On the exact solution to certain non-linear partial differential equations

This paper, based on the theory of stratifications, gives a brand-new classification of partial differential equations. To be dedicated to my beloved teacher, H. Cartan, academician of French Academy of Sciences, on the occasion of his 90th birthday. This paper was partially supported by the National Natural Science Foundation of China     

Higher order linearization in non-linear random vibration

In this paper a higher order linearization method for analyzing non-linear random vibration problems is presented. The non-linear terms of the given equation are replaced by unknown linear terms. These are in turn described by extra non-linear differential equations. The combined system of equations is then linearized to arrive at a higher degree-of-freedom equation for the original system. The method is illustrated by considering the Duffing oscillator under white noise input. The equivalent two d.o.f linear system is derived by the present method. Numerical results on steady state variance and PSD functions are obtained. These are found to be better than the simple linearization results.     

Application of complex stochastic averaging to non-linear random vibration problems

The stochastic averaging procedure in a complex-variable setting, used previously by Ariaratnam and Tarn to analyze a linear system under random parametric excitation, is extended to non-linear systems under both parametric and external random excitations. It is shown that equations for the moments of the system response, while still constituting an infinite hierarchy, form a simpler pattern compared with the corresponding equations obtained from the usual amplitude and phase formulation. From this simpler pattern, it is possible to identify those moments which tend to zero at the stationary state. Furthermore, a much smaller number of equations needs to be solved when the infinite hierarchy is truncated to calculate approximately the non-zero moments.     

The exact steady-state solution of a class of non-linear stochastic systems

In this paper exact steady state solutions are constructed for a class of non-linear systems subjected to stochastic excitation. The results are then applied to both classical and non-classical oscillator problems.     

An asymptotic solution of an initial value problem for a non-linear viscoelastic rod

Stress wave propagation in a one-dimensional materai, described by the non-linear constitutive law

$\text{?σ}\text{?t}\text{? E}\text{?ε}\text{?t}\text{+ k[σ ? E'g3 + βε}{}^2\text{] = 0}$

, is analyzed by a two-time variable perturbation method. Secular terms are “cast out” by employing special orthogonality conditions. The resulting expansion reveals a “mode coupling”, due to the non-linearity, and displays the significance of the three parameters present in the constitutive law. Numerical values for the parameters are assumed and the resulting displacements computed.     

A consistent closure method for non-linear random vibration

—A closure method is proposed for calculating moments of the state vector of a non-linear system satisfying an Itô stochastic differential equation. It is based on a finite set of moment equations and a sequence of probabilities converging to the exact distribution of the state vector that is obtained by the minimization of an objective function. Numerical results for one-dimensional diffusion processes show that the proposed closure technique is robust in the sense that resultant moments are satisfactory even when crude approximations are used for the probability of the state vector.     

On a new exact solution to Stokes’ first problem for Maxwell fluids

Stokes’ first problem for Maxwell fluids is re-examined using integral transform methods and several forms of the exact solution are presented. In the process, we call attention to, and correct, a recent case in the literature where this solution has been stated incorrectly. In addition, we note a number of analytical results and give several velocity profile plots. Finally, connections are drawn to other areas of research.     

An improved solution to a problem of magneto-thermo-elasticity

Summary This paper examines temperature distribution and displacement in an infinite elastic solid resulting from the simultaneous application of a thermal disturbance and a small perturbation due to an applied magnetic field. It is shown that the presence of the magneto-thermal coupling factor is to decrease the magnitude of the displacement as well as the magnitude of the temperature distribution. The method of similarly transformations is used for simplification and as a result the governing equations are nondimensionalized automatically. The ISML package for Numerical Inversion of Laplace Transform has been used to confirm the analytical results obtained above. The method is a considerable simplification of that by Paria [Ref. 11] and provides graphs of both displacement and temperature.

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Sommario Questo lavoro esamina la distribuzione di temperatura e lo spostamento che, in un solido elastico infinito, risultano dall'applicazione simultanea di un disturbo termico e una piccola perturbazione dovuta ad un campo magnetico applicato. Si dimostra che l'effetto del fattore di accoppiamento magnetotermico è di diminuire l'ampiezza dello spostamento e della variazione di temperatura. Per semplificare si usu il metodo delle trasformazioni di similitudine e di conseguenza le equazioni che reggono il problema risultano automaticamente adimensionali. Si è fatto uso del pacchetto ISML per l'inversione numerica della trasformata di Laplace al fine di confermare i risultati analitici sopra indicati. Il metodo è una sempliflcazione considerevole di quello di Paria (Rif. 11) e fornisce grafici sia dello spostamento sia della temperatura.

     

Iterative solution of a non-linear boundary value problem for a rotating string

The problem of determining the motion of a rotating inextensible string of length l, free at one end and fixed at the other can, under certain assumptions, be treated by solving a non-linear boundary value problem. p]In this paper, two Picard-type iterative schemes are constructed and the sequences generated are proved to converge to a positive solution of that non-linear boundary value problem. Additionally, one sequence converges to the solution from above, the other from below. Moreover, the iterative scheme can be used to solve the inverse problem of determining the angular velocity of the rotating string, given two observations (at one point in time) of position of the string.     

An exact and asymptotic analysis of a diffraction problem

The present analysis deals with diffraction of acoustic waves by an oscillating strip focusing on the exact and concise formulation of a series solution in the complex domain. The complete solution is represented by a series, the eigenfunctions of which are generalized gamma functions. An exact expression of this special function, with argument being ‘integer +1/2’, is derived. The convergence analysis of the series solution in transformed domain is discussed graphically. Finally, the scattered and total acoustic fields are obtained by exact and asymptotic evaluations of inverse Fourier transforms. The significance of the present investigation is the derivation of a high order accurate solution in a convenient form.     

An exact solution in a nonlinear theory of rods

A three dimensional nonlinear equilibrium theory of elastic rods, applicable to large displacements and small strains, and accounting for extensibility and shear deformation is developed. Integrals of the governing equations are determined for the case of specified end force and moment. A class of solutions is obtained for an initially straight, untwisted rod and compared to the classical solution. The effects of extensibility and shear deformation are discussed.     

Generalization of the equivalent linearization method for non-linear random vibration problems

The method of equivalent linearization is generalized such that the response of a non-linear oscillator subject to both parametric and external random white-noise excitations can be determined approximately. The main objective of the new method is to obtain a closed system of differential equations for certain statistical moments of the response. This can be achieved by linearizing the drift term of the corresponding Itô-equations and by replacing the square of the diffusion term by a second degree polynomial. It is found that the accuracy of the mean square amplitudes is improved considerably compared with the original equivalent linearization.