Taylor Series Solution Research Articles

F and G Taylor series solutions to the Stark and Kepler problems with Sundman transformations

The classic $$F$$ and $$G$$ Taylor series of Keplerian motion are extended to solve the Stark problem and to use the generalized Sundman transformation. Exact recursion formulas for the series coefficients are derived, and the method is implemented to high order via a symbolic manipulator. The results lead to fast and accurate propagation models with efficient discretizations. The new $$F$$ and $$G$$ Stark series solutions are compared to the Modern Taylor Series (MTS) and 8th order Runge–Kutta–Fehlberg (RKF8) solutions. In terms of runtime, the $$F$$ and $$G$$ approach is shown to compare favorably to the MTS method up to order 20, and both Taylor series methods enjoy approximate order of magnitude speedups compared to RKF8 implementations. Actual runtime is shown to vary with eccentricity, perturbation size, prescribed accuracy, and the Sundman power law. The method and results are valid for both the Stark and the Kepler problems. The effects of the generalized Sundman transformation on the accuracy of the propagation are analyzed. The Taylor series solutions are shown to be exceptionally efficient when the unity power law from the classic Sundman transformation is applied. An example low-thrust trajectory propagation demonstrates the utility of the $$F$$ and $$G$$ Stark series solutions.

Taylor Series Solution of the Electrohydrodynamic Flow Equation

In this work, the electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is considered. The differential equation of the problem is a highly nonlinear and singular boundary value problem (BVP). An analytical solution of the problem is presented using the differential transform method (DTM). In this method, the differential equation and related boundary conditions transformed into a recurrence set of equations and at the end, the coefficients of Taylor series are calculated based on the solution of this set of equations. DTM results are compared with the numerical solution of the problem and an excellent accuracy is observed. It showed that the proposed method overcame the nonlinearity and singularity of the problem without any need to restrict assumptions or linearization. Most of the scientific problems and engineering phenomena are usually defined based on the nonlinear differential equations of both initial value problems (IVP) and boundary value problems (BVP). It is important to remember that except a limited number of these problems, finding exact analytical solution is difficult. Therefore, researchers use semi-analytical techniques and numerical methods. The semi-analytical methods such as the perturbation method (PM), Adomian decomposition method (ADM), variational iteration method (VIM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and differential transform method (DTM) are useful mathematical ones, which present the analytical solutions for nonlinear differential equations. In this paper, solution of electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is presented using DTM. Already, the electrohydrodynamic flow of a fluid and its governing equations is considered (1). The electrohydrodynamic flow is important in analysis of the flow meters, accelerators, pumps and magnetohydrodynamic generators. The differential equation of the problem is the nonlinear singular boundary value problem. 2 2

Multistep Differential Transform Method for Numerical Solution of Classical Neutron Point Kinetic Equation

In this paper, the differential transform method is applied to compute the numerical solution for classical neutron point kinetic equations [5, 6, 7] in a nuclear reactor. The differential transform method (DTM) is an iterative procedure for obtaining analytic Taylor series solutions of differential equations. The new algorithm multistep differential transform method (MDTM) is applied here for solving the classical neutron point kinetic equation. The multistep DTM is treated as an algorithm in a sequence of intervals for finding simple and accurate solutions. Moreover, numerical examples with variable step reactivities, ramp reactivity, and sinusoidal reactivity are used to illustrate the preciseness and effectiveness of the proposed method.

Computing of B-series by automatic differentiation

We present an algorithm based on Automatic Differentiation for computing general B-series of vector fields $f\colon \mathbb{R}^n\rightarrow \mathbb{R}^n$. The algorithm has a computational complexity depending linearly on $n$, and provides a practical way of computing B-series up to a moderately high order $d$. Compared to Automatic Differentiation for computing Taylor series solutions of differential equations, the proposed algorithm is more general, since it can compute any B-series. However the computational cost of the proposed algorithm grows much faster in $d$ than a Taylor series method, thus very high order B-series are not tractable by this approach.

On series solutions of y ″ − f ( x ) y = 0 and applications

In this paper, we develop explicit expressions for the Taylor series coefficients in the formal Taylor series solution of the second-order linear differential equation for a given arbitrary function in terms of initial conditions. As applications, we apply our results to , and Airy’s equation and give explicit formulas for the Taylor coefficients of , which is a long-standing question.

The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method

The generalized Taylor theorem is the foundation of the homotopy analysis method proposed by Liao. This theorem is interesting but hard to understand from the mathematical point of view. Especially, there is a key parameter h whose meaning is still unknown. In the paper, we derive the generalized Taylor theorem from a usual way, that is, we prove that the generalized Taylor expansion is equivalent to a different representation of the usual Taylor expansion at different points. Therefore the meaning of the auxiliary parameter h is clarified. These results give a reasonable explanation of the parameter h and uncover the essence of the generalized Taylor theorem from which we can deeply understand the homotopy analysis method. Through the detailed analysis of some examples, we compare the series solution at the different points with the generalized Taylor series solution obtained by the homotopy analysis method.

A series solution of the non-linear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient

An extension of a series solution of the non-linear fin problem with temperature-dependent thermal conductivity [Kim and Huang 2006 J. Phys. D: Appl. Phys. 39 4894–901] is done to include a temperature-dependent heat transfer coefficient as well. It is assumed that the heat transfer coefficient is expressed in a power-law form and the thermal conductivity is a linear function of temperature. The algorithm is based on the Taylor series solution. The proposed approximate solution is successfully compared with the solutions for the Adomian decomposition method (ADM) and numerical algorithm. Also, it has been shown that the Adomian decomposition solution is just an approximation of the series solution given in this work, and therefore the present algorithm is better than ADM.

A series solution of the fin problem with a temperature-dependent thermal conductivity

In this work, a recursive algorithm to solve fin problems with temperature-dependent thermal conductivity is given. The algorithm is based on the Taylor series solution. The results are compared with the solutions obtained by the Adomian decomposition method (ADM) and with the numerical solutions as well. The results of the comparisons have shown that the ADM is just an approximation of the present study and the accuracy of the solution is also worse than the present algorithm.

Solution of the Falkner–Skan equation by recursive evaluation of Taylor coefficients

We present a computational method for the solution of the third-order boundary value problem characterized by the well-known Falkner–Skan equation on a semi-infinite domain. Numerical treatments of this problem reported in the literature thus far are based on shooting and finite differences. While maintaining the simplicity of the shooting approach, the method presented in this paper uses a technique known as automatic differentiation, which is neither numerical nor symbolic. Using automatic differentiation, a Taylor series solution is constructed for the initial value problems by calculating the Taylor coefficients recursively. The effectiveness of the method is illustrated by applying it successfully to various instances of the Falkner–Skan equation.

A Taylor-series solution in Cartesian space to GMA-system equations and its application to initial-value problems

A variable-order, variable-step Taylor-series method in Cartesian space is discussed which makes it possible to solve simultaneous first-order differential equations expressed in GMA-system canonical form with a super high-order accuracy that is comparable to the machine accuracy of the computer. In this method, the stepsize for each differential equation is calculated from the formula derived by setting the ratio of the appropriate two terms of the first few Taylor-series terms at unity, and a minimum value of them is then used as the initial value of the stepsize at each step of the integration and is halved if several criterions are satisfied. Also, when each dependent variable takes a value near zero, the integration is carried out with a very small stepsize. The Taylor-series method constructed here is first applied to the Lotka–Volterra equation to confirm the validity of the calculation algorithm and it is found that the calculated values completely agree with those by the conventional method in which the Taylor-series solution in logarithmic space is used. Second, the differential equation with a solution of cos( t) is solved and the relative error is found to instantaneously increase when the dependent variables take a value near zero and becomes equal to zero or almost zero in the other region. Finally, the differential equation for a two-body problem is solved and it is shown that the accuracies of the calculated values are kept at high levels even after integration over 625 cycles. In conclusion, the Taylor-series method constructed in Cartesian space is considered to have high applicability to simultaneous differential equations and provides high accuracies for the calculated values.

A Picard‐Maclaurin theorem for initial value PDEs

In 1988, Parker and Sochacki announced a theorem which proved that the Picard iteration, properly modified, generates the Taylor series solution to any ordinary differential equation (ODE) on ℜn with a polynomial generator. In this paper, we present an analogous theorem for partial differential equations (PDEs) with polynomial generators and analytic initial conditions. Since the domain of a solution of a PDE is a subset of ℜn, we identify one component of the domain to achieve the analogy with ODEs. The generator for the PDE must be a polynomial and autonomous with respect to this component, and no partial derivative with respect to this component can appear in the domain of the generator. The initial conditions must be given in the designated component at zero and must be analytic in the nondesignated components. The power series solution of such a PDE, whose existence is guaranteed by the Cauchy theorem, can be generated to arbitrary degree by Picard iteration. As in the ODE case these conditions can be met, for a broad class of PDEs, through polynomial projections.

A highly accurate numerical method for calculating apparent kinetic parameters of immobilized enzyme reactions: 1. Theory

To establish a highly reliable numerical method for calculation of the values of apparent kinetic parameters in various types of immobilized enzyme reactions, the performance of the shooting method combined with a variable-order, variable-step Taylor-series method has been discussed in a series of two papers. In this first paper, every formulae that are necessary to execute the numerical calculation are given and the procedure and technique for the calculation are described. As an example, the effectiveness factor is calculated for the Thiele modulus from 1 to 700 and the calculated values are found to have accuracies that are almost equal to the machine accuracy of the personal computer used. In the Taylor-series method, the step sizes are efficiently estimated from the ratio of the appropriate two of the first few terms in each Taylor-series solution to the relevant simultaneous differential equations, which contributes greatly to reduction in both the loss-of-significance error and the execution time.

An algorithm to solve the discrete HJI equation arising in the L2 gain optimization problem

A synthesis of the discrete non-linear H control law boils down to the solution of a set of algebraic and partial differential equations known as the discrete Hamilton-Jacobi-Isaacs (DHJI) equation, an extension of the discrete algebraic Ricatti equation arising in the linear discrete H control problem. Due to its non-linear nature, it is rarely possible to obtain the closed form solution for DHJI equation. This paper proposes an approximation approach to solving the DHJI equation in terms of the Taylor series. It is shown that the coefficients of the Taylor series solution for DHJI equation are governed by one discrete algebraic Ricatti equation and a sequence of linear algebraic equations, respectively. This result lends itself to a systematic algorithm to approximately synthesize a discrete non-linear H control law.

A precise time-step integration method by step-response and impulsive-response matrices for dynamic problems

In this paper, a precise time-step integration method for dynamic problems is presented. The second-order differential equations for dynamic problems are manipulated directly. A general damping matrix is considered. The transient responses are expressed in terms of the steady-state responses, the given initial conditions and the step-response and impulsive-response matrices. The steady-state responses for various types of excitations are readily obtainable. The computation of the step-response and impulsive-response matrices and their time derivatives are studied in this paper. A direct computation of these matrices using the Taylor series solutions is not efficient when the time-step size Δt is not small. In this paper, the recurrence formulae relating the response matrices at t=Δt to those at t=Δt/2 are constructed. A recursive procedure is proposed to evaluate these matrices at t=Δt from the matrices at t=Δt/2m. The matrices at t=Δt/2m are obtained from the Taylor series solutions. To improve the computational efficiency, the relations between the response matrices and their time derivatives are investigated. In addition, these matrices are expressed in terms of two symmetric matrices that can also be evaluated recursively. Besides, from the physical point of view, these matrices should be banded for small Δt. Both the stability and accuracy characteristics of the present algorithm are studied. Three numerical examples are used to illustrate the highly precise and stable algorithm. © 1997 John Wiley & Sons, Ltd.

Time Optimal Attitude Control of Asymmetric Rigid Spacecraft

Time optimal rest-to-rest reorientation of a rigid spacecraft with an arbitrary initial attitude and distinct principal moments of inertia is discussed. The effect of the gyroscopic coupling terms in Euler's equations on the control switch times is shown. A method for recursively generating coefficients for a truncated Taylor series solution of Euler's equations and the differential equations for the Euler parameter state transition matrix is shown. These solutions are used to solve for the optimal control switch times.

Explicit calculation of multicomponent equilibria for ideal adsorbed solutions

Precise multicomponent isotherms are usually required to properly analyze the dynamic behavior of adsorption separation processes. Among the most accurate and versatile isotherms for this purpose are those resulting from ideal adsorbed solution (IAS) theory. In this study, two formulations for IAS theory are developed which yield explicit relations for the adsorbed concentrations for arbitrary numbers of components. The first method involved fitting the relation between the spreading pressure and the gas-phase pressure for each individual component to a three-parameter Pade approximation which is the ratio of first- and second-order polynomials. This method can be applied to a variety of single-component isotherms, including those which deviate significantly from Langmuir isotherms. The second method is an extension to any number of adsorbates of the Taylor series solution from two adsorbates developed by LeVan and Vermeulen (1981). The second method applies when the single-component isotherms correspond closely to Langmuir isotherms and when the maximum adsorption capacities for the various adsorbates are not too dissimilar. Although this study considers only the use of IAS theory on a homogeneous surface, the methods discussed here can also be employed in certain types of heterogeneous IAS theories, such as those in which the adsorbent surface ismore » postulated to consist of several types of uniform surface patches, and IAS theory is used to determine the amounts adsorbed on each type of surface patch.« less

Convolution number

Convolution number is a new proposed name for the sequence of numbers that constitute the coefficients of polynomials and truncated Taylor expansions of functions. The arithmetic of the convolution number is the well-known arithmetic of sequences, where multiplication is the convolution of sequences, or the arithmetic of polynomials or formal power series where the multiplication is the Cauchy product. To separate the coefficients from the Taylor series formally, a Taylor transform is defined. Considering the sequence of coefficients as a single number is a new perspective which emphasizes the purely computational application, making a (digital) computer method out of a symbolic (computer) method. By this means, the whole well-known field of solution by Taylor series is cast in a simple numerical application oriented algebraic method. Convolution number analysis is an alternative computational tool to obtain numerical solutions for certain problems that would otherwise be determined by analytical analysis, or Finite Difference or Finite Element methods when analytical solution is too difficult. Taylor expansion of functions of a single variable, i.e., univariate polynomials, real or complex, are considered to develop the theory and the methods. All the methods and results are adapted to functions of two variables, i.e., bivariate polynomials, from which the extension to multivariate polynomials is then obvious. Simple programming of the four basic arithmetic operations on the convolution number is reviewed (and the square root operation), to be used as a set of subroutines, so that problems are formulated and programmed directly in terms of convolution numbers. It is shown how matrix algebra using convolution numbers as elements can be applied to vibration problems and illustrated with an example, resulting in a dynamic system matrix, although limited to small degree of freedom systems because of the large storage required. From the arithmetic, an algebra of convolution numbers is developed, considering the convolution number as a variable. An example of conformal mapping by convolution number algebra is given. A convolution function of a convolution variable is defined, with a compatibility condition analog to the Cauchy-Riemann equations of a complex function of a complex variable. The compatibility equations serve as a tool to derive some basic theorems of convolution variables which are necessary for the development of programming with convolution variables. Generally, the convolution number represents a truncated Taylor series of a function. The arithmetic is such that each coefficient has original machine accuracy and, therefore, the corresponding function could be evaluated to machine accuracy, although no theoretical general rule for the a priori required length of the convolution number and the radius of convergence is given. The well-known problems of polynomial composition and reversion of series are stated in terms of convolution number algebra. It is shown, by an example, how such problems are solved on a digital computer once the basic arithmetic routines are programmed. The well-known method to solve nonlinear ordinary differential equations by Taylor series is generalized to a simple computational solution of the integration process with the aid of a pointer in the computer storage of the convolution number. The solution process is posed in terms of a flow diagram, which is an exact copy of the analog computer diagram of a differential equation, from which the sequence of convolution number equations are programmed. Relation to the z-transform and digital filters is shown. The same example of conformal mapping is solved as a nonlinear equation, and the solution of a nonlinear ordinary differential equation by convolution number analysis is presented. In view of the applications, the radius of convergence of the function in the complex plane must be considered at all times, for which, unfortunately, no general theoretical determination can be given, and which may severely limit the advantage of a Taylor series solution. In the examples, it is shown how a practical estimate can be made. A final solution consists of the well-known method of patching up a sequence of Taylor series, similar to a sequence of high order Finite Difference solution values; the difference, however, being that the series accuracy can be tested analytically. The stability of the recursive solution routines is investigated. The convolution number and its algebra is defined for a bivariate polynomial and an example of the solution of a nonlinear partial differential equation given. Again, the solution is seriously limited by the region of convergence. Throughout, a distinctive and precise symbolic notation for convolution algebra has been attempted.

Taylor series solution of the M/M/1 queueing system

A Taylor series method for determining the transient probabilities of the classical single server queueing system is presented. The method is direct, practical and avoids Bessel function theory, Laplace transform theory and complex analysis. The resulting Taylor series are proved to converge for all time under arbitrary initial conditions. A method to obtain explicit Taylor series representations is demonstrated.

Taylor series solution of a class of diffusion problems in physiology

A Taylor series technique for obtaining pointwise bounds for the solution of a class of diffusion problems in physiology is presented. Simple analytic bounding functions are obtained using an integral representation for the solution. Computations are performed in interval arithmetic, and thus lower and upper bounds are obtained simultaneously. Higher-order Taylor coefficients are computed using simple recursion relations. The oxygen diffusion problem in spherical cells is presented as an illustrative example. For this example, the present technique is computationally more efficient than the existing ones in that it yields sharper bounds with fewer integration steps.

Direct Taylor-series solution of singular integral equations with MAPLE

Cauchy-type singular integral equations (SIEs) frequently appear in a variety of elasticity problems and, most often, in crack problems. Here we extend the classical direct numerical methods for the approximate solution of these equations to the semi-analytical/numerical (SAN) environment offered by modern computer algebra systems (CAS). The idea is simply to use Taylor-Maclaurin series in the approximate SAN solution and to reduce the problem to a purely numerical set of systems of linear equations. The approach is illustrated in the classical Lobatto-Chebyshev method with an application to the problem of a periodic array of collinear cracks. Extensive displayed SAN results for the stress intensity factors show the convergence and efficiency of the proposed SAN approach. The cases of other direct methods for SIEs are also directly applicable.