Value Problems For Differential Equations Research Articles

Existence of bounded solutions of integral boundary value problems for singular differential equations on whole lines

Boundary value problems of second-order singular differential equations with nonlinear operator Φ on whole lines are discussed. By applying the nonlinear alternative of Leray–Schauder-type fixed point theorem, some existence results of solutions for integral boundary value problems of differential equations on whole lines are established. The emphasis is put on the nonlinear operator [Φ(ρ(t)x′(t))]′ involved with the nonnegative function ρ that may satisfy ρ(0) = 0, the strictly increasing sup-multiplicative-like function Φ and the differential equations are defined on the whole line. Three examples and a remark are presented to illustrate the main theorems.

Similar Structure Algorithm for Solving Boundary Value Problem of Differential Equations

Based on the similar structure theory, we conduct rigorous mathematical derivation and proof for the boundary value problem of a class of differential equation . We put forward a new algorithm------Similar Structure Algorithm to solve such boundary value problems. The similar structure algorithm only includes arithmetic and logical operation, hence it can be oriented towards computer. According to the similar structure algorithm, an example is given to conduct a simulation experiment by the application of computer software at the end of the paper. And we also observe and analysis the variational laws of the laboratory result by changing the coefficient of the boundary condition.

A nonpolynomial collocation method for fractional terminal value problems

In this paper we propose a nonpolynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α,0<α<1. The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a nonpolynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples.

PSEUDO SPLINE METHOD FOR SOLVING CERTAIN SECOND ORDER INITIAL VALUE PROBLEM OF ORDINARY DIFFERENTIAL EQUATION

In this paper, new set of low order schemes for solving certain second order initial value problem of differential equations directly are derived. From the general method of deriving a spline function, schemes based on interpo- lation and collocation were derived. In the process of derivation we observed that it is possible to obtain the discrete scheme from the continuous scheme by collocating at a given point. Also from the process of derivation the Adams'-Stormer formula were recovered for the explicit scheme and the Cowell formula for the implicit scheme.

A New Method and Applications of the Boundary Value Problem of Differential Equation

The similar structure of solution for the boundary value problem of second order linear homogeneous differential equation has been studied. Based on the analysis of the relationship between similar structure of solution, its kernel function, the equation and boundary conditions, similar constructive method (shortened as SCM) of solution is obtained. According to the SCM, the similar structure of solution and its kernel function are constructed for the mathematical model of homogeneous reservoir which considers the influence of bottom-hole storage and skin effect under the infinite outer boundary condition. The SCM is a new and innovative way to solve boundary value problem of differential equation and seepage flow theory, which is especially used in Petroleum Engineering.

Some Generalized Gronwall-Bellman Type Impulsive Integral Inequalities and Their Applications

This paper investigates some generalized Gronwall-Bellman type impulsive integral inequalities containing integration on infinite intervals. Some new results are obtained, which generalize some existing conclusions. Our result is also applied to study a boundary value problem of differential equations with impulsive terms.

Fixed Point Theorems for a Class ofα-Admissible Contractions and Applications to Boundary Value Problem

A class ofα-admissible contractions defined via altering distance functions is introduced. The existence and uniqueness conditions for fixed points of such maps on complete metric spaces are investigated and related fixed point theorems are presented. The results are reconsidered in the context of partially ordered metric spaces and applied to boundary value problems for differential equations with periodic boundary conditions.

Existence of Solutions for Integro-differential Equations of Mixed Type in Banach Spaces

The initial value problem of differential equation can be reduced to the equivalent integral equation.Recently,this method is used successfully to discuss the existence of the solution to initial value problem of nonlinear differential equation.Using the fixed point theorem of convexpower condensing operator,we study the existence of solutions to initial value problems for nonlinear second order integro-differential equations of mixed type in Banach spaces.

Monotone iteration method for differential equations involving integral boundary conditions on the half line

By applying the monotone iterative methods, we obtain the existence of monotone positive solutions for integral boundary value problems of differential equations on the half line, and establish the iterative schemes for approximating the solutions. Our approach is based on the fixed point theorem and the monotone iterative technique. The existence of lower and upper solutions is not needed.

Singular Initial Value Problem for Certain Classes of Systems of Ordinary Differential Equations

The paper is devoted to the study of the solvability of a singular initial value problem for systems of ordinary differential equations. The main results give sufficient conditions for the existence of solutions in the right-hand neighbourhood of a singular point. In addition, the dimension of the set of initial data generating such solutions is estimated. An asymptotic behavior of solutions is determined as well and relevant asymptotic formulas are derived. The method of functions defined implicitly and the topological method (Ważewski's method) are used in the proofs. The results generalize some previous ones on singular initial value problems for differential equations.

Impulsive Boundary Value Problem for Differential Equations with Fractional Order

In this paper, we discuss some existence results for a class of impulsive boundary value problem involving fractional differential equation. Our results are based on Banach fixed point theorem and Krasnoselskii’s fixed point theorem.

Positive solutions of semi-positone Hammerstein integral equations and applications

Existence of solutions for semi-positone integral equations of Hammerstein type is obtained by using the fixed point index method. Many boundary value problems of differential equations can be transformed into these integral equations. A specific example is given.

Acoustic scattering from compact deformations of shallow-water waveguide's surfaces

In shallow-water waveguide, compact deformations of sea surface or seafloor will affect the sound propagation. The deformations include local deformation of bottom topography, rock, trench and objects sank to the bottom, floater and surface ship on the sea surface, etc, which are all the components of an ocean waveguide. In this paper, the moments method and the boundary integral equation method are presented to compute the acoustic scattering from compact surface deformations in shallow-water waveguide. The former method combines the normal-mode theory with the numerical method for solving the boundary value problem of differential equation in local area, while the latter only covers the finite integral area of the deformed surfaces. The perturbation feature of near and far field and the deformation identification feature are analyzed.

Multiplicity Result of Positive Solutions for Nonlinear Differential Equation of Fractional Order

We investigate the existence of multiple positive solutions for a class of boundary value problems of nonlinear differential equation with Caputo’s fractional order derivative. The existence results are obtained by means of the Avery‐Peterson fixed point theorem. It should be point out that this is the first time that this fixed point theorem is used to deal with the boundary value problem of differential equations with fractional order derivative.

A New Family of Phase‐Fitted and Amplification‐Fitted Runge‐Kutta Type Methods for Oscillators

In order to solve initial value problems of differential equations with oscillatory solutions, this paper improves traditional Runge‐Kutta (RK) methods by introducing frequency‐depending weights in the update. New practical RK integrators are obtained with the phase‐fitting and amplification‐fitting conditions and algebraic order conditions. Two of the new methods have updates that are also phase‐fitted and amplification‐fitted. The linear stability and phase properties of the new methods are examined. The results of numerical experiments on physical and biological problems show the robustness and competence of the new methods compared to some highly efficient integrators in the literature.

On a boundary value problem for differential equation with p-Laplacian

AbstractThe existence of a solution of a boundary value problem for differential equation with p-Laplacian is proved by the technique of lower and upper solutions. A nonlocal boundary condition and a derivative dependent nonlinearity is assumed.

Fractional boundary value problems: Analysis and numerical methods

Abstract In this paper we consider nonlinear boundary value problems for differential equations of fractional order α, 0 &lt; α &lt; 1. We study the existence and uniqueness of the solution and extend existing published results. In the last part of the paper we study a class of prototype methods to determine their numerical solution.

Optimal Control of Error Performance of Planar 3-DOF Parallel Mechanism

Abstract. The optimal control strategy is introduced to improve the motion accuracy of planar 3-DOF parallel mechanism. The optimal control of error performance of mechanism is achieved by structuring and quantizing the quadratic form index of the error performance, and taking into account both the system response and energy control in state space, so we study the problem which can be boil down to solve two-point boundary value problem of differential equation. The simulation results show that the actual motion of mechanism keep the better tracking accuracy with the global motion while dynamics parameters is changing, and the controller is ease to accomplishment in the industrial due to the simply structure and the stable change of control energy.

Existence of Solution for Boundary Value Problems of Differential Equations with Delay

In this paper , by using fixed point theorem , upper and lower solution’s method and monotone iterative technique , we prove the existence of maximum and minimum solutions of differential equations with delay , which improved and generalize the result of related paper. Keywords—Banach space , boundary value problem , differential equation , delay .

Existence results for boundary value problems of high order differential equations involving Caputo derivative

We study boundary value problems for differential equations involving Caputo derivative of order α∈(n−1,n) in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder’s inequality, a suitable singular Gronwall’s inequality and fixed point theorem via a priori estimate method. At last, three examples are given to illustrate the results.