Constructing accurate polynomial approximations for nonlinear differential initial value problems☆
Section snippets
Introduction and preliminaries
Nonlinear differential equations are frequently used to model a wide class of problems in many areas of scientific fields. Most of these equations have no analytical solution and numerical techniques have to be used to construct approximations [1].
Discrete numerical methods are probably the most widely used because of their simplicity. However, they present some drawbacks. Specifically there are questions about the preservation of qualitative properties of the solution, lack of precise error
Approximate polynomial solution of analytic differential equations
In this section we construct approximate polynomial solutions of initial value problems of the typewhere is an analytic function of two real variablesWe modify results of Section 3 of [6] in order to obtain a smaller truncation index of the infinite series solution of (18) for a given admissible error εT. By Cauchy’s inequalities [2], it follows thatwhereBy Lemma 3 and Theorem
Artificial Chebyshev initial value problem
The main objective of this paper is the construction of approximate polynomial solutions of initial value problems of the type (1) where is a non-analytic function, by approximating by a Chebyshev polynomial of degree n in x and m in y and truncating the infinite series solution ofHowever, as it was shown in the previous section, both the majorant method and the truncation process produce a reduction of the solution domain. Hence, it is suitable to
Improvement of the truncation degree of the approximating polynomial
In this section we seek to reduce the truncation degree n0 of the polynomial solution obtained after applying the majorant method to problem (63) for a given Chebyshev polynomial . From Eq. (31) the size of n0 is closely related to the distance R1 − R0 and in order to keep the domain, from (67) R0 should be close to . A practical way to reduce the size of n0 is to split the domain in N subintervals, following the ideas developed in [11, chapter 8, p. 606] to analyze the
Conclusions
This paper deals with the construction of polynomial piecewise continuous approximate solutions of initial value problems for nonlinear ordinary differential equations, with a prefixed accuracy and without a significant reduction of the existence domain. The method presented improves on some previous results.
The paper includes a constructive procedure that summarizes the method and it is illustrated with several examples.
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Cited by (6)
A comparative study of the numerical approximation of the random Airy differential equation
2011, Computers and Mathematics with ApplicationsCitation Excerpt :The paper is organized as follows. Based on the deterministic approach shown in [7], Section 2 is devoted to presenting a modification of the random Fröbenius method developed in paper [4]. Section 3 introduces the polynomial chaos method, including its application to model (1).
Polynomial approximation of nonlinear differential systems with prefixed accuracy
2011, Applied Mathematics and ComputationCitation Excerpt :Another advantage of analytic approximations is that the solution is known at any point in the interval of interest and there is no need to do interpolation with its additional error as with a discrete numerical approximation method. Disadvantages of analytic approximation methods used to be the high computational cost and the difficulty of the extension to systems of differential equations, see [4]. Some other analytic methods, such as the differential transform method [10,2], or Adomian’s decomposition method [1], do not provide accurate error estimation.
Piecewise finite series solutions of seasonal diseases models using multistage Adomian method
2009, Communications in Nonlinear Science and Numerical SimulationPiecewise finite series solution of nonlinear initial value differential problem
2009, Applied Mathematics and ComputationCitation Excerpt :Therefore, a compromise between the truncation order and time step size is necessary in order to obtain feasible accurate solutions. To illustrate the effectiveness of the proposed approach, a Riccati ODE used as a benchmark in [9] is solved and numerical comparisons with the exact solution are shown. In addition, numerical results and computation times are obtained for a second order ODE.
Solving time-invariant differential matrix Riccati equations using GPGPU computing
2014, Journal of SupercomputingOn the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials
2012, Journal of Numerical Mathematics
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This work has been partially supported by the Spanish M.C.Y.T. and FEDER grants DPI2003-07153-C02-02 and DPI2004-08383-C03-03 and the Generalitat Valenciana Grant ACOMP06/003.