Third-order Nonlinear Equation Research Articles

On Solutions of the Third-Order Ordinary Differential Equations of Emden-Fowler Type

For a linear ordinary differential equation (ODE in short) of the third order, results are presented that supplement the theory of conjugate points and extremal solutions by W. Leighton, Z. Nehari, M. Hanan. It is especially noted the sensitivity of solutions to the initial data, which makes their numerical study difficult. Similar results were obtained for the third-order nonlinear equations of the Emden-Fowler type.

A study on Darboux polynomials and their significance in determining other integrability quantifiers: A case study in third-order nonlinear ordinary differential equations

In this paper, we present a method of deriving extended Prelle-Singer method's quantifiers from Darboux Polynomials for third-order nonlinear ordinary differential equations. By knowing the Darboux polynomials and its cofactors, we extract the extended Prelle-Singer method's quantities without evaluating the Prelle-Singer method's determining equations. We consider three different cases of known Darboux polynomials. In the first case, we prove the integrability of the given third-order nonlinear equation by utilizing the Prelle-Singer method's quantifiers from the two known Darboux polynomials. If we know only one Darboux polynomial, then the integrability of the given equation will be dealt as case $2$. Likewise, case $3$ discuss the integrability of the given system where we have two Darboux polynomials and one set of Prelle-Singer method quantity. The established interconnection not only helps in deriving the integrable quantifiers without solving the underlying determining equations. It also provides a way to prove the complete integrability and helps us in deriving the general solution of the given equation. We demonstrate the utility of this procedure with three different examples.

Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus

In this paper we study the initial value problem of fully nonlinear third-order equations on the torus, that is ∂tu=F∂x3u,∂x2u,∂xu,u,x,t with F a smooth function depending on the space variable x, the time variable t, the first three derivatives of u with respect to x, and u. In particular we find conditions on u(0) and F for which one can construct a local and unique solution u. In particular if F and u(0) satisfy some conditions then the equation behaves like a diffusive one and it has a parabolic smoothing property: the solution is infinitely smooth in one direction of time and the problem is ill-posed in the other direction of time. If F and u(0) satisfy some other conditions then the equation behaves like a dispersive one. We also prove continuous dependence with respect to the data. The proof relies upon energy estimates combined with a gauge transformation [6–8] and the Bona-Smith argument (Bona and Smith, 1975).

A NECESSARY AND SUFFICIENT CONDITION FOR THE EXISTENCE OF A MOBILE SINGULAR POINT FOR A NONLINEAR DIFFERENTIAL EQUATION OF THE THIRD ORDER

A nonlinear third-order equation with a seventh-degree polynomial on the right-hand side is considered. A distinctive feature of this class of equations is the presence of movable functions, which makes these equations undecidable in quadratures. The work obtained data on the observance of movable singular points. The presented theory is a means of compiling an algorithm and writing a software complex for finding moving points.

X-ray third-order nonlinear asymmetric diffraction in the transmission geometry

For high intensity x-ray sources, the investigations of the nonlinear x-ray effects become essential. The x-ray third-order nonlinear asymmetric transmission case dynamical diffraction is investigated theoretically in perfect crystals. The three essential input parameters are taken into account: the deviation from the Bragg exact orientation, the asymmetry angle of reflecting atomic planes and the intensity of incident wave. The third-order nonlinear equations for the asymmetrical dynamical diffraction and two integrals of motion are presented. The exact solutions for the wave amplitudes, presented via Jacobi elliptic functions, have been found. The solutions are periodic functions of the depth. It is shown that the behavior of the waves inside the crystal is determined by the sign of a combined parameter, which includes the three input parameters, mentioned above. The regions, where this parameter is positive, negative or zero, are found. For positive values, the energy is concentrated both in the transmitted and diffracted waves while for the negative values, the energy is concentrated mainly in the transmitted beam. It is found that when this combined parameter is zero, the amplitudes are elementary periodic or non-periodic functions. For the periodic solutions, the nonlinear Pendellösung effect takes place, i.e. the transmitted and diffracted waves periodically exchange with their energies. For the period, called an extinction length, an exact expression is found. It is shown that for large values of the asymmetry factor, the observation of the nonlinear diffraction effects can be realized for sufficiently low intensities of the incident wave. The obtained results can be used for experimental investigation of the nonlinear Bragg diffraction and for preparation of high intensity x-ray beams with given parameters. The expression for the nonlinear extinction length allows to estimate the third-order nonlinear susceptibilities of crystals by measuring the extinction length in experiments.

The nonlinear orthotropic material model describing biaxial tensile behavior of PVC coated fabrics

The nonlinear orthotropic material model describing the biaxial tensile behavior of PVC coated fabrics is developed in this paper. Based on energy function, the general nonlinear stress-strain relationship is derived and the second-order and third-order nonlinear equations are proposed due to the in-plane stress-strain relationship assumption. The Membrane Structures Association of Japan (MSAJ) standard testing method for elastic constants of membrane materials is adopted and three types of PVC coated fabrics are tested under five different load ratios. The measured stress-strain data are used to calculate the coefficients of proposed nonlinear equations using the Least-square method of minimizing strain. Then the stress-strain curves of five different load ratios are compared among MSAJ suggested linear orthotropic model, proposed second-order nonlinear model, third-order nonlinear model, and the test data, also the 3D strain surfaces are plotted to make an intuitive comparison. Results show that both two proposed nonlinear models have a better representation of the test data than the linear model. Finally, considering the convenience and simplicity of the calculation, the second-order nonlinear model was integrated into the finite element software and further prove its adaptability. Thus it is recommended to apply the second-order nonlinear model to engineering design and numerical calculation.

On the expedient solution of the magneto-hydrodynamic Jeffery-Hamel flow of Casson fluid

The equation of magneto-hydrodynamic Jeffery-Hamel flow of non-Newtonian Casson fluid in a stretching/shrinking convergent/divergent channel is derived and solved using a new modified Adomian decomposition method (ADM). So far in all problems where semi-analytical methods are used the boundary conditions are not satisfied completely. In the present research, a hybrid of the Fourier transform and the Adomian decomposition method (FTADM), is presented in order to incorporate all boundary conditions into our solution of magneto-hydrodynamic Jeffery-Hamel flow of non-Newtonian Casson fluid in a stretching/shrinking convergent/divergent channel flow. The effects of various emerging parameters such as channel angle, stretching/shrinking parameter, Casson fluid parameter, Reynolds number and Hartmann number on velocity profile are considered. The results using the FTADM are compared with the results of ADM and numerical Range-Kutta fourth-order method. The comparison reveals that, for the same number of components of the recursive sequences over a wide range of spatial domain, the relative errors associated with the new method, FTADM, are much less than the ADM. The results of the new method show that the method is an accurate and expedient approximate analytic method in solving the third-order nonlinear equation of Jeffery-Hamel flow of non-Newtonian Casson fluid.

Численное решение одномерной задачи фильтрации несжимаемой жидкости в вязкой пористой среде

Процесс фильтрации жидкости в деформируемой пористой среде описывается системой, состоящей из уравнений сохранения массы для жидкой и твердой фаз, закона Дарси, реологического соотношения типа Максвела и закона сохранения баланса сил. Предполагается, что пороупругая среда обладает преимущественно вязкими свойствами и плотности фаз являются постоянными. В случае одной пространственной переменной переход к переменным Лагранжа позволяет свести исходную систему определяющих уравнений к одному уравнению для искомой пористости. Целью работы является численное исследование возникающей начально-краевой задачи. В пункте 1 дается постановка задачи и краткий обзор литературы по близким к данной теме работам. В пункте 2 проводится преобразование системы уравнений, в результате которого для пористости возникает нелинейное уравнение третьего порядка. В пункте 3 предложен алгоритм численного решения одномерной начально-краевой задачи. Для численной реализации используется однородная разностная схема для уравнения второго порядка с переменными коэффициентами и схема Рунге — Кутта второго порядка аппроксимации. Полученное решение удовлетворяет физическому принципу максимума. В пункте 4 рассматривается более общий случай сведения исходной системы к одному уравнению.DOI 10.14258/izvasu(2018)4-11

Travelling Wave Solutions of Some Nonlinear Third-Order Equations and Their Stabilities

Travelling Wave Solutions of Some Nonlinear Third-Order Equations and Their Stabilities

On a nonlinear third-order equation

A nonclassical nonlinear third-order partial differential equation modeling nonstationary processes in a semiconductor medium is studied. Seven classes of exact solutions of this equation given as explicit, implicit, and quadrature formulas are constructed. It is shown that, among these solutions, there are some bounded globally with respect to time and some approaching infinity on finite time intervals. These results are of interest in view of the following property of initial boundary-value problems for the given equation. Here the method of energy estimates used in many papers to justify the boundedness or unboundedness of solutions does not provide sufficient conditions for the unboundedness of solutions on finite time intervals.

Optimally Convergent HDG Method for Third-Order Korteweg–de Vries Type Equations

We develop and analyze a new hybridizable discontinuous Galerkin (HDG) method for solving third-order Korteweg-de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton-Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations.

PERIODIC MOTION AT RESONANCE IN SEMICONDUCTOR LASER EQUATIONS

A weakly nonlinear third-order equation describing the behavior of the phase of the electric field of a single-mode semiconductor laser subject to optical injection and detuning is analyzed. The method of averaging is used to explore the creation of periodic orbits in the system at resonance, when the detuning frequency is a rational multiple of the laser's natural frequency.

Conditional symmetries for ordinary differential equations and applications

We refine the definition of conditional symmetries of ordinary differential equations and provide an algorithm to compute such symmetries. A proposition is proved which provides criteria as to when the symmetries of the root system of ODEs are inherited by the derived higher-order system. We provide examples and then investigate the conditional symmetry properties of linear nth-order equations subject to root linear second-order equations. First this is considered for simple linear equations and then for arbitrary linear systems. We prove criteria when the symmetries of the root linear ODEs are inherited by the derived scalar linear ODEs and even order linear system of ODEs. Furthermore, we show that if a system of ODEs has exact solutions, then it admits a conditional symmetry subject to the first-order ODEs related to the invariant curve conditions which arises from the known solution curves. Moreover, we give examples of the conditional symmetries of non-linear third-order equations which are linearizable by second-order Lie linearizable equations. Applications to classical and fluid mechanics are presented.

Approximate periodic solution for nonlinear jerk equation as a third-order nonlinear equation via modified differential transform method

Purpose – The nonlinear jerk equation is a third-order nonlinear equation that describes some physical phenomena and in general form is given by: x = J (x, x, x). The purpose of this paper is to employ the modified (MDTM) differential transform method (DTM) to obtain approximate periodic solutions of two cases of nonlinear jerk equation. Design/methodology/approach – The approach is based on MDTM that is developed by combining DTM, Laplace transform and Padé approximant. Findings – Comparison of results obtained by MDTM with those obtained by numerical solutions indicates the excellent accuracy of solution. Originality/value – The MDTM is extended to determining approximate periodic solution of third-order nonlinear differential equations.

Disconjugacy and extremal solutions of nonlinear third-order equations

In this paper, we make an exhaustive study of the third order linear operators $u''' +Mu$, $u'''+Mu'$ and $u'''+Mu''$ coupled with $(k, 3-k)$-conjugate boundary conditions, where $k=1,2$. We obtain the optimal intervals on which the Green's functions are of one sign. The main tool is the disconjugacy theory. As an application of our results, we develop a monotone iteration method to obtain positive solutions of the nonlinear problem $u'''+Mu''+f(t,u)=0$, $u(0)=u'(0)=u(1)=0$.

Nonlinear differential equations satisfied by certain classical modular forms

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.

On the existence and multiplicity of positive periodic solutions of a nonlinear third-order equation

In this paper, we are concerned with the existence and multiplicity of positive 2 π -periodic solutions for the following nonlinear third-order problem u ‴ ( t ) + α u ″ ( t ) + β u ′ ( t ) = f ( t , u ( t ) ) . Here α , β are two positive constants, f ( t , u ) ∈ C ( R × R , R ) , f ( t + 2 π , u ) = f ( t , u ) . The proof relies on a fixed point theorem on cones.

The Kinematics of Fibre-Reinforced Fluids. An Integrable Reduction

A geometric formulation previously adopted in hydrodynamics and soliton theory is used here to investigate the kinematic conditions attendant upon the motion of an ideal fibre-reinforced fluid. Conditions are established for the existence of multiple-fibre configurations. Kinematically admissible spatial motions are obtained in which the fibres are geodesic windings on nested toroidal surfaces. In the case of purely planar motion, it is shown that the kinematic relations reduce to a third-order nonlinear equation. Remarkably, this admits a reduction to a solitonic system which is related to the classical sine-Gordon equation. The kinematic conditions in this case possess a duality property.

Existence of solutions for third-order boundary value problems on a time scale

Let T be a closed subset of R with inf T = −∞ and sup T = +∞. For the third-order nonlinear equation, u Δ3 = f( t, u, u Δ , u ΔΔ ), t ϵ T, where f: T × R 3 → R is continous, we assume solutions of initial value problems are unique and extend to T. We consider questions of the uniqueness of solutions implying the existence of solutions for conjugate boundary problems on T.

The Kinematics of the Planar Motion of Ideal Fiber-Reinforced Fluids: An Integrable Reduction and Bäcklund Transformation

We establish that the kinematic constraints on the steady planar motion of an ideal fiber-reinforced fluid can be consolidated in a single third-order nonlinear equation. Remarkably, this equation admits a solitonic reduction related to the classical sine-Gordon equation. The kinematic conditions in this case admit a novel duality property and a Backlund transformation.