Solution of the Duffing equation involving both integral and non-integral forcing terms

In this paper, a new iteration algorithm is proposed to solve the Duffing equation involving both integral and non-integral forcing terms in the reproducing kernel space. Its exact solution is represented in the form of series and the n-term approximation un(x) of the exact solution u(x) is proved to converge to the exact solution. It is worthy to note that the method used in the paper can be applied for solving nonlinear differential equations.     

Approximate solution for a variable-coefficient semilinear heat equation with nonlocal boundary conditions

This paper develops an iterative algorithm for the solution to a variable-coefficient semilinear heat equation with nonlocal boundary conditions in the reproducing space. It is proved that the approximate sequence u _n (x, t) converges to the exact solution u(x, t). Moreover, the partial derivatives of u _n (x, t) are also convergent to the partial derivatives of u(x, t). And the approximate sequence u _n (x, t) is the best approximation under a complete normal orthogonal system.     

Sufficient conditions for the existence and asymptotic behaviour of solution to a quasi-linear elliptic problem

In this article, we combine the existing regularity theory, perturbation method and the lower and upper solutions method to study the existence and asymptotic behaviour of positive solution to a boundary value problem for the p-Laplacian operator. More exactly, we study the existence and asymptotic behaviour of the positive solution to a quasi-linear elliptic problem of the form?Δ _p u=λ a(x)g(u) in D′(Ω), u>0 in Ω, lim _{x→? Ω} u(x)=0. Under some conditions on a and g, we show that there is a non-negative number Λ₀ such that for all λ∈(0, Λ₀], the problem has a solution u _λ in the sense of distribution, which is bounded from above by some positive numbers μ(λ). Such estimates and the asymptotic behaviour are important in computer programs when we know an algorithm for determining the solution.     

How to improve mamdani's approach to fuzzy control

Fuzzy control is a methodology that translates “if”-“then” rules, A_ji (x₁) &…& A_jn(x_n) → B_j(u), formulated in terms of a natural language, into an actual control strategy u(x). Implication of uncertain statements is much more difficult to understand than “and,” “or,” and “not.” So, the fuzzy control methodologies usually start with translating “if”-“then” rules into statements that contain only “and,” “not,” and “or.” the first such translation was proposed by Mamdani in his pioneer article on fuzzy control. According to this article, a fuzzy control is reasonable iff one of the rules is applicable, i.e., either the first rule is applicable (A₁₁(x₁) &…& A_1n(x_n) & B₁(u)), or the second one is applicable, etc. This approach turned out to be very successful, and it is still used in the majority of fuzzy control applications. However, as R. Yager noticed, in some cases, this approach is not ideal: Namely, if for some x, we know what u(x) should be, and add this crisp rule to our rules, then the resulting fuzzy control for this x may be different from the desired value u(x). to overcome this drawback, Yager proposed to assign priorities to the rules, so that crisp rules get the highest priority, and use these priorities while translating the rules into a control strategy u(x). In this article, we show that a natural modification of Mamdani's approach can solve this problem without adding any ad hoc priorities. © 1995 John Wiley & Sons, Inc.     

Two-sided bounds for the asymptotic behaviour of free nonlinear vibration systems with application of the differential calculus of norms

This paper deals with the free nonlinear dynamical system [xdot](t)=A x(t)+h(t, x(t)), t≥t ₀, x(t ₀)=x ₀, where A is an n×n-matrix and h a nonlinear vector function with h(t, u)=o(∥u∥). As the first novel point, a lower bound for the asymptotic behaviour on the solution x(t) is derived. Two methods are applied to determine the optimal two-sided bounds, where one of the methods is the differential calculus of norms. In this context, the second novel point enters; it consists of a new strategy to significantly reduce the computation time for the determination of the optimal constants in the two-sided bounds. The obtained results are especially of interest in engineering and cannot be obtained by the methods used so far.     

On Word Equations in One Variable

For a word equation E of length n in one variable x occurring # _x times in E a resolution algorithm of O(n+# _x log n) time complexity is presented here. This is the best result known and for the equations that feature #_x < \fracnlogn\#_{x}<\frac{n}{\log n} it yields time complexity of O(n) which is optimal. Additionally it is proven here that the set of solutions of any one-variable word equation is either of the form F or of the form F∪(uv)⁺ u where F is a set of O(log n) words and u, v are some words such that uv is a primitive word.     

Adaptive robust control of nonholonomic systems with stochastic disturbances

Based on the Brockett’s necessary condition for feedback asymptotic stabilization[1], nonholonomic systems fail to be stabilized at the origin by any static continuous state feedback though they are open loop controllable. There are two novel approaches …     

A class of hybrid collocation methods for third-order ordinary differential equations

A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x _n+j , i = 0(1)k and at an off grid point x = x _n+u , where k is the step number of the method and u is an arbitrary rational number in (x _n , x _n+k ). A predictor of order 2k ? 1 is also proposed to cater for y _n+k in the main method. Taylor series expansion is employed for the calculation of y _n+1, y _n+2, y _n+u and their higher derivatives. Evaluation of the resulting method at x = x _n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.     

A novel numerical method for a class of problems with the transition layer and Burgers’ equation

A numerical method for solving a class of quasi-linear singular two-point boundary value problems with a transition layer is presented in this paper. For the problem ? u _xx +a(u+f(x))u _x +b(x, u)=0, we develop a multiple scales method. First, this method solves the location of the transition layer, then it approximates the singular problem with reduced problems in the non-layer domain and pluses a layer corrected problem which nearly has an effect in the layer domain. Both problems are transformed into first-order problems which can be solved easily. For the problem ? u _xx +b(x, u)=0, we establish a similar method which approximate the problem with reduced problems and a two-point boundary value problem. Unsteady problems are also considered in our paper. We extend our method to solve Burgers’ equation problems by catching the transition layer with the formula of shock wave velocity and approximating it by a similar process.     

Optimum Control of a Certain Class of Servomechanism†

Linear control systems governed by the vector matrix differential equation x = A x + B u have been considered. It has been shown how to find the optimum control u so that the system, starting from an initial position x(0), is steered to a state specifying the first p coordinates of the system in time t _o fixed in advance, the values attained by the (n—p) coordinates being immaterial, where n is the dimension of the system. The optimization considered here is with regard to the norm of u supposed to belong to L _m E _r space.     

A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses

We study the partial vertex cover problem. Given a graph G=(V,E), a weight function w:V→R ⁺, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2-approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity k _u . A solution consists of a function x:V→ℕ₀ and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most x _u k _u . Our objective is to find a cover that minimizes ∑ _v∈V x _v w _v . This is the first 2-approximation for the problem and also runs in O(nlog n+m) time. Research supported by NSF Awards CCR 0113192 and CCF 0430650, and the University of Maryland Dean’s Dissertation Fellowship.     

Monotone properties of certain classes of solutions of second-order difference equations

The authors consider the difference equations

δ(a_nδx_n)=q_nx_n+1

and

δ(a_nδx_n)=q_nf(x_n+1),

where a_n > 0, q_n > 0, and f: R å R is continuous with uf(u) > 0 for u ≠ 0. They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.     

Computing median and antimedian sets in median graphs

The median (antimedian) set of a profile π=(u ₁,…,u _k ) of vertices of a graph G is the set of vertices x that minimize (maximize) the remoteness ∑ _i d(x,u _i ). Two algorithms for median graphs G of complexity O(n idim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles and will be in practice often faster than the other algorithm which in addition computes antimedian sets and remoteness functions and works in all partial cubes.     

Fast Algorithm for Computing the Upper Endpoint of Sample Variance for Interval Data: Case of Sufficiently Accurate Measurements

When we have n results x₁,...,x_n of repeated measurement of the same quantity, the traditional statistical approach usually starts with computing their sample average E and their sample variance V. Often, due to the inevitable measurement uncertainty, we do not know the exact values of the quantities, we only know the intervals x_i of possible values of x₁ In such situations, for different possible values x_i∈ x_i, we get different values of the variance. We must therefore find the range V of possible values of V. It is known that in general, this problem is NP-hard. For the case when the measurements are sufficiently accurate (in some precise sense), it is known that we can compute the interval V in quadratic time O(n²). In this paper, we describe a new algorithm for computing V that requires time O(n log(n)) (which is much faster than O(n²)).     

Why clustering in function approximation? Theoretical explanation

Function approximation is a very important practical problem: in many practical applications, we know the exact form of the functional dependence y=f(x₁,…,x_n) between physical quantities, but this exact dependence is complicated, so we need a lot of computer space to store it, and a lot of time to process it, i.e., to predict y from the given x_i. It is therefore necessary to find a simpler approximate expression g(x₁,…,x_n)≈f(x₁,…,x_n) for this same dependence. This problem has been analyzed in numerical mathematics for several centuries, and it is, therefore, one of the most thoroughly analyzed problems of applied mathematics. There are many results related to approximation by polynomials, trigonometric polynomials, splines of different type, etc. Since this problem has been analyzed for so long, no wonder that for many reasonable formulations of the optimality criteria, the corresponding problems of finding the optimal approximations have already been solved. Lately, however, new clustering‐related techniques have been applied to solve this problem (by Yager, Filev, Chu, and others). At first glance, since for most traditional optimality criteria, optimal approximations are already known, the clustering approach can only lead to non‐optimal approximations, i.e., approximations of inferior quality. We show, however, that there exist new reasonable criteria with respect to which clustering‐based function approximation is indeed the optimal method of function approximation. © 2000 John Wiley & Sons, Inc.     

Backstepping boundary control of Burgers’ equation with actuator dynamics

In this paper, we propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0,t) and u(1,t), the backstepping control also depends on u_x(0,t), u_x(1,t), u_xx(0,t) and u_xx(1,t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are sufficiently regular and the closed-loop system, including the boundary dynamics, is globally H³ stable and well posed.     

A cubically convergent iteration method for multiple roots of f(x)=0

This paper describes a cubically convergent iteration method for finding the multiple roots of nonlinear equations, f(x)=0, where f:?→? is a continuous function. This work is the extension of our earlier work [P.K. Parida, and D.K. Gupta, An improved regula-falsi method for enclosing simple zeros of nonlinear equations, Appl. Math. Comput. 177 (2006), pp. 769–776] where we have developed a cubically convergent improved regula-falsi method for finding simple roots of f(x)=0. First, by using some suitable transformation, the given function f(x) with multiple roots is transformed to F(x) with simple roots. Then, starting with an initial point x ₀ near the simple root x* of F(x)=0, the sequence of iterates {x _n }, n=0, 1, … and the sequence of intervals {[a _n , b _n ]}, with x*∈{[a _n , b _n ]} for all n are generated such that the sequences {(x _n ?x*)} and {(b _n ?a _n )} converges cubically to 0 simultaneously. The convergence theorems are established for the described method. The method is tested on a number of numerical examples and the results obtained are compared with those obtained by King [R.F. King, A secant method for multiple roots, BIT 17 (1977), pp. 321–328.].     

Exact solution of mixed problems for variable coefficient one-dimensional diffusion equation

This paper deals with diffusion problems modeled by the equation a(t)u_xx = u_t, x > 0, t > 0, u(x, 0) = c(x) together with the boundary condition u(0, t) = b(t) or u_x(0, t) = b(t). By using Fourier transforms, existence conditions and exact solutions of the above mixed problems are given.