Why pade approximation

Remark 4. Let be any sequences of distinct real numbers and ; then, is dense in [ 6 ]. Remark 6. Also, it is clear that by considering , we can obtain the classical Weierstrass theorem [ 6 ]. But the generalized version of this theorem from fractional calculus view or the theory of derivatives of arbitrary order was done by [ 7 ], because of its high applications in solving fractional ordinary differential equations integral ordinary differential equations and obtaining an operational matrix, and so forth.

For this purpose, we need some definitions in fractional calculus. Definition 7. The Caputo fractional derivative of of order with is defined as for , ,.

From Caputo fractional derivative, we have where and. However, the Caputo fractional derivative is a linear operator which means For more literature review of fractional calculus, see [ 8 ].

Theorem 8. Suppose that , for , where ; then, one has where , for all , and , -times [ 7 ]. If we consider , then we obtain a fractional Maclaurin series and in a similar manner as the arbitrary function that has an infinite Caputo differentiable at point named as analytical functions on in a fractional sense if. Also immediately by equating the coefficients of , we can obtain coefficients in a recursion formula as 7. Remark 9. Theorem 10 uniqueness. Then by 19 , we must have since both approximate the same series.

If we multiply 21 by , we obtain but the left-hand side of 22 is a polynomial of degree at most and thus is identically zero. Theorem 11 convergency. By hypothesis, is analytic in in fractional sense and consequently within a large interval, , with. Thus either a subsequence of the second row converges uniformly or else for some and all ,.

In the latter case contradicting with 24 , so the proof is completed. The algorithm for numerical approximation of solutions to the initial value problems for the fractional differential equations was implemented by MATLAB. In the case that the exact solution to a problem is known, the dependence of approximation errors on the discretization parameter is estimated in 2-norm as where is the approximated solution corresponding to the discretization parameter.

Experiment 1. We start with a simple nonlinear problem [ 8 ] where we have a nonlinear and nonsmooth right-hand side. The solution has a smooth derivative of order [ 8 ].

Thus, substituting the collocation points into 32 yields Now from 34 and its initial condition, we have algebraic equations of unknown coefficients. Thus for obtaining the unknown coefficients, we must eliminate one arbitrary equation from these equations. But because of the necessity of holding the boundary conditions, we eliminate the last equation from Finally, replacing the last equation of 34 by the equation of initial condition, we obtain a system of equations of unknowns.

By implementing the method as presented, for and also for different parameters of and , we obtain the approximate solutions. Experiment 2. Consider the nonlinear fractional integro-differential equation [ 11 ] where The exact solution of 35 is. The obtained results of our method are presented in Table 1.

The procedure for obtaining external asymptotics is nontrivial due to the presence of logarithmic components in the main elements. We describe in detail the mechanism for obtaining and evaluating both primary and secondary members of asymptotic. From Eq. After integration of Eq. After reintegration of Eq. In the resulting equation, the first compound is the principal member of the external asymptotics. To obtain the following members of the asymptotic, we will present the function as.

To calculate parameter a 2 , use the procedure of Section 4. Parameter values are determined using local asymptotic and TPPA in the respective domain. Taking into account the decomposition of the exponent in the internal domain, we will write down the local equality:. Taking into account Eq. Equalizing the coefficients in Eqs.

By substituting 39 in 38 , we get an explicit expression for the TPPA. We consider the boundary layer in hypersonic flow of viscous gas and solve a model problem which reduces to ordinary differential equations with appropriate boundary conditions.

The TPPAs parameters are calculated and relevant questions are discussed. The equations of laminar boundary layer near a semi-infinite plate in the supersonic flow of viscous perfect gas, as it is known [ 2 , 7 ], can be reduced to the form:. We solve boundary problems 40 and 41 approximately by connecting asymptotics 44 and 45 TPPA.

Boundary conditions 45 and 46 are satisfied if to put. Three parameters in asymptotics 44 are defined in the outer region if the following condition is met:. Let us add the received equations with the integrated ratios received on the basis of coincidence of TPPAs 46 and 47 ; in this case, three members in asymptotic decompositions 50 and 51 , the initial system of Eqs.

The integral relation for parameter A is obtained by multiplying Eq. Similarly, from Eq. Thus, the integral relations 52 and 55 — 47 form a nonlinear system of equations for determining the following parameters:.

Integrals of the systems 37 and 42 — 44 solution were approximated using Simpson quadrature formulas. The behavior of magnitude B proved to be highly dependent on the behavior of the exponent at large, so the integral relation had to be replaced by the local condition 52 , besides controlling the behavior of the TPPA near the maximum is more important than the weight of the exponent away from the wall.

Thus, instead of the value of B , we include the value among the parameters sought, and the value of B is expressed from Eqs.

They dealt with Lyapunov exponents which characterize the dynamics of a system near its attractor. For the Van der Pol oscillator:. The overlap of these series does not take place. So, one needs a summation procedure. Some authors [ 34 ] proposed to use PAs, but in this case one needs hundreds of perturbation series terms.

That is why we use TPPA. Using two terms from expansion 58 and one term from expansion 59 , one obtains. Below, one can see some numerical results. In Table 1 , the second column is made by calculation results by formula 4 , the third column is made by paper data [ 33 ].

One can see that TPPA gives good result for any value of used parameter. In Section 4. When the first equation of the systems 43 and 44 is solved, it becomes independent of the second equation and can be compared with the known Blasius solution see Section 3 , which was used as a test when compared to our method [ 35 , 36 , 37 , 38 , 39 , 40 ].

Of course, such a good match is due to the fact that these parameters are largely determined by local internal asymptotics, more precisely, derived from the function on the wall. The procedure of constructing the PA is much less labor-intensive than the construction of higher approximations of perturbation theory.

PA can be applied to power series but also to the series of orthogonal polynomials. PA is locally the best rational approximation of a given power series. They are constructed directly and allow for efficient analytic continuation of the series outside its circle of convergence, and their poles in a certain sense localize the singular points including the poles and their multiplicities of the function at the corresponding region of convergence and on its boundary.

PA is fundamentally different from rational approximations with fully or partially fixed poles, including the polynomial approximation, when all the poles are fixed in infinity. That is the above property of PA—effectively solving the problem of analytic continuation of power series—lies at the basis of their many successful applications in the analysis and the study of applied problems.

Currently, the PA method is one of the most promising nonlinear methods of summation of power series and the localization of its singular points. Including the reason why the theory of the PA turned into a completely independent section of approximation theory, and these approximations have found a variety of applications both directly in the theory of rational approximations, and in perturbation theory. Thus, the main advantages of PA compared with the Taylor series are as follows: Typically, the rate of convergence of rational approximations greatly exceeds the rate of convergence of polynomial approximation.

More tangible, it is property for functions of limited smoothness. Typically, the radius of convergence of rational approximation is large compared with the power series.

TPPA allows to overcome the locality of asymptotic expansions, using only a few terms of asymptotics. Wolfram Alpha » Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end.

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