Solitons and periodic solutions of coupled nonlinear evolution equations by using the sine–cosine method

In this paper, we establish exact solutions for coupled nonlinear evolution equations. The sine–cosine method is used to construct exact periodic and soliton solutions of coupled nonlinear evolution equations. Many new families of exact travelling wave solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky equations and the coupled nonlinear Klein–Gordon and Nizhnik–Novikov–Veselov equations are successfully obtained. The obtained solutions include compactons, solitons, solitary patterns and periodic solutions. These solutions may be important and of significance for the explanation of some practical physical problems.     

Stable discrete solution of coupled singular mixed partial differential problems

This paper deals with the construction of stable discrete numerical solutions of strongly coupled singular diffusion mixed partial differential problems. After discretization, using a matrix difference scheme, the resulting coupled mixed partial difference problem is treated using a discrete separation of variables method which avoids solving algebraic systems. Existence, stability of solutions, its construction, algorithm and example are given.     

On global existence of solutions to coupled matrix Riccatiequations in closed-loop Nash games

Presents comparison and global existence results for solutions of coupled matrix Riccati differential equations appearing in closed loop Nash games and in mixed H₂/H_∞-type problems. Convergence of solutions is established for the diagonal case, solutions of the corresponding algebraic equations are discussed using numerical examples     

Coupled matrix Riccati equations in minimal cost variance controlproblems

We present an algorithm for the solution of a nontrivial coupled system of algebraic Riccati equations appearing in risk sensitive control problems. Moreover, we use comparison methods to derive non-blowup conditions for the solutions of a corresponding terminal value problem for coupled systems of Riccati differential equations     

The stable discrete numerical solution of strongly coupled mixed partial differential systems

This paper is concerned with the discrete numerical solution of coupled partial differential mixed problems with non-Dirichlet coupled boundary value conditions. By application of a discrete separation of variables method, the proposed numerical solution of the problem turns out to be the exact solution of certain coupled partial difference system, appearing from the discretization of the continuous partial differential system. Our approach avoids the iterative solution of algebraic systems which appears when one uses simple discretization methods. Existence, stability and the construction of solutions are considered.     

Numerical solution of a coupled modified Korteweg–de Vries equation by the Galerkin method using quadratic B-splines

In this paper, numerical solutions of a coupled modified Korteweg–de Vries equation have been obtained by the quadratic B-spline Galerkin finite element method. The accuracy of the method has been demonstrated by five test problems. The obtained numerical results are found to be in good agreement with the exact solutions. A Fourier stability analysis of the method is also investigated.     

A note on boundary value problems for a coupled system of fractional differential equations

In this work, we establish sufficient conditions for the existence of solutions to a general class of multi-point boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann–Liouville sense. By means of a fixed point theorem, existence results for the solutions are established. We include an example to show the applicability of our results.     

A posteriori error estimation and adaptive mesh refinement for combined thermal-stress finite element analysis

Local and global error estimators and an associated h-based adaptive mesh refinement schemes are proposed for coupled thermal-stress problems. The error estimators are based on the “flux smoothing” technique of Zienkiewicz and Zhu with important modifications to improve convergence performance and computational efficiency. Adaptive mesh refinement is based on the concept of adaptive accuracy criteria, previously presented by the authors for stress-based problems and extended here for coupled thermal-stress problems. Three methods of mesh refinement are presented and numerical results indicate that the proposed method is the most efficient in terms of number of adaptive mesh refinements required for convergence in both the thermal and stress solutions. Also, the proposed method required a smaller number of active degrees of freedom to obtain an accurate solution.     

A priori error estimates of a meshless method for optimal control problems of stochastic elliptic PDEs

In this paper, we study the optimal control problems of stochastic elliptic equations with random field in its coefficients. The main contributions of this work are two aspects. Firstly, a meshless method coupled with the stochastic Galerkin method is investigated to approximate the control problems, which is competitive for high-dimensional random inputs. Secondly, a priori error estimates are derived for the solutions to the control problems. Some numerical tests are carried out to confirm the theoretical results and to demonstrate the efficiency of the proposed method.     

Least square finite element solution of stagnation point flow

A detailed analysis of the least square finite element solution of nonlinear boundary value problems is presented with reference to a particular example of nonlinear coupled differential equations governing the flow of an incompressible viscous fluid in the vicinity of a forward stagnation point at a blunt body. The numerical solutions are presented for different cases. The results obtained by the least square finite element method are in very good agreement with the results available in the literature confirming the versatility and usefulness of the application of the method to nonlinear boundary value problems governing the fluid flow problems.     

Goal-oriented error estimation and adaptivity for fluid–structure interaction using exact linearized adjoints

We develop duality-based a posteriori error estimates for functional outputs of solutions of steady fluid–structure-interaction problems. The crucial complication in obtaining these estimates pertains to the derivation of the coupled dual (exact linearized adjoint) problem owing to the free-boundary character of fluid–structure interaction. We present two approaches to derive the dual problem. In the domain-map linearization approach, the fluid subproblem is first transformed to a fixed reference domain, after which one essentially linearizes with respect to the domain transformation map. In the shape-linearization approach, fluid unknowns are fixed in the current configuration and a very weak formulation of the fluid subproblem is then linearized using shape-derivative techniques. We show that the dual problems correspond to coupled fluid–structure problems with nonstandard coupling conditions. Furthermore, we present numerical experiments that demonstrate the consistency of the dual-based error estimates and their usefulness in goal-oriented adaptive mesh-refinement.     

Transverse impact problems by higher order beam finite element

A higher order beam finite element is derived and shown to be very efficient in solving the transient dynamic problem. The finite element thus developed is then applied to impact problems concerning the response of beams to striking elastic masses. Due to the local indentation the motion of the beam is nonlinearly coupled with the motion of the mass. An efficient iterative procedure is proposed in this paper to integrate the time variable. Both elastic impact and impact with permanent indentations are considered. The finite element solutions are found in good agreement with some existing solutions.     

High-order Compact Schemes for Nonlinear Dispersive Waves

High-order compact finite difference schemes coupled with high-order low-pass filter and the classical fourth-order Runge–Kutta scheme are applied to simulate nonlinear dispersive wave propagation problems described the Korteweg-de Vries (KdV)-like equations, which involve a third derivative term. Several examples such as KdV equation, and KdV-Burgers equation are presented and the solutions obtained are compared with some other numerical methods. Computational results demonstrate that high-order compact schemes work very well for problems involving a third derivative term.     

Quadratic approximation of solutions for boundary value problems with nonlocal boundary conditions

In this paper, using the quasilinearization method coupled with the method of upper and lower solutions, we study a class of second-order nonlinear boundary value problems with nonlocal boundary conditions. We establish some sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem. An example is also included to illustrate the main result.     

Open-loop Stackelberg strategies in a linear-quadratic differential game with time delay

The open-loop Stackelberg strategies in a linear-quadratic differential game with time delay are constructed in the form of necessary and sufficient conditions. The evolution of a game with time delay is described by coupled differential equations composed of lumped and distributed parameter subsystems. Calculation of solutions is facilitated by transforming two-point boundary-value problems to terminal boundary-value ones by the introduction of Riccati-type differential equations.     

轴向运动梁横向非线性振动模型研究进展

综述了描述轴向运动梁横向非线性振动的两组数学模型的研究进展.在轴向运动梁径向和横向平面非线性振动耦合模型的基础上,总结了两组横向非线性振动模型的推导,以及在自由振动、受迫振动、参激振动工况下两组横向模型的近似解析比较的研究进展.在直接数值离散方法的基础上,总结了两组横向模型在各种工况下对平面耦合模型近似程度的研究进展.最后提出若干尚待深入研究的问题.     

A new method based on legendre polynomials for solution of system of fractional order partial differential equations

We study legendre polynomials in case of more than one variable and develop new operational matrices of fractional order integrations as well as fractional order differentiations. Based on these operational matrices, we develop a new sophisticated technique to solve a coupled system of fractional order partial differential equations. Our technique reduces the coupled system under consideration to a system of easily solvable algebraic equations without discretizing the system. As an application, we provide examples and numerical simulations demonstrating that the results obtained using the new technique matches well with the exact solutions of the problems. We also study error analysis graphically.     

A coupled gradient network approach for static and temporalmixed-integer optimization

Utilizes the ideas of artificial neural networks to propose new solution methods for a class of constrained mixed-integer optimization problems. These new solution methods are more suitable to parallel implementation than the usual sequential methods of mathematical programming. Another attractive feature of the proposed approach is that some global search mechanisms may be easily incorporated into the computation, producing results which are more globally optimal. To formulate the solution method proposed in this paper, a penalty function approach is used to define a coupled gradient-type network with an appropriate architecture, energy function and dynamics such that high-quality solutions may be obtained upon convergence of the dynamics. Finally, it is shown how the coupled gradient net may be extended to handle temporal mixed-integer optimization problems, and simulations are presented which demonstrate the effectiveness of the approach.     

A generator of dissipative methods for the numerical solution of the Schrödinger equation

In this paper a generator of hybrid methods with minimal phase-lag is developed for the numerical solution of the Schrödinger equation and related problems. The generator's methods are dissipative and are of eighth algebraic order. In order to have minimal phase-lag with the new methods, their coefficients are determined automatically. Numerical results obtained by their application to some well known problems with periodic or oscillating solutions and to the coupled differential equations of the Schrödinger type indicate the efficiency of these new methods.     

Necessary and sufficient conditions of risk‐sensitive optimal control and differential games for stochastic differential delayed equations

In this paper, we consider risk‐sensitive optimal control and differential games for stochastic differential delayed equations driven by Brownian motion. The problems are related to robust stochastic optimization with delay due to the inherent feature of the risk‐sensitive objective functional. For both problems, by using the logarithmic transformation of the associated risk‐neutral problem, the necessary and sufficient conditions for the risk‐sensitive maximum principle are obtained. We show that these conditions are characterized in terms of the variational inequality and the coupled anticipated backward stochastic differential equations (ABSDEs). The coupled ABSDEs consist of the first‐order adjoint equation and an additional scalar ABSDE, where the latter is induced due to the nonsmooth nonlinear transformation of the adjoint process of the associated risk‐neutral problem. For applications, we consider the risk‐sensitive linear‐quadratic control and game problems with delay, and the optimal consumption and production game, for which we obtain explicit optimal solutions.