Two-point Boundary Conditions Research Articles

An Adaptive Boundary Value Runge–Kutta Solver for First Order Boundary Value Problems

The use of finite difference methods which combine features of both Runge–Kutta processes and Gap schemes is proposed for developing adaptive codes for the solution of first order differential equations with two-point boundary conditions. The order conditions for the coefficients of these processes are given. A set of conditions, by which these order conditions can be reduced in number, are developed. An eighth order A-stable method which has second, fourth and sixth order A-stable methods embedded in it is given as an example. A variable order, variable step finite difference solver using embedded methods of this kind is described.

Trajectory optimization for maneuvering satellites

This paper proposes an algorithm to compute optimal trajectories for a maneuvering satellite by using a nonlinear programming representation of an optimal control problem. In this problem, a satellite must be located at a given final position with given velocity from initial position and velocity without passing a prohibited region such as the atmosphere while achieving minimum fuel consumption. Optimal control theory is applied to obtain a set of ordinary differential equations subject to two-point boundary conditions (TPB) on the adjoint system. Then an exact penalty function method is employed to obtain the optimal trajectories by solving the TPB problem as initial conditions for the adjoint system and an unknown final time are regarded as decision variables. This formulation, where the optimal control technique and the nonlinear programming method are incorporated, permits more systematic and flexible algorithm implementation.

Iterative methods for a fourth order boundary value problem

We consider a fourth order nonlinear ordinary differential equation together with two-point boundary conditions and provide a-priori error estimates on the length of the interval ( b− a) so that the Picard's iterative method, the approximate Picard's iterative method and the quasilinear iterative method convergence to the solution of the problem.

Systems of Singular Perturbation Problems with a First Order Turning Point

This paper deals with systems of singularly perturbed ordinary differential equations with a first order turning point. Two-point boundary conditions are attached. We treat interior turning points and boundary turning points and give in each case an estimate for the norm of the inverse of the differential operator. In both cases these norms tend to infinity, as the perturbation parameter a tends to zero. In the case of a boundary turning point, the norm of the inverse blows up algebraically, as $\varepsilon ^{{{ - 1} /2}} $, and in the interior turning point case, the blow-up is exponential, as $\varepsilon ^{{{ - 1} /2}} \exp ({\omega /{2\varepsilon }})$ with $\omega > 0$. For linear and quasilinear problems we prove existence (and uniqueness) results and investigate the asymptotic behavior of the solutions as $\varepsilon \to 0$. In the boundary turning point case, we show that solutions are uniformly bounded in compact subsets of the open interval and converge there uniformly (as $\varepsilon > 0$), to the solution of the reduced equation ($\varepsilon = 0$). At both boundary points, layers of height $O(\varepsilon ^{{{ - 1} /2}} )$ occur generally. In the interior turning point case the solutions generally blow up exponentially (at least left or right from the turning point).

Convergence of relaxation methods for two-point boundary-value linear problems

The solution of linear differential problems, with explicit two-point boundary conditions, can sometimes be obtained by a relaxation method of computation. This paper shows that the convergence of iterations is linked to the spectral radius value of an integral operator. The equivalence between eigenvalues research and critical lengths calculation of a differential system is demonstrated. In this context, we present a case of optimal control law calculation of a linear system. The efficiency of this preliminary convergence calculation is illustrated by a numerical example.

Contribution to error estimate

The error estimate of an approximate solution to a nonlinear ordinary differential equations of the second order is obtained. The differential equation is subject to either two-point boundary conditions or initial conditions. The independent variable interval may be finite or infinite. The theory is applied to five problems.

Linear systems with two-point boundary Lyapunov and Riccati equations

This paper extends some well-known system theories for algebraic Lyapunov and Riccati equations. These extended results deal with the existence and uniqueness properties of the solutions to matrix differential equations with two-point boundary conditions and are shown to include conventional results as special cases. Necessary and sufficient conditions are derived under which linear systems are stabilizable with periodic feedback gains derived from the two-point boundary matrix differential equations. An easy iterative method for solving the two-point boundary differential Riccati equation is given with an initial guess which is obtained from the intervalwise receding horizon control. The results in this paper are related to periodic feedback gain controls and also to the quadratic cost problem with a discrete state penalty.

NUMERICAL SOLUTION FOR FREE CONVECTION BETWEEN CONCENTRIC SPHERES AT MODERATE GRASHOF NUMBERS

Steady axisymmetric natural convection between two isothermal concentric spheres is Investigated by the method of series truncation. The flow variables and temperature distribution are expressed as series of orthogonal Gegenbauer functions and Legendre polynomials, respectively, with variable coefficients. Thus, the governing equations are reduced to three coupled sets of nonlinear ordinary differential equations of the second order with two-point boundary conditions. Numerical solutions are obtained by truncating the series at a certain stage and by using step-by-step methods. Detailed calculations are carried out for Grashof numbers in the range 1000-20,000 and for Prandtl numbers equal to 0.02, 0.7, and 6. Flow patterns, velocity components, and heat transfer results obtained are compared with published results.

Heat and mass transfer in a saturated porous wedge with impermeable boundaries

Heat and mass transfer in a two-dimensional radial flow of a viscous fluid through a saturated porous wedge-shaped region with confining walls is studied. Similarity transformations are used for the temperature, velocity and pressure, in order to reduce the describing systems to ordinary differential equations with two-point boundary conditions. Both exact and asymptotic solutions are obtained for the velocity and the temperature. It is found that two distinct solutions (jet flow type and slug flow type) exist for a given set of flow parameters. Specific results are presented for small wedge angles. It is shown that symmetric diverging solutions do not exist above a critical Rayleigh number. An application of the theory to the convection of liquid water in a crude model of a fault zone in a geothermally active area is presented.

The Asymptotic Derivation of Isola and S Responses for Chambered Diffusion Flames

We consider the asymptotic solution of the nonlinear ordinary differential equation with two-point boundary conditions which describes a steady diffusion flame in a chamber. The solution is characterized by the unburnt fuel fraction R as a function of the fuel injection rate M. The shape of the corresponding response curve depends on the Damköhler number D (representing either the constant chamber pressure or the chamber’s length) for which two critical values, $D_0 $ and $D_a $, are uncovered. For $D < D_a $ the only possibility is $R \equiv 1$ (extinguished states). When $D_a < D < D_0 $, a closed curve called an isola is also a possible response, its base lying on $R = 0$ (complete burning). As D approaches $D_a $ the isola shrinks to a point and disappears. But when D is increased towards $D_0 $ it expands until its left side swells into the end of the strip $M > 0,0 < R < 1$. Then it breaks and joins the extinguished response (which is now limited on the left) to form an S, which is the nature of the response when $D > D_0 $. The transitions between the various responses require special asymptotics which are also described herein.

Some Aspects of Oscillation and Boundary Problem Theory for Hamiltonian Systems in a $B^ * $-Algebra

For linear self-adjoins Hamiltonian differential systems in a $B^ * $-algebra the topics treated include an extension of the well-known generalized polar coordinate transformation for the finite dimensional matrix case, and the derivation of two oscillation criteria for systems that may be written as self-adjoins linear differential equations of the second order. The determination of a Green’s matrix for an incompatible boundary problem involving two-point boundary conditions is discussed, and, in particular, the established results are applied to reduce a certain type of vector boundary problem in a Hilbert space to known results for symmetrizable compact linear transformations in an associated Hilbert space.

A divided difference method for solving two-point boundary value problems

A concept of divided difference operators (DDO) is presented and combined with Newton-like methods for solving non-linear operator equations in high-dimensional spaces. The general idea of the concept lies in the approximation of local variations of non-linear operators with the aid of suitably fitted linear DDO's. The method is closely related to the conventional secant, regula falsi, and Steffensen's methods. The operator equations concerned are composed of ordinary first-order differential equations together with two-point boundary conditions. The method is tested in a numerical example originating in an optimal control problem for the classical Van der Pol system.

Intermediate equations of the theory of diffusion processes with two-point boundary conditions

Intermediate equations of the theory of diffusion processes with two-point boundary conditions

Calculation of the flow between two rotating spheres by the method of series truncation

A numerical method is described for solving three coupled sets of nonlinear ordinary differential equations of the second order which arise in the study of the steady axially symmetric motion of an incompressible viscous fluid contained between two concentric rotating spheres. The flow variables are expressed as series of orthogonal Gegenbauer functions with variable coefficients, thus reducing the equations of motion to ordinary differential equations with two-point boundary conditions. The boundary conditions for the stream function are utilized to obtain an integral condition which permits one of the sets of equations to be solved using step-by-step methods. Numerical solutions are obtained for values up to 2000 of the Reynolds number based on the radius of the outer sphere. Results for the stream function and the torque required to rotate the spheres are compared with those obtained by previous investigators.

A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for the general second-order non-linear differential equation yN = f(x, y, y′) subject to mixed two-point boundary conditions. An interesting feature of our method is that each discretization of the differential equation at an interior grid point is based on just three evaluations of f. We establish, under appropriate conditions, O(h4)-convergence of the finite difference scheme. In the case of linear differential equations, our finite difference scheme leads to tridiagonal linear systems. Numerical examples are considered to demonstrate computationally the fourth order of the method.

Similarity solutions for convection of groundwater adjacent to horizontal impermeable surfaces with axisymmetric temperature distribution

The axisymmetric buoyancy‐induced groundwater flow adjacent to horizontal impermeable surfaces, wall temperature being a power function of radius, is discussed in this paper. With the boundary layer simplifications the governing nonlinear partial differential equations can be transformed into a coupled pair of nonlinear ordinary differential equations with two‐point boundary conditions that can be integrated numerically by established techniques. Simple algebraic expressions for boundary layer thickness and heat transfer rate are obtained. Applications to free convective flow in a liquid‐dominated geothermal system at high Rayleigh numbers are discussed.

An iterative method for determining the current-potential curves of a spherical probe in a slightly ionized continuum plasma with [formula omitted

A desk analogue computer has been used to draw normalized curves for positive and negative particle concentrations (n +,n −) and potential (y) vs the distance from a spherical probe surface for a slightly ionized continuum plasma. The governing equations are three simultaneous differential equations leading to a two-point boundary condition problem. An iterative analogue method has been developed which enables the positive and negative particle currents (J +,J −) to the probe, for particular y p < 7 and ρ p < 50 , to be found to a numerical accuracy of about 1 digit in the second decimal place.

A solution to the distributed parameter model of a continuous grinding mill at steady state

A distributed parameter model of an open-circuit size reduction device at steady state is considered. The model is applicable to tumbling or vibratory mills. Material transport is described in terms of a size-dependent axial convective velocity and a size-dependent axial dispersion coefficient. For n size fractions, the model consists of a set of n coupled ordinary differential equations of second order with two-point boundary conditions. An explicit solution to the model equations is derived for the case in which these equations are linear with constant coefficients. This corresponds to the circumstance in which the axial convective velocities, axial dispersion coefficients and the quantities used to describe the size reduction process are independent of the axial position or of the dependence of the material holdup or size distribution on axial position in the device. These assumptions are not too restrictive and the solution appears to be a useful approximation for a broad range of material-mill combinations. Applications to other process unit operations are suggested.

Bifurcation in Difference Approximations to Two-Point Boundary Value Problems

Numerical methods for bifurcation problems of the form \begin{equation}\tag {$\ast $} Ly = \lambda f(y),\quad By = 0,\end{equation} where $f(0) = 0$ and $f’(0) \ne 0$, are considered. Here y is a scalar function, $\lambda$ is a real scalar, L is a linear differential operator and $By = 0$ represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if $(\ast )$ is replaced by an appropriate difference scheme, then there exists a unique branch of nontrivial solutions of the discrete problem in a neighborhood of a branch of nontrivial solutions of $(\ast )$ bifurcating from the trivial solution and that the discrete branch converges to the continuous one. Error estimates are derived and an illustrative numerical example is included.

Bifurcation in difference approximations to two-point boundary value problems

Numerical methods for bifurcation problems of the form ( ∗ ) L y = λ f ( y ) , B y = 0 , \begin{equation}\tag {$\ast $} Ly = \lambda f(y),\quad By = 0,\end{equation} where f ( 0 ) = 0 f(0) = 0 and f ′ ( 0 ) ≠ 0 f’(0) \ne 0 , are considered. Here y is a scalar function, λ \lambda is a real scalar, L is a linear differential operator and B y = 0 By = 0 represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if ( ∗ ) (\ast ) is replaced by an appropriate difference scheme, then there exists a unique branch of nontrivial solutions of the discrete problem in a neighborhood of a branch of nontrivial solutions of ( ∗ ) (\ast ) bifurcating from the trivial solution and that the discrete branch converges to the continuous one. Error estimates are derived and an illustrative numerical example is included.