Comparison for L of numerical results (NR) from paper by [33] with TPPA formulate (60).
\\n\\n
Dr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\\n\\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\\n\\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\\n\\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\\n\\nThank you all for being part of the journey. 5,000 times thank you!
\\n\\nNow with 5,000 titles available Open Access, which one will you read next?
\\n\\nRead, share and download for free: https://www.intechopen.com/books
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Preparation of Space Experiments edited by international leading expert Dr. Vladimir Pletser, Director of Space Training Operations at Blue Abyss is the 5,000th Open Access book published by IntechOpen and our milestone publication!
\n\n"This book presents some of the current trends in space microgravity research. The eleven chapters introduce various facets of space research in physical sciences, human physiology and technology developed using the microgravity environment not only to improve our fundamental understanding in these domains but also to adapt this new knowledge for application on earth." says the editor. Listen what else Dr. Pletser has to say...
\n\n\n\nDr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\n\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\n\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\n\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\n\nThank you all for being part of the journey. 5,000 times thank you!
\n\nNow with 5,000 titles available Open Access, which one will you read next?
\n\nRead, share and download for free: https://www.intechopen.com/books
\n\n\n\n
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"640",leadTitle:null,fullTitle:"Modern Arthroscopy",title:"Modern Arthroscopy",subtitle:null,reviewType:"peer-reviewed",abstract:"Modern Arthroscopy will assist practitioners to stay current in the rapidly changing field of arthroscopic surgery. 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An important drawback of asymptotic methods is the local character of solutions obtained [1, 2, 3, 4]. Since the constructed series are often asymptotic, a simple increase in the number of terms does not remove this drawback. Essence of the problem consists of divergence of obtained series. There exist a lot of approaches to these problems [5, 6]. The method of analytic continuation (e.g., the Euler transform or generalized Euler transform [7, 8, 9, 10, 11, 12]) requires a priori information about the singularities of the searched function in the complex domain [4, 9]. These methods are useful if a large number of terms of the series are known. In this case, it is possible to use the Domb-Sykes plot [5, 8]. But usually only a few terms of asymptotic series are known, and to get information from them, the method of Padé approximations (PAs) is useful [1, 2, 5, 13, 14, 15]. PAs yield meromorphic continuations of functions defined by power series and can be used even in cases where analytic continuations are inapplicable. If a PAs converges to the given function, then roots of the denominator tend to points of singularities. One-point PAs give possibilities to improve convergence of series [16, 17, 18, 19, 20]. Two-point PAs (TPPAs) allow matching asymptotics in transition zones and are widely used in mechanics and physics [1, 2, 4, 14, 21, 22, 23, 24]. Overcoming the mentioned limitations of asymptotic methods for practically important problem is the purpose of this chapter. We consider at the beginning (Section 2) the mathematical bases of asymptotic methods and the use of Padé approximants for the summations of the asymptotic series. Section 3 discusses the method of combining of internal and external asymptotics (matching method) by means of Padé approximants. In the Section 4, the methods of solving specific problems of mathematical physics and mechanics of fluid and gas are demonstrated. Section 5 presents a discussion of the obtained.
\nWe suppose that by the result of the asymptotic study, one obtains the following series:
\nAs is known, the radius of convergence \n
Recast the function f in terms of \n
A natural generalization of Euler transformation looks as follows:
\nwhere \n
“The coefficients of the Taylor series in the aggregate have a lot more information about the values of features than its partial sums. It is only necessary to be able to retrieve it, and some of the ways to do this is to construct a Padé approximant” [11]. Padé approximants (PAs) allow us to transform of power series to a fractional-rational function. Let us define PAs, following Baker and Graves-Morris [25].
\nSuppose we are given the power series:
\nPAs can be written as the following expression:
\nwhose coefficients are determined from the condition
\nEquating coefficients near the same powers \n
We note some properties of the PAs [5, 13, 19]. If the PAs at the chosen m and n exists, then it is unique.
1. If the PAs sequence converges to some function, the roots of its denominator tend to the poles of the function. This allows for a sufficiently large number of terms to determine the pole and then perform an analytical continuation.
2. PAs gives meromorphic continuation of a given power series.
3. PAs of the inverse function is treated as the PAs function inverse itself. This property is called duality and is more exactly formulated as follows. Let
\n
4. Diagonal PAs are invariant under fractional linear transformations of the argument. Suppose that the function is given by their expansion (3). Consider the linear fractional transformation that preserves the origin \t\n
5. Diagonal PAs are invariant under fractional linear transformations of functions. Let us analyze a function (3). Let
\n
If \n
\n
provided that there is \n
6. PAs can get the upper and lower bounds for \n
\n
Typically, this estimate is valid for the function itself, that is, \n
7. Diagonal and close to them a sequence of PAs often possesses the property of autocorrection [17, 18]. It consists of the following. To determine the coefficients of the numerator and denominator of PAs, we have to solve systems of linear algebraic equations. This is an ill-posed procedure, so the coefficients of PAs can be determined with large errors. However, these errors in a certain sense are of self-consistent, so the accuracy of PAs is high. This is the radical difference the PAs from the Taylor series, the calculation error of which only increases with increasing number of terms.
Autocorrection property is verified for a number of special functions. At the same time, even for elliptic functions, the so-called Froissart doublets phenomenon arises [26]. Thus, in general, having no information about the location of the poles of the PAs, but relying solely on the very PAs (computed exactly as you wish), we cannot say that you have found a good approximated function. Now consider the question: In what sense the available mathematical results on the convergence of the PAs can facilitate the solution of practical problems? Gonchar’s theorem [16] states: If none of the diagonal PAs \n
This method was originally proposed by Slepyan and Yakovlev for the inversion of the integral transformations. Here is a description of this method, following [26].
\nSuppose that the Laplace transform of a function of a real variable f(t) is
\nTo obtain an approximate expression for the inverse transform, it is necessary to clarify the behavior of the transform to the vicinity of the points s = 0 and s = ∞ and to determine whether the nature and location of its singular points are on the exact boundary of the regularity or near it. Then the transform F(s) is replaced by the function F\n0(s), approximated the exact inversion and satisfying the following conditions:
1. Functions F\n0(s) and F(s) are asymptotically equivalent at s → ∞ and s → 0, that is,
\n
2. Singular points of functions F\n0(s) and F(s), located on the exact boundary of the regularity, coincide.
The free parameters of the function F\n0(s) are chosen so as to satisfy the conditions of the good approximation of F(s) in the sense of minimum relative error for all real values s ≥ 0:
\n\nCondition (8) is achieved by variation of free parameters \n
or \n
Constructed in such a way function F\n0\n(s) is called asymptotically equivalent function for F(s) (AEF). Let’s dwell on the terminology. In the following sections, we will use the symbols of ordinal relations. We will give strict definitions of these concepts.
\nLet’s consider the function f(x). To describe the ordinal relationships with respect to another function φ(x), enter the following definitions:
\nDefinition 1. Let us say that f(x) is a value of order φ(x) at \n
if \n
Definition 2. Let us say that f(x) is a value of order less than φ(x) at \n
if \n
Here A is a finite number, and ε, δ are infinitely small.
\nDefinition 3. Let us say that f (x) is asymptotically equal to φ(x) at \n
Here we use the term “asymptotically equivalent function.” Other terms (“reduced method of matched asymptotic expansions” [28], “quasi-fractional approximants” (QAs) [29], and “mimic function” [30]) are also used.
\nThe analysis of numerous examples confirms “complementarity principle”: if for \n
TPPA is a rational function of the form:
\n\nk coefficients which are determined from the condition
\nand the remaining coefficients from a similar condition for \n
For boundary-value problems, we assume that there exist two asymptotics for limit values of the parameter. In this case, the method of matching of asymptotic expansions is usually used [4]. However, for correct application of the matching method, it is necessary to know the matching point or, at least, the domain of overlapping of asymptotics. An exact description of the transition layer 0 <ε< ∞ exists only in the cases where solutions with different behaviors on opposite sides of the layer can be matched by a special function (e.g., the Airy function).
\nFor the matching of nonoverlapping asymptotics, a method based on TPPAs has recently been developed. In [15, 21, 23], this method was applied for the construction of thermal profiles in a boundary layer of gas. In [2, 6], this method allowed one to examine the heat exchange in hypersonic boundary layers.
\nTwo-point Padé approximations (TPPAs) are defined in Section 3.2 [see formulas (2)–(4)]. As an example of application of TPPAs, we consider the Airy boundary-value problem [4, 10, 31]:
\nwith boundary conditions
\nThis boundary-value problem has the form in terms of Airy function U(s):
\nThe asymptotic solution for problems (13) and (14) has the form:
\nThe interior asymptotic (s → 0) has the form of a power function:
\nThe exterior asymptotic has the form of an exponential function:
\nas \n
The transition layer is defined by the domain, where \n
Airy function approaches with TPPA:
\nThe TPPA (19) preserves three terms of the asymptotics at both ends and provides accuracy with relative error:
\nParameters a and b are obtained from the integral equations (relations). The relations (20) and (21) can be obtained by multiplying Eq. (18) by \n
This is the first integral relation.
\nThis is the next integral relation.
\nSubstituting in Eqs. (20) and (21) instead of U (4) interpolation Ua\n (7), calculate using quadrature integration formulas a = 0.7287 and b = 0.7922.
\nIn the same manner, integral relations with weights U, U’\n can be obtained by part integration. Multiplying Eq. (18) by \n
This is the first integral relation for the second method of producing it:
\nAnd this is the next integral relation for the second method of producing it:
\nUsing Eq. (19), from Eqs. (22) and (23), we calculate а = 0.7277, and b = 0.7966.
\nFrom the given example, it follows that the features of the asymptotic connection method are the ambiguity of the algorithm, the freedom to choose both the form of TPPAs, integral relations, and methods for calculating the parameters of the TPPAs. The question of choosing integral relations is, in fact, a question of controlling the asymptotic approximation using weights selected to obtain integral relations. Choosing the weight allows you to achieve acceptable accuracy in a particular area of the boundary layer: a weight equal to 1 means that the uniform influence of the entire layer is taken into account; a weight equal to \n
In the illustrated example (5), Eq. (18) TPPA represents a modified (quasi-fractional) two-point Padé approximant (10) by an exponential weight function, the choice of which is dictated by a kind of exterior asymptotics. Evidently, the TPPAs are not panacea. For example, one of the “bottlenecks” of the TPPAs method is related to the presence of logarithmic components in numerous asymptotic expansions. This problem is the most essential for the TPPAs, because, as a rule, one of the limits \n
where \n
Consider the Blasius equation (45), which describes laminar boundary layers on a flat plate:
\nwhere \n
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. (25) follows:
\nAfter integration of Eq. (27) by the coordinate ζ follows:
\nAfter reintegration of Eq. (28) by the coordinate
\nfollows:
\nsubject to boundary conditions
\nLet us make a limit transition \n
\n\n
We use the mean theorem in the last equation
\nIn 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
\nwhere \n
If \n
In the external domain, where \n
where \n
To calculate parameter \n
At that, in external domain, \n
Therefore,
\nand
\nType of generalized and normalized TPPA of order (4,4):
\nSpecies of TPPA taking into account four nontrivial parameters:
\nTherefore,
\nParameter 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:
\nTaking into account Eq. (33) in the external domain, we will write down the second local equality:
\nEqualizing the coefficients in Eqs. (36) and (37) at the same degrees ζ, we get
\nTherefore, the TPPA has the form:
\nAfter systems (31)–(33) are solved, we will obtain
\nBy substituting (39) in (38), we get an explicit expression for the TPPA.
\nWe 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:
\nwhere
\n\nM is the Mach number, \n
The boundary conditions at the wall are
\nAt external boundary of layer is
\nInterior asymptotic expansions are for \n
where two constants \n
Exterior asymptotics for \n
where three constants are unknown: c, A, and B.\n
\nWe solve boundary problems (40) and (41) approximately by connecting asymptotics (44) and (45) TPPA
\nBoundary conditions (45) and (46) are satisfied if to put
\nWe complement the last equalities (50) and (51) with a normalizing condition:
\nFollowing the procedure of the previous section, we will calculate the coefficients at ζ and ζ\n2 in asymptotic expansions (44) and, equating them with the corresponding expressions from Eqs. (46) and (47), we will obtain equalities, from which values \n
Three parameters in asymptotics (44) are defined in the outer region if the following condition is met:
\nA priori at large M numbers, it is known that the temperature profile is non-monotonic and has a maximum within the layer at point \n
From the convexity condition of the temperature profile in the vicinity of the point \n
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. (40) and (41), with boundary conditions (42) and (43), by using the technique stated in the previous sections.
\nThe integral relation for parameter A is obtained by multiplying Eq. (40) by
\nand integrating from 0 to ∞ taking into account Eq. (48):
\nSimilarly, from Eq. (41), we get
\nThus, the integral relations (52) and (55)–(47) form a nonlinear system of equations for determining the following parameters:
\nIntegrals 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. (50) and (51).
\nAs an example of TPPA (see Section 3.2) used for matching of limiting asymptotics, consider the paper by Grasman et al. [33]. They dealt with Lyapunov exponents which characterize the dynamics of a system near its attractor. For the Van der Pol oscillator:
\nSimilar to the asymptotic approximation of amplitude and period, expressions are derived for the nonzero Lyapunov exponent \n
The overlap of these series does not take place. The authors of [33] remark: “Such an overlap comes within reach if in the regular expansion a large number of terms is included.” This is not correct, because the obtained series is asymptotic; so, with increasing of number of terms, the results will be worst. 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
\n\nExpression (60) has a pole at \n
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.
\n\n\n | \nL (4) | \nL (NR) | \n
---|---|---|
1 | \n1.057 | \n1.0648 | \n
5 | \n1.513 | \n1.4724 | \n
10 | \n1.685 | \n1.6358 | \n
25 | \n1.759 | \n1.7398 | \n
50 | \n1.768 | \n1.7691 | \n
Comparison for L of numerical results (NR) from paper by [33] with TPPA formulate (60).
In Section 4.4, the problem was solved for several variants of the Mach number and the heating temperature: \n
\nM\n | \n\nT\ns\n | \n\n\n | \n\nT\n1\n | \n
---|---|---|---|
5 | \n3 | \n0.426 | \n1.340 | \n
10 | \n3 | \n0.744 | \n9.127 | \n
10 | \n5 | \n0.637 | \n7.929 | \n
10 | \n7 | \n0.531 | \n6.676 | \n
15 | \n3 | \n0.806 | \n22.00 | \n
15 | \n5 | \n0.756 | \n20.82 | \n
15 | \n7 | \n0.712 | \n19.75 | \n
TPPAs parameters for different Mach numbers M, temperature T\nS, and n = 1 values.
If \n
\nM\n | \n5 | \n10 | \n10 | \n
---|---|---|---|
\nT\ns\n | \n3 | \n3 | \n5 | \n
\na\n2\n | \n0.17 | \n0.74 | \n0.56 | \n
\n\n | \n0.729 | \n0.714 | \n0.798 | \n
\nc\n | \n1.45 | \n1.42 | \n1.38 | \n
\nT\n1\n | \n0.56 | \n8.69 | \n8.16 | \n
TPPAs parameters for different Mach numbers M, temperature T\nS, and n = 0.76 values.
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.
\nThus, 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. For example, the function \n
Typically, the radius of convergence of rational approximation is large compared with the power series. Thus, for the function \n
PA can establish the position of singularities of the function.
TPPA allows to overcome the locality of asymptotic expansions, using only a few terms of asymptotics. Unfortunately, the situations when both asymptotic limits have the form of power expansions are rarely encountered in practice, so we have to resort to other methods of AEFs construction, for example, the method quasirational approximation which is described in [23]. The method of combination (combining method) of asymptotics by using TPPA is alternative to the well-known matching method [6]; it is useful in local domains of transition layers where asymptotics are not uniform. This method was tested on well-known problems of mathematical physics, in particular, problems of fluid dynamics. The main advantage of the method is that it has an analytic form.
\nAn important drawback of asymptotic methods is the local character of solutions obtained [1, 2, 3, 4]. Since the constructed series are often asymptotic, a simple increase in the number of terms does not remove this drawback. Essence of the problem consists of divergence of obtained series. There exist a lot of approaches to these problems [5, 6]. The method of analytic continuation (e.g., the Euler transform or generalized Euler transform [7, 8, 9, 10, 11, 12]) requires a priori information about the singularities of the searched function in the complex domain [4, 9]. These methods are useful if a large number of terms of the series are known. In this case, it is possible to use the Domb-Sykes plot [5, 8]. But usually only a few terms of asymptotic series are known, and to get information from them, the method of Padé approximations (PAs) is useful [1, 2, 5, 13, 14, 15]. PAs yield meromorphic continuations of functions defined by power series and can be used even in cases where analytic continuations are inapplicable. If a PAs converges to the given function, then roots of the denominator tend to points of singularities. One-point PAs give possibilities to improve convergence of series [16, 17, 18, 19, 20]. Two-point PAs (TPPAs) allow matching asymptotics in transition zones and are widely used in mechanics and physics [1, 2, 4, 14, 21, 22, 23, 24]. Overcoming the mentioned limitations of asymptotic methods for practically important problem is the purpose of this chapter. We consider at the beginning (Section 2) the mathematical bases of asymptotic methods and the use of Padé approximants for the summations of the asymptotic series. Section 3 discusses the method of combining of internal and external asymptotics (matching method) by means of Padé approximants. In the Section 4, the methods of solving specific problems of mathematical physics and mechanics of fluid and gas are demonstrated. Section 5 presents a discussion of the obtained.
\nWe suppose that by the result of the asymptotic study, one obtains the following series:
\nAs is known, the radius of convergence \n
Recast the function f in terms of \n
A natural generalization of Euler transformation looks as follows:
\nwhere \n
“The coefficients of the Taylor series in the aggregate have a lot more information about the values of features than its partial sums. It is only necessary to be able to retrieve it, and some of the ways to do this is to construct a Padé approximant” [11]. Padé approximants (PAs) allow us to transform of power series to a fractional-rational function. Let us define PAs, following Baker and Graves-Morris [25].
\nSuppose we are given the power series:
\nPAs can be written as the following expression:
\nwhose coefficients are determined from the condition
\nEquating coefficients near the same powers \n
We note some properties of the PAs [5, 13, 19]. If the PAs at the chosen m and n exists, then it is unique.
1. If the PAs sequence converges to some function, the roots of its denominator tend to the poles of the function. This allows for a sufficiently large number of terms to determine the pole and then perform an analytical continuation.
2. PAs gives meromorphic continuation of a given power series.
3. PAs of the inverse function is treated as the PAs function inverse itself. This property is called duality and is more exactly formulated as follows. Let
\n
4. Diagonal PAs are invariant under fractional linear transformations of the argument. Suppose that the function is given by their expansion (3). Consider the linear fractional transformation that preserves the origin \t\n
5. Diagonal PAs are invariant under fractional linear transformations of functions. Let us analyze a function (3). Let
\n
If \n
\n
provided that there is \n
6. PAs can get the upper and lower bounds for \n
\n
Typically, this estimate is valid for the function itself, that is, \n
7. Diagonal and close to them a sequence of PAs often possesses the property of autocorrection [17, 18]. It consists of the following. To determine the coefficients of the numerator and denominator of PAs, we have to solve systems of linear algebraic equations. This is an ill-posed procedure, so the coefficients of PAs can be determined with large errors. However, these errors in a certain sense are of self-consistent, so the accuracy of PAs is high. This is the radical difference the PAs from the Taylor series, the calculation error of which only increases with increasing number of terms.
Autocorrection property is verified for a number of special functions. At the same time, even for elliptic functions, the so-called Froissart doublets phenomenon arises [26]. Thus, in general, having no information about the location of the poles of the PAs, but relying solely on the very PAs (computed exactly as you wish), we cannot say that you have found a good approximated function. Now consider the question: In what sense the available mathematical results on the convergence of the PAs can facilitate the solution of practical problems? Gonchar’s theorem [16] states: If none of the diagonal PAs \n
This method was originally proposed by Slepyan and Yakovlev for the inversion of the integral transformations. Here is a description of this method, following [26].
\nSuppose that the Laplace transform of a function of a real variable f(t) is
\nTo obtain an approximate expression for the inverse transform, it is necessary to clarify the behavior of the transform to the vicinity of the points s = 0 and s = ∞ and to determine whether the nature and location of its singular points are on the exact boundary of the regularity or near it. Then the transform F(s) is replaced by the function F\n0(s), approximated the exact inversion and satisfying the following conditions:
1. Functions F\n0(s) and F(s) are asymptotically equivalent at s → ∞ and s → 0, that is,
\n
2. Singular points of functions F\n0(s) and F(s), located on the exact boundary of the regularity, coincide.
The free parameters of the function F\n0(s) are chosen so as to satisfy the conditions of the good approximation of F(s) in the sense of minimum relative error for all real values s ≥ 0:
\n\nCondition (8) is achieved by variation of free parameters \n
or \n
Constructed in such a way function F\n0\n(s) is called asymptotically equivalent function for F(s) (AEF). Let’s dwell on the terminology. In the following sections, we will use the symbols of ordinal relations. We will give strict definitions of these concepts.
\nLet’s consider the function f(x). To describe the ordinal relationships with respect to another function φ(x), enter the following definitions:
\nDefinition 1. Let us say that f(x) is a value of order φ(x) at \n
if \n
Definition 2. Let us say that f(x) is a value of order less than φ(x) at \n
if \n
Here A is a finite number, and ε, δ are infinitely small.
\nDefinition 3. Let us say that f (x) is asymptotically equal to φ(x) at \n
Here we use the term “asymptotically equivalent function.” Other terms (“reduced method of matched asymptotic expansions” [28], “quasi-fractional approximants” (QAs) [29], and “mimic function” [30]) are also used.
\nThe analysis of numerous examples confirms “complementarity principle”: if for \n
TPPA is a rational function of the form:
\n\nk coefficients which are determined from the condition
\nand the remaining coefficients from a similar condition for \n
For boundary-value problems, we assume that there exist two asymptotics for limit values of the parameter. In this case, the method of matching of asymptotic expansions is usually used [4]. However, for correct application of the matching method, it is necessary to know the matching point or, at least, the domain of overlapping of asymptotics. An exact description of the transition layer 0 <ε< ∞ exists only in the cases where solutions with different behaviors on opposite sides of the layer can be matched by a special function (e.g., the Airy function).
\nFor the matching of nonoverlapping asymptotics, a method based on TPPAs has recently been developed. In [15, 21, 23], this method was applied for the construction of thermal profiles in a boundary layer of gas. In [2, 6], this method allowed one to examine the heat exchange in hypersonic boundary layers.
\nTwo-point Padé approximations (TPPAs) are defined in Section 3.2 [see formulas (2)–(4)]. As an example of application of TPPAs, we consider the Airy boundary-value problem [4, 10, 31]:
\nwith boundary conditions
\nThis boundary-value problem has the form in terms of Airy function U(s):
\nThe asymptotic solution for problems (13) and (14) has the form:
\nThe interior asymptotic (s → 0) has the form of a power function:
\nThe exterior asymptotic has the form of an exponential function:
\nas \n
The transition layer is defined by the domain, where \n
Airy function approaches with TPPA:
\nThe TPPA (19) preserves three terms of the asymptotics at both ends and provides accuracy with relative error:
\nParameters a and b are obtained from the integral equations (relations). The relations (20) and (21) can be obtained by multiplying Eq. (18) by \n
This is the first integral relation.
\nThis is the next integral relation.
\nSubstituting in Eqs. (20) and (21) instead of U (4) interpolation Ua\n (7), calculate using quadrature integration formulas a = 0.7287 and b = 0.7922.
\nIn the same manner, integral relations with weights U, U’\n can be obtained by part integration. Multiplying Eq. (18) by \n
This is the first integral relation for the second method of producing it:
\nAnd this is the next integral relation for the second method of producing it:
\nUsing Eq. (19), from Eqs. (22) and (23), we calculate а = 0.7277, and b = 0.7966.
\nFrom the given example, it follows that the features of the asymptotic connection method are the ambiguity of the algorithm, the freedom to choose both the form of TPPAs, integral relations, and methods for calculating the parameters of the TPPAs. The question of choosing integral relations is, in fact, a question of controlling the asymptotic approximation using weights selected to obtain integral relations. Choosing the weight allows you to achieve acceptable accuracy in a particular area of the boundary layer: a weight equal to 1 means that the uniform influence of the entire layer is taken into account; a weight equal to \n
In the illustrated example (5), Eq. (18) TPPA represents a modified (quasi-fractional) two-point Padé approximant (10) by an exponential weight function, the choice of which is dictated by a kind of exterior asymptotics. Evidently, the TPPAs are not panacea. For example, one of the “bottlenecks” of the TPPAs method is related to the presence of logarithmic components in numerous asymptotic expansions. This problem is the most essential for the TPPAs, because, as a rule, one of the limits \n
where \n
Consider the Blasius equation (45), which describes laminar boundary layers on a flat plate:
\nwhere \n
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. (25) follows:
\nAfter integration of Eq. (27) by the coordinate ζ follows:
\nAfter reintegration of Eq. (28) by the coordinate
\nfollows:
\nsubject to boundary conditions
\nLet us make a limit transition \n
\n\n
We use the mean theorem in the last equation
\nIn 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
\nwhere \n
If \n
In the external domain, where \n
where \n
To calculate parameter \n
At that, in external domain, \n
Therefore,
\nand
\nType of generalized and normalized TPPA of order (4,4):
\nSpecies of TPPA taking into account four nontrivial parameters:
\nTherefore,
\nParameter 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:
\nTaking into account Eq. (33) in the external domain, we will write down the second local equality:
\nEqualizing the coefficients in Eqs. (36) and (37) at the same degrees ζ, we get
\nTherefore, the TPPA has the form:
\nAfter systems (31)–(33) are solved, we will obtain
\nBy substituting (39) in (38), we get an explicit expression for the TPPA.
\nWe 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:
\nwhere
\n\nM is the Mach number, \n
The boundary conditions at the wall are
\nAt external boundary of layer is
\nInterior asymptotic expansions are for \n
where two constants \n
Exterior asymptotics for \n
where three constants are unknown: c, A, and B.\n
\nWe solve boundary problems (40) and (41) approximately by connecting asymptotics (44) and (45) TPPA
\nBoundary conditions (45) and (46) are satisfied if to put
\nWe complement the last equalities (50) and (51) with a normalizing condition:
\nFollowing the procedure of the previous section, we will calculate the coefficients at ζ and ζ\n2 in asymptotic expansions (44) and, equating them with the corresponding expressions from Eqs. (46) and (47), we will obtain equalities, from which values \n
Three parameters in asymptotics (44) are defined in the outer region if the following condition is met:
\nA priori at large M numbers, it is known that the temperature profile is non-monotonic and has a maximum within the layer at point \n
From the convexity condition of the temperature profile in the vicinity of the point \n
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. (40) and (41), with boundary conditions (42) and (43), by using the technique stated in the previous sections.
\nThe integral relation for parameter A is obtained by multiplying Eq. (40) by
\nand integrating from 0 to ∞ taking into account Eq. (48):
\nSimilarly, from Eq. (41), we get
\nThus, the integral relations (52) and (55)–(47) form a nonlinear system of equations for determining the following parameters:
\nIntegrals 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. (50) and (51).
\nAs an example of TPPA (see Section 3.2) used for matching of limiting asymptotics, consider the paper by Grasman et al. [33]. They dealt with Lyapunov exponents which characterize the dynamics of a system near its attractor. For the Van der Pol oscillator:
\nSimilar to the asymptotic approximation of amplitude and period, expressions are derived for the nonzero Lyapunov exponent \n
The overlap of these series does not take place. The authors of [33] remark: “Such an overlap comes within reach if in the regular expansion a large number of terms is included.” This is not correct, because the obtained series is asymptotic; so, with increasing of number of terms, the results will be worst. 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
\n\nExpression (60) has a pole at \n
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.
\n\n\n | \nL (4) | \nL (NR) | \n
---|---|---|
1 | \n1.057 | \n1.0648 | \n
5 | \n1.513 | \n1.4724 | \n
10 | \n1.685 | \n1.6358 | \n
25 | \n1.759 | \n1.7398 | \n
50 | \n1.768 | \n1.7691 | \n
Comparison for L of numerical results (NR) from paper by [33] with TPPA formulate (60).
In Section 4.4, the problem was solved for several variants of the Mach number and the heating temperature: \n
\nM\n | \n\nT\ns\n | \n\n\n | \n\nT\n1\n | \n
---|---|---|---|
5 | \n3 | \n0.426 | \n1.340 | \n
10 | \n3 | \n0.744 | \n9.127 | \n
10 | \n5 | \n0.637 | \n7.929 | \n
10 | \n7 | \n0.531 | \n6.676 | \n
15 | \n3 | \n0.806 | \n22.00 | \n
15 | \n5 | \n0.756 | \n20.82 | \n
15 | \n7 | \n0.712 | \n19.75 | \n
TPPAs parameters for different Mach numbers M, temperature T\nS, and n = 1 values.
If \n
\nM\n | \n5 | \n10 | \n10 | \n
---|---|---|---|
\nT\ns\n | \n3 | \n3 | \n5 | \n
\na\n2\n | \n0.17 | \n0.74 | \n0.56 | \n
\n\n | \n0.729 | \n0.714 | \n0.798 | \n
\nc\n | \n1.45 | \n1.42 | \n1.38 | \n
\nT\n1\n | \n0.56 | \n8.69 | \n8.16 | \n
TPPAs parameters for different Mach numbers M, temperature T\nS, and n = 0.76 values.
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.
\nThus, 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. For example, the function \n
Typically, the radius of convergence of rational approximation is large compared with the power series. Thus, for the function \n
PA can establish the position of singularities of the function.
TPPA allows to overcome the locality of asymptotic expansions, using only a few terms of asymptotics. Unfortunately, the situations when both asymptotic limits have the form of power expansions are rarely encountered in practice, so we have to resort to other methods of AEFs construction, for example, the method quasirational approximation which is described in [23]. The method of combination (combining method) of asymptotics by using TPPA is alternative to the well-known matching method [6]; it is useful in local domains of transition layers where asymptotics are not uniform. This method was tested on well-known problems of mathematical physics, in particular, problems of fluid dynamics. The main advantage of the method is that it has an analytic form.
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