Analysis of moderately-thick and deep shells subjected to thermal and mechanical loadings

An analytical solution to a boundary value problem of moderately-thick and deep doubly curved shells of rectangular planform subjected to thermal and mechanical loadings is presented. Sanders' kinematic relations that incorporate transverse shear deformations in moderately deep shell theory are considered in the shell formulations, yielding five coupled second order partial differential equations in five unknowns displacements. These are then solved in conjunction with admissible boundary conditions. The numerical results presented herein should serve as bench-mark solutions for future comparisons.     

Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space,disturbed by Gaussian processes. Part I

Nonlinear evolution differential equations with unbounded linear operators of disturbance by Gaussian random processes are considered in an abstract Hilbert space. For the Cauchy problem for the differential equations, the sufficient existence and uniqueness conditions for their solutions are proved and the sufficient conditions for the equivalence of the probability measures generated by these solutions are derived. Moreover, the corresponding Radon–Nikodym densities are calculated explicitly in terms of the coefficients or characteristics of the considered differential equations.     

Legendre wavelets method for approximate solution of fractional-order differential equations under multi-point boundary conditions

In this paper, Legendre wavelet collocation method is applied for numerical solutions of the fractional-order differential equations subject to multi-point boundary conditions. The explicit formula of fractional integral of a single Legendre wavelet is derived from the definition by means of the shifted Legendre polynomial. The proposed method is very convenient for solving fractional-order multi-point boundary conditions, since the boundary conditions are taken into account automatically. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. Several numerical examples are solved to demonstrate the validity and applicability of the presented method.     

Solidification and control of a liquid metal column

In this paper, two different 1D mechanistic models for the solidification of a pure substance are presented. The first model is based on the two-domain approach, resulting in 2 partial differential equations (PDEs) and one ordinary differential equation (ODE) with 2 boundary conditions, 2 interface conditions, and one initial condition: the Stefan problem.In the second model, the metal column is considered as one-domain, and one PDE is valid for the whole domain. The result is one PDE with two boundary conditions.The models are implemented in MATLAB, and the ODE solver ode23s is used for solving the systems of equations. The models are developed in order to simulate and control the dynamic response of the solidification rate. The control scheme is based on a linear PI controller.     

A numerical algorithm for avascular tumor growth model

Avascular tumor growth model for multicellular spheroids is considered. The model is a moving boundary problem and consists of three types of cells. The governing equations are nonlinear hyperbolic and/or parabolic differential equations. Comparisons of the numerical solutions with the solution of the recent studies are done. The effect of the necrotic region on the tumor growth is discussed.     

A Roadmap to Well Posed and Stable Problems in Computational Physics

All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier–Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.     

Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method

During the past few decades, the idea of using differential quadrature methods for numerical solutions of partial differential equations (PDEs) has received much attention throughout the scientific community. In this article, we proposed a numerical technique based on polynomial differential quadrature method (PDQM) to find the numerical solutions of two-dimensional sine-Gordon equation with Neumann boundary conditions. The PDQM reduced the problem into a system of second-order linear differential equations. Then, the obtained system is changed into a system of ordinary differential equations and lastly, RK4 method is used to solve the obtained system. Numerical results are obtained for various cases involving line and ring solitons. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions that exist in literature. It is shown that the technique is easy to apply for multidimensional problems.     

Computational algorithm for solving problem of optimal boundary-control with nonseparated boundary conditions

The problem of optimal boundary-control with nonseparated boundary conditions is considered in the case where the motion of a plant is described by a system of nonlinear ordinary differential equations with a corresponding initial condition, which thereafter is taken as a control action. First, the system of corresponding nonlinear Euler–Lagrange equations is described and in order to solve it, the quasi-linearization method (the first method) is taken. After that, the quasi-linearization method is used with the aim to solve just the optimization problem with boundary control and the nonseparated boundary condition; as a result, the initial nonlinear problem is reduced to the solution of a corresponding linearly quadratic optimization problem (the second method). By the particular example (of oil production) we demonstrate that the second method converges considerably faster and its accuracy is five times higher than the accuracy of the first method. The numerical results reinforce the compliance of the constructed mathematical model with practice.     

Lyapunov matrices for a class of time-delay systems with piecewise-constant kernel

In this contribution, we study time-delay systems with distributed delay whose kernel is piecewise constant. It is shown that the problem of finding a Lyapunov matrix for this class of time-delay systems can be reduced to the computation of solutions to an auxiliary system of linear differential equations with boundary conditions. The problem of solving the auxiliary system is then reduced to the solution of a system of linear algebraic equations. It is established that the auxiliary system with boundary conditions admits a unique solution if and only if there exists a unique Lyapunov matrix.     

Analytical Solution of the Minimum Time Slew Maneuver Problem for an Axially Symmetric Spacecraft in the Class of Conical Motions

The conventional problem of the time-optimal slew of a spacecraft considered as a solid body with a single symmetry axis subject to arbitrary boundary conditions for the attitude and angular velocity is considered in the quaternion statement. By making certain changes of variables, the original dynamic Euler equations are simplified, and the problem turns into the optimal slew problem for a solid body with a spherical distribution of mass containing one additional scalar differential equation. For this problem, a new analytical solution in the class of conical motions is found; in this solution, the initial and terminal attitudes of the space vehicle belong to the same cone realized under a bounded control. A modification of the optimal slew problem in the class of generalized conical motions is made that makes it possible to obtain its analytical solution under arbitrary boundary conditions for the attitude and angular velocity of the spacecraft. A numerical example of a spacecraft’s conical motion and examples demonstrating the proximity of the solutions of the conventional and modified optimal slew problems of an axially symmetric spacecraft are discussed.     

Fractional Differential Mathematical Models of the Dynamics of Nonequilibrium Geomigration Processes and Problems with Nonlocal Boundary Conditions

The analytical solutions of boundary-value problems with nonlocal boundary conditions are presented for two fractional differential mathematical models of the dynamics of a geomigration process non-equilibrium in time. The models based on the equations with the Caputo and Hilfer derivatives of fractional order are considered.     

Feedback stabilization of rotating Timoshenko beam with adaptive gain

The problem of boundary feedback stabilization of rotating Timoshenko beam, arising from control of flexible robot arms, is studied in this paper. First, under gain adaptive direct strain feedback controls, a counterexample is given to show that the corresponding closed loop system is not asymptotically stable, which is contrary to traditional conjecture. The counterexample given in this paper also exemplifies an interesting result: certain two two-order linear partial differential equations with five homogeneous boundary conditions have non-trivial solutions. Then, with an additional boundary feedback control, the related energy of the closed loop system is proved to be strongly stable, or more precisely, the configuration of the beam can be exponentially stabilized with some suitable non-linear boundary feedback controls with adaptive gain.     

Analytical solution of the optimal slew problem for an axisymmetric spacecraft in the class of conical motions

The traditional problem is discussed of an optimal spacecraft slew in terms of minimum energy costs. The spacecraft is considered as a rigid body with one symmetry axis under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft in the quaternion formulation. Using substitutions of variables, the original problem is simplified (in terms of dynamic Euler equations) to the optimal slew problem for a rigid body with a spherical mass distribution. The simplified problem contains one additional scalar differential equation. A new analytical solution is presented for this problem in the class of conical motions, leading to constraints on the initial and final values of the angular velocity vector. In addition, the optimal slew problem is modified in the class of conical motions to derive an analytical solution under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft. A numerical example is given for the conical motion of the spacecraft, as well as examples showing the closeness of the solutions of the traditional and modified optimal slew problems for an axisymmetric spacecraft.     

Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order

We investigate the existence of nontrivial solutions for a multi-point boundary value problem for fractional differential equations. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray–Schauder nonlinear alternative. As an application, some examples to illustrate our results are given.     

L2‐absolute and input‐to‐state stabilities of equations with nonlinear causal mappings

Nonlinear scalar equations with causal mappings are considered. These equations include differential, difference, differential‐delay, integro‐differential and other traditional equations. Estimates for the L²‐norms of solutions are established. These estimates give us explicit conditions for the absolute and input‐to‐state stabilities of the considered equations. The Aizerman‐type problem from the theory of absolute stability is also discussed. The suggested approach enables us to consider various classes of systems from the unified point of view. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical Solution to a Class of Inverse Problems for Parabolic Equation

A class of inverse problems for parabolic equation is considered. In particular, boundary value problems with nonlocal conditions are reduced to such class of problems. The proposed numerical approach is based on the method of lines to reduce the problem to a system of ordinary differential equations. To solve this system, an analog of the transfer method for boundary conditions is applied. The results of numerical experiments are given.     

A series solution algorithm for initial boundary-value problems with variable properties

A multiple infinite trigonometric cum polynomial series method for solving initial-boundary value problems governed by hyperbolic differential equations with variable coefficients is developed. The method proposed herein can be easily applied to a broad class of engineering systems including those cases where boundary conditions may vary with time. In the proposed mathematical technique, the solution form is assumed as a combination of infinite Fourier series and polynomial series of nth order, where n is the order of the differential equation. The coefficients of the polynomial series are obtained as functions of undetermined Fourier series coefficients by satisfying the initial-boundary conditions. The variable coefficients are expanded in appropriate half-range sine or cosine series. Insertion of the above Fourier-polynomial series solutions into the differential equation and application of orthogonality conditions leads to a linear summation equation which can be solved in open form. However, the authors have developed a closed-form series solution consisting of a highly efficient algorithm. The major advantage of this technique is the development of a solution algorithm, coupled with the multiple infinite trigonometric cum polynomial series solutions, leading to fast converging series solutions. A representative initial and boundary value problem governed by hyperbolic partial differential equations of variable coefficients is presented herein to demonstrate the efficiency and accuracy of the method.     

Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations

We consider the solution of partial differential equations for initial/boundary conditions using the decomposition method. The partial solutions obtained from the seperate equations for the highest-ordered linear operator terms are shown to be identical when the boundary conditions are general, and asymptotically equal when the boundary conditions in one independent variable are independet of other variables.     

A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators

We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP–SAT). Recently it was shown that SBP–SAT finite difference methods can yield superconvergent functional output if the boundary conditions are imposed such that the discretization is dual consistent. We generalize these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalized to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.     

Remarks on the paper “Optimal control of a class of linear multivariable systems with integral quadratic energy constraint ”

An optimal periodic control problem for a system described by differential equations is considered. Control units are assumed to generate control actions with the square-integrable derivative. The above problem is approximated by a sequence of discretized problems containing trigonometric polynomials, which approximate the state and control variables, and the functions in the criterion and differential equations. The conditions for a sequence of optimal solutions to discretized problems, which are to be a generalized minimizing sequence for the basic problem, are given. Extensions to more general problem formulations are presented. The possibility of application is illustrated by the example of an optimal periodic control problem for a chemical reactor.