First-order Initial Value Problems Research Articles

Numerical Solution of Stiff and Oscillatory Differential Equations Using a Block Integrator

This paper presents the derivation and implementation of a block integrator for the solution of stiff and oscillatory first-order initial value problems of Ordinary Differential Equations (ODEs). The integrator was derived by collocation and interpolation of the combination of power series and exponential function to generate a continuous implicit Linear Multistep Method (LMM). The basic properties of the derived integrator were investigated and the integrator was implemented on some sampled stiff and oscillatory problems. From the results obtained, it is obvious that the block integrator gives better approximation than some existing ones.

Contractivity-preserving explicit Hermite–Obrechkoff ODE solver of order 13

A new optimal, explicit, Hermite–Obrechkoff method of order 13, denoted by HO(13), that is contractivity-preserving (CP) and has nonnegative coefficients is constructed for solving nonstiff first-order initial value problems. Based on the CP conditions, the new 9-derivative HO(13) has maximum order 13. The new method usually requires significantly fewer function evaluations and significantly less CPU time than the Taylor method of order 13 and the Runge–Kutta method DP(8,7)13M to achieve the same global error when solving standard \(N\)-body problems.

The new class of implicit [formula omitted]-stable hybrid Obrechkoff method for the numerical solution of first order initial value problems

In a recent paper, Shokri et al. (2011) [1] introduce a new class of hybrid Obrechkoff methods for the numerical solution of second-order initial value problems. In this work, we will derive L-stable hybrid Obrechkoff methods for the numerical integration of stiff first-order initial value problems. The numerical results obtained by the new method for some problems show its superiority in efficiency, accuracy and stability.

A Model for Predicting Flowing Gas Temperature and Pressure Profiles in Buried Pipeline

Steady sate flow of natural gas in buried pipeline is predicted with both the heat transfer between the flowing gas and the surrounding and the Joule-Thompson effect. The steady-state flow continuity, momentum and energy equations constitute the governing equations. As a constant gas mass flux (or gas mass flow rate) distribution along the pipeline is obtained from the steady-state continuity equation, the mathematical model describing steady-state gas flow in pipeline may be reduced to a second-order ODE system of first-order initial-value problem with gas pressure and temperature as the dependent variables. The forth-order Runge-Kutta method is used to solve this ODE system. Comparison between the predicted results and the observed field data are very good

Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations

This paper focuses on the derivation of diagonally implicit two-point block backward differentiation formulas (DI2BBDF) for solving first-order initial value problem (IVP) with two fixed points. The method approximates the solution at two points simultaneously. The implementation and the stability of the proposed method are also discussed. A performance of the DI2BBDF is compared with the existing methods.

Analysis of the factorization method for elliptic differential equations

This paper presents a numerical analysis of the method of factorization for elliptic boundary value problems. A second-order elliptic boundary value problem is transformed into a decoupled system of first-order initial value problems. This elaborates the decoupled system's occupying operator, the solution of the problem, and the affine part of the operator. The solution of the problem is thus obtained by solving the factorized system using the finite-element method. A comparative study is made with the help of an example that illustrates many of its properties.

Singular and nonsingular first-order initial value problems

Under barrier strip type arguments we investigate the existence of global solutions to the initial value problem x ′ = f ( t , x , x ′ ) , x ( 0 ) = A , where the scalar function f ( t , x , p ) may be singular at x = A .

A one-step 7-stage Hermite–Birkhoff–Taylor ODE solver of order 11

A one-step 7-stage Hermite–Birkhoff–Taylor method of order 11, denoted by HBT(11)7, is constructed for solving nonstiff first-order initial value problems y ′ = f ( t , y ) , y ( t 0 ) = y 0 . The method adds the derivatives y ′ to y ( 6 ) , used in Taylor methods, to a 7-stage Runge–Kutta method of order 6. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution to order 11 leads to Taylor- and Runge–Kutta-type order conditions. These conditions are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than the Dormand–Prince DP87 and a larger unscaled interval of absolute stability than the Taylor method, T11, of order 11. HBT(11)7 is superior to DP87 and T11 in solving several problems often used to test higher-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. Numerical results show the benefit of adding high-order derivatives to Runge–Kutta methods.

Positive solutions for nonlinear operator equations and several classes of applications

In this paper, we study a class of nonlinear operator equations x = Ax + x0 on ordered Banach spaces, where A is a monotone generalized concave operator. Using the properties of cones and monotone iterative technique, we establish the existence and uniqueness of solutions for such equations. In particular, we do not demand the existence of upper-lower solutions and compactness and continuity conditions. As applications, we study first-order initial value problems and two-point boundary value problems with the nonlinear term is required to be monotone in its second argument. In the end, applications to nonlinear systems of equations and to nonlinear matrix equations are also considered.

Uniqueness and existence results for initial value problems on time scales through a reciprocal problem and applications

This paper is devoted to deriving the existence, uniqueness and approximation of solutions to a first-order initial value problem on an interval of an arbitrary time scale, taking values in a different time scale. We obtain the results considering dynamic equations whose nonlinear part is nonnegative and satisfies a type of “inverse” Carathéodory’s conditions and discontinuity. Several examples are given to illustrate the applicability of these results.

Initial-value technique for self-adjoint singular perturbation boundary value problems

We have developed an initial-value technique for self-adjoint singularly perturbed two-point boundary value problems. The original problem is reduced to its normal form, and the reduced problem is converted into first-order initial-value problems. These initial-value problems are solved by the cubic spline method. Numerical illustrations are given at the end to demonstrate the efficiency of our method. Graphs are also depicted in support of the results.

Numerical solution of fuzzy differential equations by Nyström method

In this paper, some numerical procedures for solving fuzzy first-order initial value problem have been investigated. Sufficiently conditions for stability and convergence of the proposed algorithms are given and their applicability is illustrated with examples.

Solution of initial value problems by the differential quadrature method with Hermite bases

The differential quadrature method (DQM) is used to solve the first-order initial value problem. The initial condition is given at the beginning of the interval. The derivative of a space-independent variable at a sampling grid point within the interval can be defined as a weighted linear sum of the given initial conditions and the function values at the sampling grid points within the defined interval. Hermite polynomials have advantages compared with Lagrange and Chebyshev polynomials, and so, unlike other work, they are chosen as weight functions in the DQM. The proposed method is applied to a numerical example and it is shown that the accuracy of the quadrature solution obtained using the proposed sampling grid points is better than solutions obtained with the commonly used Chebyshev–Gauss–Lobatto sampling grid points.

Nadler’s type principle with high order of convergence

Recently Proinov [P.D. Proinov, A generalization of the Banach contraction principle with high order of convergence of successive approximations, Nonlinear Analysis (2006), doi:10.1016/j.na.2006.09.008] generalized Banach contraction principle with high order of convergence. We extend some results of Proinov to the case of multi-valued maps from a complete metric space X into the space of all nonempty proximinal closed subsets of X . Our results not only generalize Nadler’s fixed-point theorem (in the case when T is a mapping from a complete metric space X into the space of all nonempty proximinal closed subsets of X ) but also gives high order of convergence. As an application, we obtain an existence theorem for first-order initial value problem.

Study of the initial value problems appearing in a method of factorization of second-order elliptic boundary value problems

In [J. Henry, A.M. Ramos, Factorization of second order elliptic boundary value problems by dynamic programming, Nonlinear Analysis. Theory, Methods & Applications 59 (2004) 629–647] we presented a method for factorizing a second-order boundary value problem into a system of uncoupled first-order initial value problems, together with a nonlinear Riccati type equation for functional operators. A weak sense was given to that system but we did not perform a direct study of those equations. This factorization utilizes either the Neumann to Dirichlet (NtD) operator or the Dirichlet to Neumann (DtN) operator, which satisfy a Riccati equation. Here we consider the framework of Hilbert–Schmidt operators, which provides tools for a direct study of this Riccati type equation. Once we have solved the system of Cauchy problems, we show that its solution solves the original second-order boundary value problem. Finally, we indicate how this techniques can be used to find suitable transparent conditions.

Existence of extremal solutions by approximation to a first-order initial dynamic equation with Carathéodory's conditions and discontinuous non-linearities*

This paper is devoted to the study of the existence of extremal solutions to a first-order initial value problem on an interval of an arbitrary time scale. We prove the existence of extremal solutions for problems satisfying Carathéodory's conditions. Moreover, they are approximated uniformly by a sequence of lower and upper solutions to this problem, respectively. We also can warrant the existence and approximation of extremal solutions for the problem by relaxing their continuity properties.

Numerical strategies for the system of first-order IVPS using the RK–Butcher algorithm

In this article, a new method of analysis for first-order initial-value type ordinary differential equations using the Runge–Kutta (RK)–Butcher algorithm is presented. To illustrate the effectiveness of the RK–Butcher algorithm, 10 problems have been considered and compared with the RK method based on arithmetic mean, and with exact solutions of the problems, and are found to be very accurate. Stability analysis for the first-order initial-value problem (IVP) has been discussed. Error graphs for the first-order IVPs are presented in a graphical form to show the efficiency of this RK–Butcher method. This RK–Butcher algorithm can be easily implemented in a digital computer and the solution can be obtained for any length of time.

General elastic wave scattering problems using an impedance operator approach - II. Two-dimensional isotropic validation and examples

SUMMARY Highly accurate solutions to general wave scattering problems can be calculated using an impedance operator to factorize the wave equation by direction of energy flow. The classical second-order partial differential wave equation for a heterogeneous region bounded in the vertical direction is recast as a set of two first-order initial value problems: a spectral expansion converts these to ordinary differential equations, which can then solved to a very high degree of precision. We demonstrate this technique for both P-SV and SH problems and validate our results by comparison with a finite-difference calculation. In its most straightforward form, when the wavefields inside and outside the heterogeneous region are matched using a common Fourier expansion, the method is computationally expensive; a reformulation in which the coupling is carried out via a boundary integral technique is much more efficient, however.

Boundedness and Uniqueness of Solutions to Dynamic Equations on Time Scales

In this work, we investigate the boundedness and uniqueness of solutions to systems of dynamic equations on time scales. We define suitable Lyapunov-type functions and then formulate appropriate inequalities on these functions that guarantee all solutions to first-order initial value problems are uniformly bounded and/or unique. Several examples are given. †cct@maths.unsw.edu.au

Factorization of second-order elliptic boundary value problems by dynamic programming

We present a method to factorize a second-order boundary value problem in a cylindrical domain in a system of uncoupled first-order initial value problems, together with a nonlinear Riccati-type equation for functional operators. This uncoupling is obtained by a space invariant embedding technique along the axis of the cylinder. This method can be viewed as an infinite-dimensional generalization of the block Gauss LU factorization.