Mechanical Differential Equations Research Articles

Group analysis tasks. Conservation laws

Differential equations of mechanics are derived from the laws of conservation of mass, momentum and energy. Other conservation laws can be obtained using the symmetries of these equations. Any conservation law can be derived from one using canonical operators, special cases of which are symmetry operators. The calculation of canonical conservation law operators is equivalent to a direct method of finding a divergent form of equations that are valid by virtue of the equations of mechanics. Using the example of one-dimensional equations of gas dynamics, all zero-order conservation laws are obtained by the direct method. For special equations of state, an infinite set of conservation laws have been obtained, many of which are new.

Block Elements in Boundary-Value Problems for Sets of Differential Equations of Mechanics and Physics in Non-Classical Domains

Block Elements in Boundary-Value Problems for Sets of Differential Equations of Mechanics and Physics in Non-Classical Domains

Modern interpretation of Saint-Venant’s principle and semi-inverse method

Relevance. The progressive development of views on the Saint-Venant formulated principles and methods underlying the deformable body mechanics, the growth of the mathematical analysis branch, which is used for calculation and accumulation of rules of thumb obtained by the mathematical results interpretation, lead to the fact that the existing principles are being replaced with new, more general ones, their number is decreasing, and this field is brought into an increasingly closer relationship with other branches of science and technology. Most differential equations of mechanics have solutions where there are gaps, quick transitions, inhomogeneities or other irregularities arising out of an approximate description. On the other hand, it is necessary to construct equation solutions with preservation of the order of the differential equation in conjunction with satisfying all the boundary conditions. Thus, the following aims of the work were determined: 1) to complete the familiar Saint-Venants principle for the case of displacements specified on a small area; 2) to generalize the semi-inverse Saint-Venants method by finding the complement to the classical local rapidly decaying solutions; 3) to construct on the basis of the semi-inverse method a modernized method, which completes the solutions obtained by the classical semi-inverse method by rapidly varying decaying solutions, and to rationalize asymptotic convergence of the solutions and clarify the classical theory for a better understanding of the classic theory itself. To achieve these goals, we used such methods , as: 1) strict mathematical separation of decaying and non-decaying components of the solution out of the plane elasticity equations by the methods of complex variable theory function; 2) construction of the asymptotic solution without any hypotheses and satisfaction of all boundary conditions; 3) evaluation of convergence. Results. A generalized formulation of the Saint-Venants principle is proposed for the displacements specified on a small area of a body. A method of constructing asymptotic analytical solutions of the elasticity theory equations is found, which allows to satisfy all boundary conditions.

Simplified variational iteration method for solving ordinary differential equations and eigenvalue problems

A simplified variational iteration method is proposed to solve high-order homogeneous or nonhomogeneous linear ordinary differential equation and ordinary differential equation eigenvalue problems more efficiently and conveniently. The simplification includes two aspects: (1) explicitly deducing the general form of the differential equation for the identification of the general Lagrange multiplier while avoiding the complexity of variational calculations during identification and (2) simplifying the iterative expressions to reduce the computational work of each iteration. Three ordinary differential equations in mechanics are solved by this simplified variational iteration method, which proves that it is valid and more concise than traditional methods. To make the method more practical, it is suggested that some complicated analytical derivations be executed numerically, thereby achieving a simplified semi-analytical variational iteration method that can be easily implemented by computer programs. The method is then used to numerically solve two complex ordinary differential equation problems derived from the continuum analysis of tall building structures: a sixth-order nonhomogeneous ordinary differential equation with complex boundary conditions and a sixth-order ordinary differential equation eigenvalue problem. Numerical computer programs are developed for these two problems, and corresponding examples are provided to verify the accuracy and efficiency of the simplified variational iteration method in solving complex ordinary differential equation problems.

Application of Lie Group Analysis to the Plastic Torsion of Rod with the Saint Venant–Mises Yield Criterion

Based on Lie group and Lie algebra theory, the basic principles of Lie group analysis of differential equations in mechanics are analyzed, and its validity in theory of plasticity is explained by example. For the plastic torsion of rod with variable cross section that consists in non-linear Saint Venant-Mises yield criterion, the 10-dimensional Lie algebra admitted by the equilibrium equation and yield criterion is completely solved, and invariants and group invariant solutions relative to different sub-algebras are given. At last, physical explanations of each group invariant solution are discussed by some types of transformations.

On the Convergence of Random Differential Quadrature (RDQ) Method and Its Application in Solving Nonlinear Differential Equations in Mechanics

Differential Quadrature (DQ) is one of the efficient derivative approximation techniques but it requires a regular domain with all the points distributed only along straight lines. This severely restricts the DQ while solving the irregular domain problems discretized by the random field nodes. This limitation of the DQ method is overcome in a proposed novel strong-form meshless method, called the random differential quadrature (RDQ) method. The RDQ method extends the applicability of the DQ technique over the irregular or regular domains discretized using the random field nodes by approximating a function value with the fixed reproducing kernel particle method (fixed RUM), and discretizing a governing differential equation by the locally applied DQ method. A superconvergence condition is developed for the RDQ method, which gives more than O(h(p+1)) function value convergence for the uniform as well as random field nodes scattered in the domain. The RDQ method convergence analysis is carried out, and the superconvergence condition is verified by solving several 1D, 2D and elasticity problems. The applicability of the RDQ method to solve the nonlinear governing differential equations is successfully demonstrated by solving the fixed-fixed and cantilever beams for deflection due to the nonlinear electrostatic loading. It is concluded that the RDQ method effectively handles the irregular or regular domains discretized by the uniform or random field nodes, with good convergence rates.

Possibilities of finite calculus in computational mechanics

AbstractThe expression ‘finite calculus’ refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a space–time domain of finite size. The governing equations resulting from this approach are different from those of infinitesimal calculus theory and they incorporate new terms which depend on the dimensions of the balance domain. The new governing equations allow the derivation of naturally stabilized numerical schemes using any discretization procedure. The paper discusses the possibilities of the finite calculus method for the finite element solution of convection–diffusion problems with sharp gradients, incompressible fluid flow and incompressible solid mechanics problems and strain localization situations. Copyright © 2004 John Wiley & Sons, Ltd.

Book Review: Partial Differential Equations in Mechanics. By A. P. C. Selvadurai

Book Review: Partial Differential Equations in Mechanics. By A. P. C. Selvadurai

Partial differential equations in mechanics 1: fundamentals, Laplace's equation, diffusion equation, wave equation

Partial differential equations in mechanics 1: fundamentals, Laplace's equation, diffusion equation, wave equation

Partial differential equations in mechanics 2: the biharmonic equation, Poisson's equation

Partial differential equations in mechanics 2: the biharmonic equation, Poisson's equation

Linear and non-linear deflection analysis of thick rectangular plates—II. Numerical applications

Variational methods are widely used for the solution of complex differential equations in mechanics for which exact solutions are not possible. The finite difference method, although well known as an efficient numerical method, was applied in the past only for the analysis of linear and non-linear thin plates. In this paper the suitability of the method for the analysis of non-linear deflection of thick plates is studied for the first time. While there are major differences between small deflection and large deflection plate theories, the former can be treated as a particular case of the latter, when the centre deflection of the plate is less than or equal to 0.2–0.25 of the thickness of the plate. The finite difference method as applied here is a modified finite difference approach to the ordinary finite difference method generally used for the solution of thin plate problems. In this analysis thin plates are treated as a particular case of the corresponding thick plate when the boundary conditions of the plates are taken into account. The method is first applied to investigate the deflection behaviour of clamped and simply supported square isotropic thick plates. After the validity of the method is established, it is then extended to the solution of rectangular thick plates of various aspect ratios and thicknesses. Generally, beginning with the use of a limited number of mesh sizes for a given plate aspect ratio and boundary conditions, a general solution of the problem including the investigation of accuracy and convergence was extended to rectangular thick plates by providing more detailed functions satisfying the rectangular mesh sizes generated automatically by the program. Whenever possible results obtained by the present method are compared with existing solutions in the technical literature obtained by much more laborious methods and close agreements are found. The significant number of results presented here are not currently available in the technical literature. The submatrices involved in the formation of the finite difference equations from the governing differential equations are generated directly by the computer program. The subroutine SOLINV using the change of variable technique illustrated elsewhere takes care of the solution of the general system. Simplicity in formulation and quick convergence are the obvious advantages of the finite difference formulation developed to compute small and large deflection analysis of thick plates in comparison with other numerical methods requiring extensive computer facilities.

Solutions of the equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $$\end{document}. Exact solutions of some partial differential equations in mechanics and physics

AbstractThe solutions of the equation \documentclass{article}\pagestyle{empty}\begin{document}$ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $\end{document}, for L̂ a linear operator are derived. Different forms for L̂ whether it is time independent or time dependent and self‐commutative (or not) at different times are considered separately. By using the results obtained, exact solutions of some partial differential equations are found for the first time.

Numerical Solution of Multiparameter Eigenvalue Problems

AbstractIn this paper we describe an algorithm for the computation of eigenvalue curves and corresponding eigenvectors of multiparameter eigenvalue problems. After input of an algebraic simple eigenvalue with eigenvector we use continuation methods to compute the eigenvalue curve starting from the given eigenvalue. We indicate algebraic bifurcation equations for the computation of branching algebraic simple eigenvalue curves. For numerical examples we treated several ordinary differential equations in mechanics with finite differences.

Miscellanea from the history of celestial mechanics

The KS-transformation introduced by P. Kustaanheimo and E. Stiefel into celestial mechanics is derived straight from the ‘Kepler Formulas’. There follow the treatment of the inverse Newton problem comprising the derivation of the differential equations of mechanics by J. Hermann and L. Euler and also remarks concerning the fundamental papers by Euler about the planet problem and then-body problem. The conclusion is a simple example given by A. Voss and H. Liebmann, for the differential equations of mechanics with non-holonomic condition, which is of ‘pseudoplanetary’ quality.

On the Algebraic Integrals of a System of Differential Equations of Mechanics

On the Algebraic Integrals of a System of Differential Equations of Mechanics

On a class of differential equations in mechanics of continua

On a class of differential equations in mechanics of continua

Differential Equations in Mechanics and in Physics

In accepting the honour of being your President, I was told that the office carried only a single duty—that of delivering an address at a then distant date: even the selection of the topic could be deferred for a time. So there lay before me a primrose path of dalliance. I had some hesitation about the selection of a topic. One decision, however, was immediate; my determination was instantly framed to avoid any discussion of the place of mathematics in education, an unending subject, of which perhaps you have heard only too much. It would have been possible to select some special topic, such as functionality, in its historical development; and you would have had a mere mathematical lecture. I have preferred to choose a general subject: my text seems highly technical in phrase, but I shall not keep very closely to it; my purpose rather is to fall back upon my own experience over a wide range of teaching and to exhibit for your consideration some reflexions based on that experience. Moreover, technical as its title is, my address will contain no mathematical symbols. Though my work as a teacher and writer is largely associated with differential equations—probably in the course of a year I receive (with a request for solution) a greater number of insoluble equations than any one here—the reason for its selection is not to be found in any specially personal relation. The reason is that the particular processes affect many subjects in mathematics, and consequently allow general comparisons to be made, as well as particular inferences to be drawn: they constitute a method as powerful in operation and effective in results as any within our domain.