The paper presents models of car dynamics with varying complexity.Joint coordinates and homogenous transformations are used to model the motionof a car. Having formulated the models of the car, we discuss the influence of thecomplexity of the model on numerical efficiency of integrating the equationsdescribing car dynamics. Methods with both constant and adaptive step size havebeen applied. The results of numerical calculations are presented and conclusionsare formulated.

This paper investigates the local existence of the nonnegative solution and the finite time blow-up of solutions and boundary layer profiles of diffusion equations with nonlocal reaction sources; we also study the global existence and that the rate of blow-up is uniform in all compact subsets of the domain, the blow-up rate of ${\left|u\left(t\right)\right|}_{\infty }$ is precisely determined.

The stochastic finite element method (SFEM) is employed for solving stochastic one-dimension time-dependent differential equations with random coefficients. SFEM is used to have a fixed form of linear algebraic equations for polynomial chaos coefficients of the solution process. Four fixed forms are obtained in the cases of stochastic heat equation with stochastic heat capacity or heat conductivity coefficients and stochastic wave equation with stochastic mass density or elastic modulus coefficients. The relation between the exact deterministic solution and the mean of solution process is numerically studied.

This paper studies the existence of global solutions to the initial-boundary value problem for some nonlinear degenerate wave equations by means of compactness method and the potential well idea. Meanwhile, we investigate the decay estimate of the energy of the global solutions to this problem by using a difference inequality.

In the present work, we consider a class of nonlinear optimal control problems, which can be called “optimal control problems in mechanics.” We deal with control systems whose dynamics can be described by a system of Euler-Lagrange or Hamilton equations. Using the variational structure of the solution of the corresponding boundary-value problems, we reduce the initial optimal control problem to an auxiliary problem of multiobjective programming. This technique makes it possible to apply some consistent numerical approximations of a multiobjective optimization problem to the initial optimal control problem. For solving the auxiliary problem, we propose an implementable numerical algorithm.

We study the first-order nonhomogenous wave equation. We extend the convolution theorem into a general case with a double convolution as the nonhomogenous term. The uniqueness and continuity of the solution are proved and we provide some examples in order to validate our results.

We implement a relatively new analytical technique, the variational iteration decomposition method (VIDM), for solving the eighth-order boundary value problems. The proposed method is an elegant combination of variational iteration method and decomposition method. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Numerical work is given to check the efficiency of the method. Comparisons are made to confirm the reliability and accuracy of the technique. The technique can be used as an alternative for solving nonlinear boundary value problems.