Series Solutions Of Differential Equations Research Articles

Application of q-HATM over Fractional in Time Model for Advection-Dispersion Equation with Variable Migration Parameters

In this article, an extended and more generalized method of homotopy analysis method (HAM), known as q-homotopy analysis transform method (q-HATM) is employed to develop solutions for highly nonlinear form of various time-fractional advection-dispersion models (TFADE). The proposed methodology of q-HATM is a conjunction of two different strong methodologies, that are HAM and Laplace transform technique (LTT) which makes the scheme much more capable to develop the convergent series solutions for differential equations with high nonlinearity. In this work, two different examples are constructed for the nonlinear time-fractional form of dispersion models corresponding to distance dependent dispersion and velocity components and also for the time dependent decay rate system and solutions are constructed by using q-HATM in generalized form. The graphical comparison is made for different types of variable functions and also for the different fractional order indexes. Prominent impact of fractional derivative indexes is significantly observed over both the examples of dispersion models. Hence, proposed examples shows that the q-HATM is much convenient to handle the nonlinear systems of fractional models.

A New Approach Using Integral Transform to Solve Cancer Models

The objective of this work is to investigate analytical solutions of some models of cancer tumors using the Laplace residual power series method (LRPSM). The proposed method was effective and required simple calculations to find the analytic series solution, utilizing computer software such as the Mathematica package. Figures and graphs of the attained analytical Maclaurin solutions are presented to depict the procedure. The outcomes we obtained in this research showed the applicability and strength of the proposed approach in studying numerical series solutions of differential equations of fractional orders.

A new series solution associated with the quantum harmonic oscillator

In the non-relativistic case, the problem of quantum harmonic oscillator is solved by the aid of the Schrödinger equation. An additional term of potential energy that is related with the square of the spatial coordinate is also included in the equation. A second order linear differential equation is obtained in the stationary state. The direct solution gives the wave function in terms of the parabolic cylinder functions. This solution leads to the Hermite polynomials when the quantization of energy is considered. However, the derived expression of the wave function does not yield the correct functions for the limiting case, in which the potential energy is zero. We put forth an alternative solution for the second order linear differential equation. First of all, a new series expression is suggested as an ansatz for the power series solution of differential equations. The coefficients of the series are obtained in terms of recurrence relations. Some computational simulations are given.

Improved Numerical Computation to Solve Lane-Emden type Equations

Lane-Emden equation is also of fundamental importance in mathematical physics, celestial mechanics,and computer science. It can be used to describe stellar structures, equilibrium density distribution in polytrophicisothermal gas, thermal behavior in mutual attraction of its molecules. An improved numerical method is developed for solving Lane-Emden type differential equations. The method is based on power series solutions of differential equations and Maclaurin series expansion. A python program is written to carry out numerical calculations. Five examples are solved and shown in this paper, the solutions obtained by the program are compared with the exact solutions of differential equation, an excellent agreement is found between them. The present method improves runtime.

Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

A study on the convergence conditions of generalized differential transform method

This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non‐linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.

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.

High precision series solutions of differential equations: Ordinary and regular singular points of second order ODEs

A subroutine for a very-high-precision numerical solution of a class of ordinary differential equations is provided. For a given evaluation point and equation parameters the memory requirement scales linearly with precision P, and the number of algebraic operations scales roughly linearly with P when P becomes sufficiently large. We discuss results from extensive tests of the code, and how one, for a given evaluation point and equation parameters, may estimate precision loss and computing time in advance.Program summaryProgram title: seriesSolveOde1Catalogue identifier: AEMW_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEMW_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 991No. of bytes in distributed program, including test data, etc.: 488116Distribution format: tar.gzProgramming language: C++Computer: PC’s or higher performance computers.Operating system: Linux and MacOSRAM: Few to many megabytes (problem dependent).Classification: 2.7, 4.3External routines: CLN — Class Library for Numbers [1] built with the GNU MP library [2], and GSL — GNU Scientific Library [3] (only for time measurements).Nature of problem:The differential equation (1)−s2(d2dz2+1−ν+−ν−zddz+ν+ν−z2)ψ(z)+1z∑n=0Nvnznψ(z)=0, is solved numerically to very high precision. The evaluation point z and some or all of the equation parameters may be complex numbers; some or all of them may be represented exactly in terms of rational numbers.Solution method: The solution ψ(z), and optionally ψ′(z), is evaluated at the point z by executing the recursion (2)Am+1(z)=s−2(m+1+ν−ν+)(m+1+ν−ν−)∑n=0NVn(z)Am−n(z),(3)ψ(m+1)(z)=ψ(m)(z)+Am+1(z), to sufficiently large m. Here ν is either ν+ or ν−, and Vn(z)=vnzn+1. The recursion is initialized by (4)A−n(z)=δn0zν,for n=0,1,…,N(5)ψ(0)(z)=A0(z).Restrictions: No solution is computed if z=0, or s=0, or if ν=ν− (assuming Reν+≥Reν−) with ν+−ν− an integer, except when ν+−ν−=1 and v0=0 (i.e. when z is an ordinary point for z−ν−ψ(z)).Additional comments: The code of the main algorithm is in the file seriesSolveOde1.cc, which “#include” the file checkForBreakOde1.cc. These routines, and the programs using them, must “#include” the file seriesSolveOde1.cc.Running time: On a Linux PC that is a few years old, at y=10 to an accuracy of P=200 decimal digits, evaluating the ground state wavefunction of the anharmonic oscillator (with the eigenvalue known in advance); (cf. Eq. (6)) takes about 2 ms, and about 40 min at an accuracy of P=100000 decimal digits.

A Fast and Accurate Algorithm for Computing SLERP

Abstract The SLERP of quaternions is a common operation in keyframe animation. The operation can be a significant bottleneck in an animation-heavy application. The standard implementation of SLERP for an FPU typically involves trigonometric function evaluations, divisions, and branching. Commonly available SIMD implementations will modify the standard implementation and obtain a moderate speed-up. This paper provides a novel approach to computing SLERP, using only multiplications and additions. The algorithm is based on ideas from Chebyshev polynomials, power series solutions for differential equations, and error balancing using the Chebyshev Equioscillation Theorem and the associated Remez Algorithm. Implementations are provided for the FPU and SIMD. When SLERPing a pair of quaternions in parallel using four time samples, performance measurements show a 10-fold acceleration per SLERP over the standard SLERP implementation on an FPU. Source code is available online.

Singular Complex Periodic Solutions of van der Pol’s Equation

The solutions that are studied here arise when limit cycle solutions of van der Pol’s equation with parameter $\epsilon $ are continued analytically into the complex $\epsilon $-plane. These analytic continuations are described, for small $\epsilon $, by the perturbation expansions in powers of $\epsilon $ which were developed at length by Andersen, Dadfar, and Geer [SIAM J. Appl. Math., 42(1982), pp. 678–693 and 44(1984), pp. 881–895]. We show that the moving singularities that they found are caused by a family of singular complex periodic solutions. These occur along two infinitely long branches in the complex $\epsilon $-plane; half of one being plotted in Fig. 2. Numerical integration is needed to obtain these singular solutions. As Fig. 4 for the large value $\epsilon = 10 + 3.042i$ shows, the complex solution is primarily real over most of the cycle. And this primarily real part behaves much like the main part of the real relaxation oscillation, falling from near 2 at its start toward 1. But the rapid transition from 1 to -2 that then occurs prior to the start of the next half cycle now takes place via a complex singularity, rather than smoothly as in the real case. An asymptotic description of the singular complex periodic solutions for large values of $| \epsilon |$ is given using singular perturbation theory and matched asymptotic expansions. It shares many features with the corresponding theory for the real case, and, as Table 1 shows, is numerically accurate. Moving complex singularities similar to those studied here are likely to occur with other perturbation series solutions of differential equations and to hinder their effectiveness.

Dirichlet series solutions of differential equations

In this paper we obtained Dirichlet series solutions, convergent in a half plane, of an ordinary linear differential equation of nth order whose coefficients possess Dirichlet series expansions convergent also in a half plane, by a simple application of Banach's fixed point theorem. We also proved the existence ofn linearly independent solutions of such an equation.

On a uniqueness problem

We prove a uniqueness theorem for series which are a broad generalization of Dirichlet series. In particular, we obtain a uniqueness theorem for series solutions of differential equations.

Studies on doubly asymptotic series solutions for differential equations in unbounded domains

Studies on doubly asymptotic series solutions for differential equations in unbounded domains

Hermite series solutions of differential equations

A standard approach both for the numerical solution as well as the deduction of certain theoretical properties of partial differential equations is their replacement by appropriate difference equations. Another standard approach is the use of Fourier transforms. These two approaches could be combined if the transformed equation could be replaced by a difference equation which could then be solved. IJnfortunately, this combination is usually of little use for numerical considerations since the Fourier transform of a good approximation to the solution of an equation is not gcncrally a good approximation to the Fourier transform of the solution and vise versa However, there is one type of approximating function which is well qualified to give a good approximation both to the solution and to its Fourier transform, and that is a partial sum of a Hermite Series. This is true because the Hermite functions are eigenfunctions of the Fourier transform corresponding to eigenvalues of modules 1. Thus solutions in terms of Hermite series of the transformed equation may be interesting from a numerical point of view. The transformed equation of a partial differential equation with constant coefficients is just an equation of the form

The Application of Lie Series to Reactor Theory

Lie series are power series solutions of differential equations, whose expansion coefficients can be derived from algebraic recursion relations. The method of deriving these relations is illustrate...

1415. Series Solution of Differential Equations

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.