Nonlinear Two-point Boundary Value Research Articles

Solving Singular Boundary Value Problems by Optimal Homotopy Asymptotic Method

In this paper, optimal homotopy asymptotic method (OHAM) for the semianalytic solutions of nonlinear singular two-point boundary value problems has been applied to several problems. The solutions obtained by OHAM have been compared with the solutions of another method named as modified adomain decomposition (MADM). For testing the success of OHAM, both of the techniques have been analyzed against the exact solutions in all problems. It is proved by this paper that solutions of OHAM converge rapidly to the exact solution and show most effectiveness as compared to MADM.

Approximate Solutions of Singular Two-Point BVPs Using Legendre Operational Matrix of Differentiation

Exact and approximate analytical solutions of linear and nonlinear singular two-point boundary value problems (BVPs) are obtained for the first time by the Legendre operational matrix of differentiation. Different from other numerical techniques, shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The accuracy of the technique is demonstrated through several linear and nonlinear test examples.

Optimal Ascent Guidance for Air-Breathing Launch Vehicle Based on Optimal Trajectory Correction

An optimal guidance algorithm for air-breathing launch vehicle is proposed based on optimal trajectory correction. The optimal trajectory correction problem is a nonlinear optimal feedback control problem with state inequality constraints which results in a nonlinear and nondifferentiable two-point boundary value problem (TPBVP). It is difficult to solve TPBVP on-board. To reduce the on-board calculation cost, the proposed guidance algorithm corrects the reference trajectory in every guidance cycle to satisfy the optimality condition of the optimal feedback control problem. By linearizing the optimality condition, the linear TPBVP is obtained for the optimal trajectory correction. The solution of the linear TPBVP is obtained by solving linear equations through the Simpson rule. Considering the solution of the linear TPBVP as the searching direction for the correction values, the updating step size is generated by linear search. Smooth approximation is applied to the inequality constraints for the nondifferentiable Hamiltonian. The sufficient condition for the global convergence of the algorithm is given in this paper. Finally, simulation results show the effectiveness of the proposed algorithm.

New Spectral Second Kind Chebyshev Wavelets Algorithm for Solving Linear and Nonlinear Second-Order Differential Equations Involving Singular and Bratu Type Equations

A new spectral algorithm based on shifted second kind Chebyshev wavelets operational matrices of derivatives is introduced and used for solving linear and nonlinear second-order two-point boundary value problems. The main idea for obtaining spectral numerical solutions for these equations is essentially developed by reducing the linear or nonlinear equations with their initial and/or boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. Convergence analysis and some efficient specific illustrative examples including singular and Bratu type equations are considered to demonstrate the validity and the applicability of the method. Numerical results obtained are compared favorably with the analytical known solutions.

A new superconvergent method for systems of nonlinear singular boundary value problems

A new superconvergent method based on a sextic spline is described and analysed for the solution of systems of nonlinear singular two-point boundary value problems (BVPs). It is well known that the optimal orders of convergence could not be achieved using standard formulation of a sextic spline for the solution of BVPs. Based on the method used in our earlier research papers [J. Rashidinia and M. Ghasemi, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math. 235 (2011), pp. 2325–2342; J. Rashidinia, M. Ghasemi, and R. Jalilian, An o(h 6) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algorithms 55(4) (2010), pp. 403–428], we construct a new O(h 8) locally superconvergent method for the solution of general nonlinear two-point BVPs up to order 6. The error bounds and the convergence properties of the method have been proved theoretically. Then, the method is extended to solve the system of nonlinear two-point BVPs. Some test problems are given to demonstrate the applicability and the superconvergent properties of the proposed method numerically. It is shown that the method is very efficient and applicable for stiff BVPs too.

Nodal solutions of second-order two-point boundary value problems

Abstract We shall study the existence and multiplicity of nodal solutions of the nonlinear second-order two-point boundary value problems, u ″ + f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . The proof of our main results is based upon bifurcation techniques. Mathematics Subject Classifications: 34B07; 34C10; 34C23.

Existence of solutions for a fully nonlinear fourth-order two-point boundary value problem

In this paper, we investigate the existence of solutions of a fully nonlinear fourth-order differential equation $$x^{(4)}=f(t,x,x',x'',x'''),\quad t\in [0,1]$$ with one of the following sets of boundary value conditions; $$x'(0)=x(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0,$$ $$x(0)=x'(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0.$$ By using the Leray-Schauder degree theory, the existence of solutions for the above boundary value problems are obtained. Meanwhile, as an application of our results, an example is given.

An O(h 6 ) numerical solution of general nonlinear fifth-order two point boundary value problems

A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard sextic spline for the solution of fifth order two point boundary-value problems gives only O(h 2) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. O(h 6) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior of the orders of convergence.

Multiple positive solutions for nonlinear third order general two-point boundary value problems

Multiple positive solutions for nonlinear third order general two-point boundary value problems

Existence and iteration of monotone positive solutions for a third-order two-point boundary value problem

The existence of nondecreasing positive solutions for the nonlinear third-order two-point boundary value problem u‴(t) + q(t)f(t, u(t), u′(t)) = 0, 0 < t < 1, u(0) = u″(0) = u′ (1) = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.

Successive approximation procedure of optimal tracking control for nonlinear similar composite systems

The optimal tracking control (OTC) problem for a class of affine nonlinear composite systems with similar structure is considered. By using a modeling technique, the nonlinear similar composite system is first transformed into some quasi-decoupled subsystems. Then the high-order, strongly coupled, nonlinear two-point boundary value (TPBV) problem is transformed into a sequence of linear decoupled TPBV problems through a successive approximation procedure. The obtained OTC law consists of an accurate linear term and a nonlinear compensation term which is the limit of the adjoint vector sequence. A suboptimal tracking control law is obtained by truncating a finite iterative result of the adjoint vector sequence as its nonlinear compensation term.

Positive solutions of two-point boundary value problems for n-order p-Laplacian operator

In this paper, we study the existence of positive solutions for the non-linear two-point singular boundary value problem for higher order with p-Laplacian operator. By using the fixed-point index theory, the existence of a positive solution and several positive solutions for non-linear singular boundary value problem with p-Laplacian operator are obtained.

Optimal output tracking control for nonlinear systems via successive approximation approach

This paper deals with the optimal output tracking control (OOTC) problem of a class of nonlinear systems whose reference input to be tracked is produced by a generalized linear exosystem. To solve the nonlinear OOTC problem, the nonlinear two-point boundary value (TPBV) problem derived from the necessary conditions of the OOTC problem is transformed into two decoupled iterative sequences of linear differential equations. The OOTC law obtained consists of accurate feedforward and feedback terms and a nonlinear compensation term. The former can be found by solving a Sylvester equation and a Riccati equation, and the latter can be approximated using a successive approximation approach (SAA). A reduced-order reference input observer is constructed to make the feedforward control law physically realizable. A simulation example is employed to illustrate the validity of the results.

An non-uniform mesh cubic spline TAGE method for non-linear singular two-point boundary value problems

We report a non-uniform mesh cubic spline method of accuracy for the solution of non-linear singular two-point boundary value problems. The application of two-parameter alternating group explicit (TAGE) and Newton–TAGE iteration methods which are suitable for use on parallel computers is discussed. The error analysis for TAGE iteration method is discussed in detail. The numerical results confirm the utility of proposed cubic spline TAGE iteration methods.

Existence of solutions and positive solutions to a fourth-order two-point BVP with second derivative.

Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of integral equations. The main conditions of our results are local. In other words, the existence of the solution can be determined by considering the "height" of the nonlinear term on a bounded set. This class of problems usually describes the equilibrium state of an elastic beam which is simply supported at both ends.

The Decomposition Method For Solving Of A Class Of Singular Two-Point Boundary Value Problems

In this paper, a class of nonlinear singular two-point boundary value problems are solved by using Adomian's decomposition methods. The approximate solution of this problem is calculated in the form of series with easily computable components. Finally, we give a nonlinear numerical example.

Complexity of Nonlinear Two-Point Boundary-Value Problems

We study upper and lower bounds on the worst-case ε-complexity of nonlinear two-point boundary-value problems. We deal with general systems of equations with general nonlinear boundary conditions, as well as with second-order scalar problems. Two types of information are considered: standard information defined by the values or partial derivatives of the right-hand-side function, and linear information defined by arbitrary linear functionals. The complexity depends significantly on the problem being solved and on the type of information allowed. We define algorithms based on standard or linear information, using perturbed Newton's iteration, which provide upper bounds on the ε-complexity. The upper and lower bounds obtained differ by a factor of log log 1/ ε. Neglecting this factor, for general problems the ε-complexity for the right-hand-side functions having r( r⩾2) continuous bounded partial derivatives turns out to be of order (1/ ε) 1/ r for standard information, and (1/ ε) 1/( r+1) for linear information. For second-order scalar problems, linear information is even more powerful. The ε-complexity in this case is shown to be of order (1/ ε) 1/( r+2) , while for standard information it remains at the same level as in the general case.

Bifurcation Analysis of a Class of Parametrized Two-Point Boundary Value Problems

In this paper, we study the solution manifold M of a class of nonlinear parametrized two-point boundary value problems. Typical representatives of this class are the shell equations of Bauer, Reiss, Keller [2] and Troger, Steindl [29]. The boundary value problems are formulated as an abstract operator equation T(x,λ)=0 in appropriate Banach spaces. By exploiting the equivariance of T , we obtain detailed information about the structure of M. Moreover, we show how these theoretical results can be used to compute efficiently interesting parts of M with numerical standard techniques. Finally, we present numerical results for the shell equations given in [2] and [29].

A parameter expansion method for two-point nonlinear singularly-perturbed boundary value problems

This paper presents a parameter expansion method for two-point nonlinear singularly perturbed boundary value problem for second order ordinary differential equation. Newton's linearization scheme is used to linearize the nonlinear problems. In Okoro et al. [1] linear problems were treated. The numerical results obtained for some examples show that the parameter expansion becomes more accurate as the value of e approaches zero when compared with the standard collocation method, at no extra computational effort. For favourable problems to exponential fitting reported in [2], the proposed method in this paper is less accurate. It is however more general in application than the exponential fitting.

Non-Linear Filtration Algorithms Based on the Least Square Method and their Application to Power System Processes

Abstract Having no information about the probabilistic laws of noises least square method has been applied to solve nonlinear filtration prob lem. It introduces the variational methods and the solution of nonline ar two-point boundary value system. Two recursive techniques for sol ving this system have been proposed and investigated. One of them based on the disturbance theory has been applied for estimation of the unk nown power system model parameters and for filtration of measurement noise simultaneously. Besides the estimates of unknown parameters and power system state, the algorithm calculates the speed and acceleration in changes of these estimates.