Approximate solution to the nonlinear problem of an underground pipeline deformation

Some geometrically nonlinear mathematical model is constructed of a pipeline as a shell in a strongly viscous medium. We find a universal procedure for reducing the two-dimensional equations of motion to the one-dimensional equations for long bent pipes. We also constructed a difference scheme and propose a software complex for numerical analysis of the equations under study. Some numerical experiments are conducted for a particular case of a bent pipeline, the deformations of the pipe wall are found, and the displacement of its centerline is calculated.     

Numerical solution of a nonlinear time-optimal control problem

Nonlinear systems with a stationary (i.e., explicitly time independent) right-hand side are considered. For time-optimal control problems with such systems, an iterative method is proposed that is a generalization of one used to solve nonlinear time-optimal control problems for systems divided by phase states and controls. The method is based on constructing finite sequences of simplices with their vertices lying on the boundaries of attainability domains. Assuming that the system is controllable, it is proved that the minimizing sequence converges to an ɛ-optimal solution after a finite number of iterations. A pair {T, u(·)} is called an ɛ-optimal solution if |T − T _opt| − ɛ, where T _opt is the optimal time required for moving the system from the initial state to the origin and u is an admissible control that moves the system to an ɛ-neighborhood of the origin over the time T.     

Numerical solution of the nonlinear seepage problem in a domain with an unknown boundary

A finite-element algorithm is developed for the problem of headless steady nonlinear seepage (boundary-value problem for a nonlinear elliptic equation in a domain with an unknown boundary) in a multicomponent medium with a piecewise-linear boundary. Numerical solution results are reported for a number of problems. The effects of the form of the nonlinearity on the characteristics of the seepage process are considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 83–90, 1987.     

Numerical solution of the Goursat problem by a nonlinear trapezoidal formula

In this paper the solution of the Goursat problem is obtained by the use of a nonlinear Trapezoidal formula based on geometric means. The numerical results indicate the new strategy to be superior.     

Numerical solution for a nonlinear,one-dimensional problem of thermoelasticity

A numerical solution for a nonlinear, one-dimensional boundary-value problem of thermoelasticity for the elastic half-space is presented. The resulting equations are discussed and the numerical method is investigated for stability. Comparison with other existing numerical schemes is carried out. The obtained results clearly indicate the process of shock formation. The presented figures show the effects of different nonlinear coupling constants on the distributions of the mechanical displacement and temperature in the medium. A special case is briefly discussed.     

Numerical solution of a Cauchy problem for nonlinear reaction diffusion processes

In this paper, the authors propose a numerical method to compute the solution of the Cauchy problem: _wt-(w^m_wx)x=w^p$w_t-(w^m{w}_x{)}_x=w^p$, the initial condition is a nonnegative function with compact support, m>0$m>0$, p?m+1$p?m+1$. The problem is split into two parts: a hyperbolic term solved by using the Hopf and Lax formula and a parabolic term solved by a backward linearized Euler method in time and a finite element method in space. The convergence of the scheme is obtained. Further, it is proved that if m+1?p<m+3$m+1?p<m+3$, any numerical solution blows up in a finite time as the exact solution, while for p>m+3$p>m+3$, if the initial condition is sufficiently small, a global numerical solution exists, and if p?m+3$p?m+3$, for large initial condition, the solution is unbounded.     

An inertial manifold in the problem of nonlinear oscillations of an infinite panel

For the problem of nonlinear oscillations of an infinite panel in supersonic gas flow, we prove the existence of a finite-dimensional invariant manifold, that exponentially attracts trajectories of the system and contains the maximal attractor.Translated from Ukrainskii Matematicheskii. Zhurnal, Vol. 42, No. 9, pp. 1291–1293, September, 1990.     

Properties of the attractor in the problem of the nonlinear oscillations of an infinite panel

In the problem of the nonlinear oscillations of an infinite panel in a supersonic gas flow, the existence of a maximal attractor and the finiteness of its fractal dimension are proved. The dimension is estimated from above in terms of the parameters of the problem. Cases are indicated when the attractor has a regular structure.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 108–115, 1988.     

Numerical solution of the Dirichlet problem for nonlinear parabolic equations by a probabilistic approach

A number of new layer methods for solving the Dirichlet problemfor semilinear parabolic equations are constructed by usingprobabilistic representations of their solutions. The methodsexploit the ideas of weak sense numerical integration of stochasticdifferential equations in a bounded domain. Despite their probabilisticnature these methods are nevertheless deterministic. Some convergencetheorems are proved. Numerical tests are presented.     

Numerical solution of the boundary-value problem for a nonlinear diffusion equation in image processing

Diffusion filtering methods involve solving an initial boundary-value problem for the diffusion equation in which the initial condition is specified by a function representing the filtered image. The output of this filter is the solution u(x, y, t) of the initial boundary-value problem at some fixed time t = T. In a previous study we have proposed a new version of the diffusion filtering method that ensures improved noise removal due to inclusion of a dependence of the diffusion coefficient on local image intensity. The present study analyzes the resulting finite-difference method for the initial boundary-value problem, examines its numerical implementation, and analyzes its efficiency on prototype and real images. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 35–43, 2006.     

Numerical solution of a nonlinear parabolic control problem by a reduced SQP method

We consider a control problem for a nonlinear diffusion equation with boundary input that occurs when heating ceramic products in a kiln. We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.     

Numerical solution of the nonlinear Klein-Gordon equation

A numerical method is developed to solve the nonlinear one-dimensional Klein-Gordon equation by using the cubic B-spline collocation method on the uniform mesh points. We solve the problem for both Dirichlet and Neumann boundary conditions. The convergence and stability of the method are proved. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L₂, L_∞ and Root-Mean-Square errors (RMS) in the solutions show the efficiency of the method computationally.     

Numerical solution of an inverse problem of nonisothermal sorption dynamics

We consider the inverse problem for a mathematical model of sorption dynamics that incorporates diffusion, intradiffusion kinetics, and the heat balance. Two numerical methods are proposed. Their efficiency is investigated in a computer experiment. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 56–63, 2008.