Truncated Power Series Research Articles

Searching for laurent solutions of truncated systems of linear differential equations with the use of EG-eliminations

Laurent solutions of systems of linear ordinary differential equations with the truncated power series coefficients are considered. The Laurent series in the solutions are also truncated. We use induced recurrent systems for constructing the solutions and have previously proposed an algorithm for the case when the induced system has a non-singular leading matrix. The algorithm finds the maximum possible number of terms of the series in the solutions that are invariant with respect to any prolongation of the original system. Below we present advances in extending our algorithm to the case when the leading matrix is singular using algorithm EG as an auxiliary tool. The implementation of the algorithm as a Maple procedure and examples of its usage are presented.

Multilayered neural network for power series‐based approximation of fractional delay differential equations

This paper trains a multilayered neural network (MLNN) for solving fractional delay differential equations (FDDEs), including nonlinear and singular types. The proposed methodology involves replacing the unknown functions in the equations with a truncated power series expansion. Subsequently, a collection of algebraic equations is solved utilizing an iterative minimization technique that leverages the capabilities of the MLNN architecture. The outcomes demonstrate that the MLNN architecture provides the required accuracy and strong stability compared to several numerical methods.

Couple New Iterative Method with Pade Approximation to Solve the Nonlinear Wave-Like Equations with Variable Coefficients

In this paper, some cases of wave-like equation have been solved by modified new iterative method as an infinite power series but it is impossible to compute the infinite series, therefore, the truncated power series was approximated by Pade approximation. Pade approximation approximates a truncated power series as ratio of two polynomials to reduce calculations and shorten the power series while maintaining accuracy. These cases are shown the successful use of solution approximation by Pade approximation compared to power series expansion.

Power-series solutions of fractional-order compartmental models

Compartmental models based on coupled differential equations of fractional order have been widely employed in the literature for modeling. An abstraction of these models is given by a system with polynomial vector field. We investigate the use of power series for solving generic polynomial differential equations in any dimension, with Caputo fractional derivative. As is well known, power series convert a continuous formulation into a discrete system of difference equations, which are easily solved by recursion. The novelty of this paper is that we rigorously prove that the series converge on a neighborhood of the initial instant, which is an analogue of the Cauchy–Kovalevskaya theorem. Besides, these series are proved to be continuous with respect to the fractional index. For applications, a general-purpose symbolic implementation of truncated power series is developed, and its execution is illustrated for the fractional SIR epidemiological model.

Searching for Laurent Solutions of Systems of Linear Differential Equations with Truncated Power Series in the Role of Coefficients

Searching for Laurent Solutions of Systems of Linear Differential Equations with Truncated Power Series in the Role of Coefficients

Searching for Laurent Solutions of Systems of Linear Differential Equations with Truncated Power Series in the Role of Coefficients

Systems of linear ordinary differential equations with the coefficients in the form of infinite formal power series are considered. The series are represented in a truncated form, with the truncation degree being different for different coefficients. Induced recurrent systems and literal designations for unspecified coefficients of the series are used as a tool for studying such systems. An algorithm for constructing Laurent solutions of the system is proposed for the case where the determinant of the leading matrix of the induced system is not zero and does not contain literals. The series included in the solutions are still truncated. The algorithm finds the maximum possible number of terms of the series that are invariant with respect to any prolongations of the truncated coefficients of the original system. The implementation of the algorithm as a Maple procedure and examples of its usage are presented.

Gravity inversion by second order approximation applied to the Styrian Basin (Austria)

AbstractAn approximative second order gravity inversion scheme by a truncated power series expansion is applied to derive the thickness of the Neogene, mostly clastic, sedimentary section of the Styrian Basins in South–East Austria, which are sub‐basins of the Pannonian Basin System. These sub‐basins with a derived thickness of up to 4 km are of interest for geothermal exploitation because of the increased geothermal gradient and heat flow observed in the Pannonian Basin in general and a geothermal gradient of 4°–5°/100 m measured in some wells in the Styrian Basin. The Styrian Basin also has been an area for hydrocarbon exploration in the past 50 years, with oil and gas show encountered in several exploration wells and one sub‐commercial gas discovery. The Miocene and Plio–Pleistocene volcanism in the Styrian Basin caused by Miocene crustal thinning is discussed in terms of the influence to the gravity inversion taking the aeromagnetic field into account. The volcanism is of relevance for the geothermal prospectivity but poses problems for the single layer‐based gravity inversion scheme. Results are discussed from a computational side comparing observed and calculated gravity fields but also the match with well data is discussed. In terms of gravity inversion methodology, the presented can be viewed as an approximative fast‐track approach.

Approximate symmetry of time-fractional partial differential equations with a small parameter

In the sense of Riemann–Liouville fractional derivative, an approximate symmetry method of (1+1)-dimensional time-fractional partial differential equations (PDEs) with a small parameter is proposed. By first inserting the power series expansion of dependent variable in the small parameter into the fractional PDEs and then separating the equation with respect to the small parameter to get a coupled system of fractional PDEs, we define the pth-order approximate symmetries by the Lie symmetries of the first p+1 coupled fractional PDEs. After performing symmetry reductions for the coupled system of fractional PDEs, we find that zero-order reduced equation coincides with the fractional ordinary differential equation corresponding to the fractional PDE without containing the small parameter while the solutions of higher-order reduced equation can be obtained by solving linear variable coefficient fractional ordinary differential equations. We first exemplify the whole procedure of computing approximate symmetry by studying a time-fractional KdV equation with a small parameter. Specifically, we obtain the higher-order approximate symmetries and perform approximate symmetry reductions by means of an optimal system of one-dimensional approximate Lie subalgebras. Moreover, we construct an explicit power series solution and show the dynamic behaviors of the truncated power series solutions by the evolutional figures. Then in order to make the proposed approximate symmetry method be further convincing, we study an anomalous diffusion equation with a small parameter and construct an analytical solution in terms of the Mittag-Leffler function.

Solution processes for second-order linear fractional differential equations with random inhomogeneous parts

The novelty of this paper is to derive a mean square solution of a second-order (Caputo) fractional linear differential equation in which the coefficients and initial conditions are random variables, and the forcing term is a second order stochastic process. Using the so-called mean square calculus and assuming mild conditions on the random variables of the equation together with an exponential growth condition on the forcing term, a mean square convergent generalized power series solution is constructed. As a result of this convergence, the sequences of the mean and correlation obtained from the truncated power series solution are convergent as well. The theory is illustrated with several examples in which different kind of distributions on the input parameters are assumed.

Artificial neural network for solving the nonlinear singular fractional differential equations

This paper proposes an artificial neural network (ANN) architecture for solving nonlinear fractional differential equations. The proposed ANN algorithm is based on a truncated power series expansion to substitute the unknown functions in the equations in this approach. Then, a set of algebraic equations is resolved using the ANN technique in an iterative minimization process. Finally, numerical examples are provided to demonstrate the usefulness of the ANN architectures. The results verify that the suggested ANN architecture achieves high accuracy and good stability.

Failing parametrizations: what can go wrong when approximating spectral submanifolds

Invariant manifolds provide useful insights into the behavior of nonlinear dynamical systems. For conservative vibration problems, Lyapunov subcenter manifolds constitute the nonlinear extension of spectral subspaces consisting of one or more modes of the linearized system. Conversely, spectral submanifolds represent the spectral dynamics of non-conservative, nonlinear problems. While finding global invariant manifolds remains a challenge, approximations thereof can be simple to acquire and still provide an effective framework for analyzing a wide variety of problems near equilibrium solutions. This approach has been successfully employed to study both the behavior of autonomous systems and the effects of non-autonomous forcing. The current computation strategies rely on a parametrization of the invariant manifold and the reduced dynamics thereon via truncated power series. While this leads to efficient recursive algorithms, the problem itself is ambiguous, since it permits the use of various approaches for constructing the reduced system to which the invariant manifold is conjugated. Although this ambiguity is well known, it is rarely discussed and usually resolved by an ad hoc choice of method, the effects of which are mostly neglected. In this contribution, we first analyze the performance of three popular approaches for constructing the conjugate system: the graph style parametrization, the normal form parametrization, and the normal form parametrization for “near resonances.” We then show that none of them is always superior to the others and discuss the potential benefits of tailoring the parametrization to the analyzed system. As a means for illustrating the latter, we introduce an alternative strategy for constructing the reduced dynamics and apply it to two examples from the literature, which results in a significantly improved approximation quality.

The time-fractional (2+1)-dimensional Hirota–Satsuma–Ito equations: Lie symmetries, power series solutions and conservation laws

In this paper, we concentrate on the Lie symmetries and conservation laws of the time-fractional (2+1)-dimensional Hirota–Satsuma–Ito equations which contain both the integer-order partial differential equations (PDEs) and the fractional PDEs with the mixed derivative of Riemann–Liouville fractional derivative and second order x-derivative. Therefore, based on a new prolongation formula of the infinitesimal generator in the case of mixed derivatives, we find the admitted Lie symmetries and reduce the equations to time-fractional (1+1)-dimensional PDEs by means of an optimal system of one-dimensional Lie subalgebras. In particular, we construct an explicit power series solution with fractional degrees and show the dynamic behaviors of the truncated power series solutions by the evolutional figures. Moreover, we give a conservation law formula using the idea of nonlinear self-adjointness for the fractional case and obtain several nontrivial conservation laws by means of the obtained Lie symmetries.

Multipole infinite series expansion of the Coulomb repulsion term using analytical spherical Bessel-like functions

Special functions are particular mathematical functions which arise in the solutions of various classical problems in physics. The functions are quite useful in mathematical analysis. Mostly, they find applications in applied mathematics, theoretical physics, and engineering. Special functions are usually uniquely described by a generating function, a Rodrigues’ formula, and a recurrence relation. In this work, we derive analytical spherical Bessel-like functions corresponding to the infinite degree power series expansion of the functions obtained in the multipole series expansion of the Coulomb repulsion term. The convergence of the truncated power series expansion of the functions with the corresponding derived analytical functions, as well as the comparison of analytical functions of the first and second kinds, is analyzed. Rodrigues’ formula and a recurrence relation for obtaining these analytical functions are also presented. It is shown that the higher order even and odd spherical Bessel-like functions of the first kind are derivable from the lowest order functions of the first and second kinds, respectively.

Uniformization and Constructive Analytic Continuation of Taylor Series

We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point, and additional analyticity properties. This situation occurs frequently in applications. The question of the optimal procedure was open, and we formulate it as a well-posed mathematical problem. Its solution leads to a practical method which provides dramatic accuracy improvements over existing techniques. Our procedure is based on uniformization of Riemann surfaces. As an application, we show that our procedure can be implemented for solutions of a wide class of nonlinear ODEs. We find a new uniformization method, which we use to construct the uniformizing maps needed for special functions, including solution of the Painlev'e equations P_I-P_V. We also introduce a new rigorous and constructive method of regularization, elimination of singularities whose position and type are known. If these are unknown, the same procedure enables a highly sensitive resonance method to determine the position and type of a singularity. In applications where less explicit information is available about the Riemann surface, our approach and techniques lead to new approximate, but still much more precise reconstruction methods than existing ones, especially in the vicinity of singularities, which are the points of greatest interest.

On Existence and Uniqueness of Formal Power Series Solutions of Algebraic Ordinary Differential Equations

Given an algebraic ordinary differential equation (AODE), we propose a computational method to determine when a truncated power series can be extended to a formal power series solution. If a certain regularity condition on the given AODE or on the initial values is fulfilled, we compute all of the solutions. Moreover, when the existence is confirmed, we present the algebraic structure of the set of all formal power series solutions.

The HVT Technique and the "Uncertainty" Relation for Central

The quantum mechanical hypervirial theorems (HVT) technique is used to treat the so-called "uncertainty" relation for quite a wide class of central potential wells, including the (reduced) Poeschl-Teller and the Gaussian one.It is shown that this technique is quite suitable in deriving an approximate analytic expression in the form of a truncated power series expansion for the dimensionless product $P_{nl}\equiv <r^2>_{nl}<p^2>_{nl}/\hbar^2$, for every (deeply) bound state of a particle moving non-relativistically in the well, provided that a (dimensionless) parameter s is sufficiently small. Numerical results are also given and discussed.

Numerical solutions of wavelet neural networks for fractional differential equations

Neural network has good self‐learning and adaptive capabilities. In this paper, a wavelet neural network is proposed to be used to solve the value problem of fractional differential equations (FDE). We construct a wavelet neural network (WNN) with the structure 1 ×N× 1 based on the wavelet function and give the conditions for the convergence of the given algorithm. This method uses the truncated power series of the solution function to transform the original differential equation into an approximate solution, then, using WNN, update the parameters, and finally get the FDE solution. Simulation results prove the validity of WNN.

Procedures for Constructing Truncated Solutions of Linear Differential Equations with Infinite and Truncated Power Series in the Role of Coefficients

In this paper, we propose a package for symbolic construction of exponential–logarithmic solutions to linear differential equations whose coefficients are represented in incomplete form, i.e., as power series for which only a finite number of initial terms is known. The series involved in the solutions are also represented in incomplete form. For each such series, the maximum possible number of initial terms is constructed, which are uniquely determined by known terms of coefficients of a given equation. Additionally, the truncation degree of each series in a solution should not exceed a value specified by the user. It ensures the termination of the computation even when any number of the terms of the series involved in the solution can be defined by known terms of the coefficients of a given equation.

Construction of a method for representing an approximation model of an object as a set of linear differential models

This paper has demonstrated the need to use models not only at the stage of theoretical research and design operations but also when studying existing objects. The techniques to build them on the basis of identification methods have been analyzed. The identification methods have been shown when determining the parameters of processes and objects. The difficulty of defining the models' structures has been emphasized.A method has been proposed to determine the structure of an arbitrary object's model as the approximating set of linear differential models. The data on the object's response to external impact have been used as source data. Demonstrating the method's feasibility employed a set of standard links and a standard external influence in the form of a stepped function as a model. This approach helps assess the adequacy of the obtained approximation results based on the precise solutions available. In a general case, there are no specific requirements for the form of an external influence and an object's reaction.The data that reflect the object's response should allow their approximation using a polynomial. That makes it possible to represent them following a Laplace transform in the form of a truncated power series in the image domain. The transfer function is written in a general form as a rational fraction. It underlies a Padé approximant of the truncated power series.The comparison of the available accurate calculation results and those derived on the basis of the built model has shown good agreement. In the cases under consideration, the computation error did not exceed the 5 % value permissible for engineering calculations. This is also the case when using the approximation of original data over a limited period.The response of the resulting model to the external influence that simulates a real pulse was investigated. The comparison with precise results showed a discrepancy not exceeding the value permissible for engineering calculations (<5 %)

Truncated Series and Formal Exponential-Logarithmic Solutions of Linear Ordinary Differential Equations

The approach we used earlier to construct Laurent and regular solutions enables one, in combination with the well-known Newton polygon algorithm, to find formal exponential-logarithmic solutions of linear ordinary differential equations the coefficients of which have the form of truncated power series. (Thus, only incomplete information about the original equation is available.) The series involved in the solution are also represented in truncated form. For these series, the combined approach proposed enables one to obtain the maximum possible number of terms.