu, Andrea Deaconescu, and Tudor Deaconescu. "Experimental Research on the Hysteretic Behaviour of Pressurized Artificial Muscles Made from Elastomers with Aramid Fibre Insertions." Actuators 9, no. 3 (September 11, 2020): 83. http://dx.doi.org/10.3390/act9030083.

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Inherent hysteresis behaviour of pressurized artificial muscles is complicated to understand and handle, calling for experimental research that allows the modelling of this phenomenon. The paper presents the results of the experimental study of the hysteretic behaviour of a small-size pneumatic muscle. The specific hysteresis loops were revealed by isotonic and isometric tests. Starting from hypothesis according to that the tube used for the pneumatic muscle is made entirely of aramid fibres enveloped by an elastomer material that merely ensures their airtightness, the paper presents the hysteresis curves that describe the radial and axial dimensional modifications as well as the variation of the developed forces for different feed pressures. The obtained third-degree polynomial equations underlie the configuration of high-performance positioning systems.

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Journal articles on the topic 'Third degree equations'

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