Invariant imbedding and multiple shooting for the solution of unstable linear boundary value problems

Invariant imbedding has been used to solve unstable linear boundary value problems for a few years. First this method is derived using the theory of characteristics; there the boundary value problem has to be imbedded in a problem of double dimension. If the corresponding Riccati equation has a critical length, one has to repeat the algorithm. A relation between this repeated invariant imbedding and multiple shooting is shown. In examples invariant imbedding, repeated invariant imbedding, multiple shooting and the superposition principle are compared.     

The reconstruction of groundwater parameters from head data in an unconfined aquifer

The estimation of groundwater flow parameters from head measurements and other ancillary data is fundamental to the process of modelling a groundwater system. In an unconfined aquifer, the problem is more complex because the governing equation for the well heads, the Boussinesq equation, is non-linear. We consider here a new method that allows for the simultaneous computation of the unconfined groundwater parameters as the unique minimum of a convex functional.     

Invariant imbedding approach for the solution of the minimal surface equation

A new combined technique based on the application of a linearization procedure either (i), the combination of Outer- and Picard-approximation or (ii) the combination of Newton- and Picard-approximation, and invariant imbedding is proposed for obtaining a numerical solution of the minimal surface equation. The existence of inverses of certain matrices appearing in the invariant imbedding equations and the stability of the algorithm are investigated. The minimal surface equation under various boundary conditions and the subsonic fluid flow problem are chosen as test examples for illustrating the method. The numerical results indicate that the proposed method can be used efficiently for solving elliptic problems of a highly nonlinear nature.     

Some relationships between Green's functions and invariant imbedding

The method of invariant imbedding has been used to resolve the solution of linear two-point boundary-value problems into contributions associated with the homogeneous equation with homogeneous boundary conditions, with inhomogeneous boundary conditions, and with an inhomogeneous source term in the equation. The relationship between the Green's function and the invariant imbedding equations is described, and it is shown that the Green's function can be determined from an initial-value problem. Several numerical examples are given which illustrate the efficacy of the initial-value algorithm.This work was supported by the US Atomic Energy Commission.     

An invariant imbedding method for computation of eigenlengths of singular two-point boundary-value problems

In this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular TPBVP with the singularity at the origin. The invariant imbedding yields a first-order nonlinear equation called a Riccati equation and also gives the initial conditions at the origin for this equation. With the aid of Theorem 8 in Section 3 we numerically integrate the Riccati equation to “blowup” which gives our computed eigenlength.In closing, we would like to comment on the numerical merits of the integration-to-blowup technique. On the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results. The feature of a variable step size is essential as the value of z approaches the actual eigenlength. However, it is desirable to have a priori estimate or bounds of the eigenlength similar to those of Boland and Nelson [2] for the nonsingular case. The singular system, however, presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds. Hopefully an investigation of the matrix R(z) will yield these results.     

Groundwater response to tidal fluctuation in a sloping leaky aquifer system

The effect of tidal fluctuation on groundwater flow is an important issue from many aspects in coastal areas. This paper develops a new analytical solution to describe the groundwater fluctuation in a sloping coastal aquifer system which comprises an upper unconfined aquifer, a lower confined aquifer, and an aquitard in between. The solution is allowed to investigate the effects of bottom slope and leakage as well as aquifer parameters on head fluctuations in both unconfined and confined aquifers. The research result indicates that the effect of the bottom angle on the groundwater fluctuation and time lag is significant in the unconfined aquifer and not negligible if the leakage in the confined aquifer is large. In addition, the joint effects of aquifer parameters and bottom angle on groundwater fluctuation and time lag are also discussed.     

The method of lines for Poisson's equation with nonlinear or free boundary conditions

Summary The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.This work was supported by the U.S. Army Research Office under Grant DA-AG29-76-G-0261     

The conversion of Fredholm integral equations to equivalent cauchy problems

A new method is developed for converting various classes of Fredholm integral equations into equivalent initial value problems. In contrast with previous methods, which accomplished this by imbedding the equation, with respect to some parameter, in a family of similar ones, our approach is parameter free. To effect the conversion the integral equation is first shown to be equivalent to a two point boundary value problem. The application of various invariant imbedding algorithms completes the task. An extensive examination of linear equations is made, and it is shown that our procedure leads to a substantial reduction of dimensionality over previous methods. New techniques for solving critical length and continuation problems are another important consequence of our approach.     

The conversion of Fredholm integral equations to equivalent Cauchy problems— II. computation of resolvents

A new method is given for computing the resolvent of a large class of Fredholm integral equations. The technique is based on converting the integral equation satisfied by the resolvent to a family of two point boundary value problems. The application of invariant imbedding then gives an equivalent Cauchy problem satisfied by the resolvent kernel. The procedure is compared to previous ones based on the Bellman—Krein equation. It is shown that our method requires fewer equations to integrate if the number of output points on each axis exeeds the bank of the kernel.     

Invariant imbedding for block tridiagonal systems

The invariant imbedding technique is applied to the block tridiagonal systems in a general setting using, when necessary, the generalized Moore-Penrose inverse of a matrix. Three examples are given, and the existence and the stability of solutions are discussed.     

Solution of some simple problems in the conduction and radiation of heat by invariant imbedding

In recent years the use of invariant imbedding in the solution of a variety of problems has been increasing. In this paper, application of this method to problems in heat conduction and radiation is demonstrated. No previous knowledge of invariant imbedding is assumed. The method is applied to several relatively simple problems. The initial value problem obtained by the method is numerically stable. Sample calculations are presented which demonstrate the accuracy of the algorithm.     

An optimistic decision-making in fuzzy environment

Bellman and Zadeh have originated three systems of multistage decision processes in a fuzzy environment: deterministic, stochastic and fuzzy systems. In this article, we consider an optimization problem with an optimistic criterion on a fuzzy system. By making use of minimization–maximization expectation in a fuzzy environment, we derive a recursive equation for the fuzzy decision process through invariant imbedding approach. By illustrating a three-state, two-decision and two-stage model, we give an optimal solution through dynamic programming. The optimal solution is also verified by the method of multistage fuzzy decision tree-table.     

Wave splitting of the Timoshenko beam equation in the time domain

In recent years, wave splitting in conjunction with invariant imbedding and Green's function techniques has been applied with great success to a number of interesting inverse and direct scattering problems. The aim of the present paper is to derive a wave splitting for the Timoshenko equation, a fourth order PDE of importance in beam theory. An analysis of the hyperbolicity of the Timoshenko equation and its, in a sense, less physical relatives-the Euler-Bernoulli and the Rayleigh equations-is also provided.     

Three-dimensional non-Darcy flow analysis using a hybrid numerical model

A hybrid numerical model is developed for the simulation of three-dimensional, unsteady non-Darcy flow through an unconfined aquifer. The major problem in analysing flow through unconfined aquifers is that they involve two boundaries, namely a surface of seepage and a free surface, the location of which is not known beforehand. The model that is presented here determines these boundaries via a two stage modelling technique. In the first stage a one-dimensional finite difference model is used to estimate the surface of seepage height whereas in the second stage a vertically integrated finite element model determines the free surface solution within the flow domain. A comparison between numerical and experimental results is included which indicates the sensitivity of the numerical solution to the selected aquifer parameters, particularly to those associated with the determination of the height of the surface of seepage.     

Application of invariant imbedding to linear multipoint boundary value problems with general boundary conditions

Two extensions of the usual application of invariant imbedding to the solution of linear boundary value problems are presented. The invariant imbedding formulation of a linear two point boundary value problem in which functional relationships are given between the variables at either one or both of the boundary points is presented. Also, extension of invariant imbedding to linear multipoint boundary value problems is given. Using these extensions singly or in combination, a general multipoint boundary value of linear ordinary differential equations can be solved. In addition, the problems of infinite initial conditions and / or indeterminate initial derivatives are resolved. Numerical examples demonstrate the feasibility and accuracy of the method.     

Numerical integration of a class of singular perturbation problems

In this paper, we discuss an approximate method for the numerical integration of a class of linear, singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This method requires a minimum of problem preparation and can be implemented easily on a computer. We replace the original singular perturbation problem by an approximate first-order differential equation with a small deviating argument. Then, we use the trapezoidal formula to obtain the three-term recurrence relationship. Discrete invariant imbedding algorithm is used to solve a tridiagonal algebraic system. The stability of this algorithm is investigated. The proposed method is iterative on the deviating argument. Several numerical experiments have been included to demonstrate the efficiency of the method.The authors wish to express their sincere thanks to Dr. S. M. Roberts for his comments and valuable suggestions.     

Coefficient identification in Euler-Bernoulli equation from over-posed data

A method for solving the inverse problem for coefficient identification in the Euler-Bernoulli equation from over-posed data is presented. The original inverse problem is replaced by a minimization problem. The method is applied to the problem for identifying the coefficient in the case when it is a piece-wise polynomial function. Several examples are elaborated and the numerical results confirm that the solution of the imbedding problem coincides with the direct simulation of the original problem within the second order of approximation.     

On Finite Markov Chain Imbedding and Its Applications

The finite Markov Chain Imbedding technique has been successfully applied in various fields for finding the exact or approximate distributions of runs and patterns under independent and identically distributed or Markov dependent trials. In this paper, we derive a new recursive equation for distribution of scan statistic using the finite Markov chain imbedding technique. We also address the problem of obtaining transition probabilities of the imbedded Markov chain by introducing a notion termed Double Finite Markov Chain Imbedding where transition probabilities are obtained by using the finite Markov chain imbedding technique again. Applications for random permutation model in chemistry and coupon collector’s problem are given to illustrate our idea.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.