Selected generalized integration techniques for analog computation

SIMULATION ◽  
1967 ◽  
Vol 9 (1) ◽  
pp. 21-28 ◽  
Author(s):  
Arthur Hausner
Keyword(s):  
Differential Equations ◽  
Rational Functions ◽  
Analog Computer ◽  
Great Accuracy ◽  
Analog Computation ◽  
Major Disadvantage ◽  
Integration Techniques ◽  
High Degree

Generalized integration is a technique for generating ex plicit functions on an analog computer by solving the appropriate differential equations they satisfy. Setting up the solution of differential equations using the parametric technique is first reviewed. Two theorems regarding the capability of linear equipment in generating sums and products are stated, and their usefulness is illustrated with examples. Applications of the technique to generating high-degree oscillatory polynomials and rational functions (which require nonlinear equipment) are also described. The major advantage of the technique is achievement of great accuracy with minimum equipment in some cases. The major disadvantage is that, with time, errors may sometimes increase and may not be bounded.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals Efficient approximations for stationary single-channel Ca2+ nanodomains across length scales

2020 ◽  
Author(s):  
Y Chen ◽  
C Muratov ◽  
V Matveev
Keyword(s):  
Parameter Space ◽  
Single Channel ◽  
Asymptotic Series ◽  
Rational Functions ◽  
Great Accuracy ◽  
Model Parameters ◽  
Relevant Parameter ◽  
Buffer Concentration ◽  
Computationally Efficient ◽  
Wide Range

ABSTRACTWe consider the stationary solution for the Ca2+ concentration near a point Ca2+ source describing a single-channel Ca2+ nanodomain, in the presence of a single mobile Ca2+ buffer with one-to-one Ca2+ binding. We present computationally efficient approximants that estimate stationary single-channel Ca2+ nanodomains with great accuracy in broad regions of parameter space. The presented approximants have a functional form that combines rational and exponential functions, which is similar to that of the well-known Excess Buffer Approximation and the linear approximation, but with parameters estimated using two novel (to our knowledge) methods. One of the methods involves interpolation between the short-range Taylor series of the buffer concentration and its long-range asymptotic series in inverse powers of distance from the channel. Although this method has already been used to find Padé (rational-function) approximants to single-channel Ca2+ and buffer concentration, extending this method to interpolants combining exponential and rational functions improves accuracy in a significant fraction of the relevant parameter space. A second method is based on the variational approach, and involves a global minimization of an appropriate functional with respect to parameters of the chosen approximations. Extensive parameter sensitivity analysis is presented, comparing these two methods with previously developed approximants. Apart from increased accuracy, the strength of these approximants is that they can be extended to more realistic buffers with multiple binding sites characterized by cooperative Ca2+ binding, such as calmodulin and calretinin.STATEMENT OF SIGNIFICANCEMathematical and computational modeling plays an important role in the study of local Ca2+ signals underlying vesicle exocysosis, muscle contraction and other fundamental physiological processes. Closed-form approximations describing steady-state distribution of Ca2+ in the vicinity of an open Ca2+ channel have proved particularly useful for the qualitative modeling of local Ca2+ signals. We present simple and efficient approximants for the Ca2+ concentration in the presence of a mobile Ca2+ buffer, which achieve great accuracy over a wide range of model parameters. Such approximations provide an efficient method for estimating Ca2+ and buffer concentrations without resorting to numerical simulations, and allow to study the qualitative dependence of nanodomain Ca2+ distribution on the buffer’s Ca2+ binding properties and its diffusivity.

New Integration Techniques for Chemical Kinetic Rate Equations: Part II—Accuracy Comparison

1986 ◽  
Vol 108 (2) ◽  
pp. 348-353 ◽  
Author(s):  
K. Radhakrishnan
Keyword(s):  
Differential Equations ◽  
Time Derivative ◽  
Iterative Solution ◽  
Chemical Kinetic ◽  
Conservation Equation ◽  
General Purpose ◽  
Rate Equations ◽  
Arbitrary System ◽  
Kinetic Rate ◽  
Integration Techniques

A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general-purpose codes EPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREK1D, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas-phase chemical reactions at constant pressure, and include all three combustion regimes: induction, heat release, and equilibration. The comparisons show that LSODE is the most efficient code—in the sense that it requires the least computational work to attain a specified accuracy level—currently available for chemical kinetic rate equations. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.

scholarly journals Simplified error bounds for turning point expansions

2020 ◽  
pp. 1-32
Author(s):  
T. M. Dunster ◽  
A. Gil ◽  
J. Segura
Keyword(s):  
Differential Equations ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Turning Point ◽  
Linear Differential Equations ◽  
Airy Functions ◽  
Elementary Functions ◽  
Varying Coefficient ◽  
Linear Differential ◽  
High Degree

Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coefficient functions, and were simpler than previous approximations, in particular being computable to a high degree of accuracy. Here we present explicit error bounds for these expansions which only involve elementary functions, and thereby provide a simplification of the bounds associated with the classical expansions of Olver.

Using Pontryagin's Maximum Principle to Solve One-Dimensional Optimization Problems, with and without Constraints, on an Iterative Analog Computer

SIMULATION ◽  
1965 ◽  
Vol 4 (6) ◽  
pp. 382-389 ◽  
Author(s):  
Hans L. Steinmetz
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Time Domain ◽  
Pontryagin’S Maximum Principle ◽  
Analog Computer ◽  
Solution Time ◽  
Pontryagin's Maximum Principle ◽  
Computer Technique ◽  
One Dimensional ◽  
Time Required

An analog computer technique is presented which enables application of Pontryagin's maximum prin ciple to the problem of optimizing control systems. The key problem in using Pontryagin's maximum principle is the extremization of the Hamiltonian function at every instant of time. Since the analog computer is an excellent differential equation solver, it is of advantage to convert this task into a dynamic problem. The technique used to do this is based upon the steepest ascent method. The method is applied to a one-dimensional control problem; higher-di mensional control problems can be treated using the same approach. The argument that an analog computer can solve differential equations with only one independent variable, corresponding to machine time, is true only in a technical sense. In practice it is feasible for cer tain types of problems to integrate one set of differ ential equations sufficiently fast enough so that, while integrating another set of differential equations at a much slower rate, the solution error associated with this approach remains within acceptable limits. When using the analog computer in this way, one time domain always corresponds to the solution time required for solving the differential equations de scribing the system; a second time domain corre sponds to the solution time required for solving an auxiliary set of differential equations which has no direct relationship with the system. Technological improvements and innovations made in the analog computer field during the recent past have contributed to the successful application of this approach.

Digital simulation of analog computation and Church's thesis

1989 ◽  
Vol 54 (3) ◽  
pp. 1011-1017 ◽  
Author(s):  
Lee A. Rubel
Keyword(s):  
Digital Simulation ◽  
Analog Computer ◽  
General Purpose ◽  
Algebraic Differential Equation ◽  
Analog Computation ◽  
Church’S Thesis ◽  
Black Boxes ◽  
Church's Thesis ◽  
Research Questions ◽  
The Given

Church's thesis, that all reasonable definitions of “computability” are equivalent, is not usually thought of in terms of computability by a continuous computer, of which the general-purpose analog computer (GPAC) is a prototype. Here we prove, under a hypothesis of determinism, that the analytic outputs of a C∞ GPAC are computable by a digital computer.In [POE, Theorems 5, 6, 7, and 8], Pour-El obtained some related results. (The proof there of Theorem 7 depends on her Theorem 2, for which the proof in [POE] is incorrect, but for which a correct proof is given in [LIR]. Also, the proof in [POE] of Theorem 8 depends on the unproved assertion that a solution of an algebraic differential equation must be analytic on an open subset of its domain. However, this assertion was later proved in [BRR].) As in [POE], we reduce the problem to a problem about solutions of certain systems of algebraic differential equations (ADE's). If such a system is nonsingular (i.e. if the “separant” does not vanish along the given solution), then the argument is very easy (see [VSD] for an even simpler situation), so that the essential difficulties arise from singular systems. Our main tools in handling these difficulties are drawn from the excellent (and difficult) paper [DEL] by Denef and Lipshitz. The author especially wants to thank Leonard Lipshitz for his kind help in the preparation of the present paper.We emphasize here that our proof of the simulation result applies only to the GPAC as described below. The GPAC's form a natural subclass of the class of all analog computers, and are based on certain idealized components (“black boxes”), mostly associated with the technology of past decades. One can easily envisage other kinds of black boxes of an input-output character that would lead to different kinds of analog computers. (For example, one could incorporate delays, or spatial integrators in addition to the present temporal integrators, etc.) Whether digital simulation is possible for these “extended” analog computers poses a rich and challenging set of research questions.

scholarly journals PRODUCT INTEGRATION FOR WEAKLY SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS

2011 ◽  
Vol 16 (1) ◽  
pp. 153-172 ◽  
Author(s):  
Arvet Pedas ◽  
Enn Tamme
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
Discrete Version ◽  
Product Integration ◽  
Weakly Singular ◽  
Integration Techniques ◽  
Singular Kernels ◽  
Convergence Estimates ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

On the basis of product integration techniques a discrete version of a piecewise polynomial collocation method for the numerical solution of initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels is constructed. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms.

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