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Mathematical Problems in Engineering
Volume 2010, Article ID 874540, 23 pages
Research Article

Approximate Ad Hoc Parametric Solutions for Nonlinear First-Order PDEs Governing Two-Dimensional Steady Vector Fields

Department of Engineering Sciences, University of Patras, GR 26504, Greece

Received 20 April 2010; Accepted 3 November 2010

Academic Editor: Oleg V. Gendelman

Copyright © 2010 M. P. Markakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Through a suitable ad hoc assumption, a nonlinear PDE governing a three-dimensional weak, irrotational, steady vector field is reduced to a system of two nonlinear ODEs: the first of which corresponds to the two-dimensional case, while the second involves also the third field component. By using several analytical tools as well as linear approximations based on the weakness of the field, the first equation is transformed to an Abel differential equation which is solved parametrically. Thus, we obtain the two components of the field as explicit functions of a parameter. The derived solution is applied to the two-dimensional small perturbation frictionless flow past solid surfaces with either sinusoidal or parabolic geometry, where the plane velocities are evaluated over the body's surface in the case of a subsonic flow.

1. Introduction

First-order PDEs, which mostly appear in fluid mechanics, describe the motion of ideal as well as of real fluids [13] and govern even the electrostatic plasma oscillation [4]. As is well known, there is no complete general theory concerning the derivation of exact analytical solutions for such equations. However, general solutions can be obtained for the quasilinear forms by means of the subsidiary Lagrange equations ([1, Section 2.6.a], Appendix A). We also mention Charpit’s method for the general nonlinear case that yields to complete and general solutions [1, Section 2.6.b]. These solutions involve arbitrary functions of specific expressions of the dependent and independent variables. Furthermore, appropriate transformations of the dependent and (or) independent variables [1, Section 2.1], combined in several cases with the introduction of auxiliary functions (like stream functions), can occasionally linearize the original equation or more generally reduce it to a solvable form, like a quasilinear one, or even to a nonlinear ODE.

In our previous work [5], four simplified forms of the full two-dimensional nonlinear steady small perturbation equation in fluid mechanics [6] were treated analytically. As far as the three of the considered cases are concerned, closed form solutions have been derived for the two dependent variables of the equation, which represent the dimensionless velocities 𝑢, 𝑣 of a perturbed frictionless flow past a solid body surface, while in the fourth case, a parametric solution was obtained with regard to these velocity resultants. We note that the components 𝑢, 𝑣 are parallel to the 𝑥1, 𝑥2 axes of the Cartesian plane, respectively (see Figure 1 in Section 4, where a wavy surface is represented), with 𝑥1 being the direction of the uniform velocity of the steady flow. The extracted closed form solutions provide 𝑣 as a specific expression of 𝑢, as well as an equation for 𝑢 involving an unknown arbitrary function. The analytical method was based on the introduction of a convenient ad hoc assumption, originally due to Pai [7], by means of which the original (simplified) equations, as well as the irrotanional relation, take a quasilinear form integrated by the Lagrange method. Thus, the above-mentioned solution (including the unknown function) for 𝑢 is obtained, together with an ordinary differential equation, which, after a further analytical treatment, provides the exact or approximate (depending on the case) solutions 𝑣(𝑢). However, it should be mentioned that only in the first, more simplified case [5, Equation (9)] of the general equation, the unknown function can be defined by the use of the boundary condition of the problem, resulting in a transcendental equation for 𝑢 (or 𝑣). Furthermore, no investigation has been performed in [5] with regard to the effectiveness of the obtained formulas (the expressions extracted in the application [5, Section 5] concerning the above-mentioned simplified case and the parametric solutions derived for one of the other examined cases [5, Equation (8)]) to evaluate the perturbed flow field.

Figure 1: Orientation of the perturbed plane field with respect to the body’s surface.

In the present work (Section 2), we firstly treat a steady three-dimensional PDE concerning a general weak irrotational vector field. By taking into account the three irrotationality conditions and using the ad hoc assumption introduced in [5], the Lagrange method (see Appendix A) finally results in a system of two nonlinear ODEs for the two unknown functions introduced by the ad hoc assumption. These functions represent the field’s components 𝑢2, 𝑢3, while the first component 𝑢1=𝑢 stands for the independent variable. In Section 3, we proceed into the integration of the first ODE, which corresponds to the plane problem (𝑢1,𝑢2) (the second involves also 𝑢3). The herein developed methodology consists of a functional transformation of the dependent variable, in combination with an appropriate split of the resulting equation by using an arbitrary function, which eventually is eliminated. By this technique, we finally derive an Abel equation, which admits a parametric solution. Thus, we obtain the field’s components 𝑢1(=𝑢) and 𝑢2 as explicit expressions of a parameter 𝜏. In several steps of the analysis developed in Section 3, the established, in Appendix C (linear), approximations based on the weakness of the field (𝑢11) have been used. Additionally, some limitations imposed by the analysis (see Cases P-1, P-2 in Appendix D) affect the domain of the physical parameter(s) of the problem, for which the extracted solution is valid.

Then in Section 4 we apply the obtained parametric solution in the plane case of the full small perturbation equation, simplified forms of which were investigated in [5]. Here, by combining the extracted parametric formulae with the boundary condition concerning the flow tangential to the solid surface, a transcendental equation is derived, involving 𝜏, 𝜉1, 𝜉2, where 𝜉1, 𝜉2 represent the plane coordinates on the body’s surface. Then, for a given pair (𝜉1,𝜉2), the solution of this equation yields 𝜏(𝜉1,𝜉2), and hence the “surface” perturbed flow velocity field (𝑢1,𝑢2), can be evaluated (the perturbed velocity components 𝑢1, 𝑢2 refer to the 𝑥1, 𝑥2 cartesian plane). Moreover, by expanding in Taylor series and taking into account the small perturbation, the perturbed velocities can be approximately obtained within a thin zone over the surface. In addition, under the mentioned limitations, we deduce that the obtained results hold true for subsonic flows as well.

Finally, by means of the extracted formulas, graphic representations of the perturbed field versus 𝑥1(=𝜉1) are obtained, concerning a sinusoidal as well as a parabolic boundary, and the results are compared to the solution of the linearized equation.

2. The Analytical Procedure

2.1. Transformation of the Governing Equations

Consider an irrotational field 𝑢=(𝑢1,𝑢2,𝑢3) satisfying the following PDE:𝐴0𝑖𝑗+𝐴𝜅𝑖𝑗𝑢𝜅+𝐴𝑖𝑗𝜅𝜆𝑢𝜅𝑢𝜆𝑢𝑖,𝑗+𝐴033+𝐴333𝑢3𝑢3,3=0,𝑖,𝑗,𝜅,𝜆=1,2,(2.1) where summation convention has been adopted and𝐴𝑖𝑗𝜅𝜆=𝐴𝑖𝑗𝜆𝜅𝑢,𝑖,𝑗,𝜅,𝜆=1,2,𝑖,𝑗=𝜕𝑢𝑖𝜕𝑥𝑗,𝑢3,3=𝜕𝑢3𝜕𝑥3,𝑖,𝑗=1,2,(2.2) with (𝑥1,𝑥2,𝑥3) being the Cartesian space coordinates. Equation (2.1) is assumed dimensionless and properly scaled, while the coefficients 𝐴 (with the respective upper and subindexes) represent constants or functions of one or more parameters. In this paper, we investigate the case where𝐴𝑖𝑖12=𝐴𝑖𝑗𝜅𝜅=0,𝑖𝑗,𝑖,𝑗,𝜅=1,2,(2.3a) as well as the case where𝐴2𝑖𝑖=𝐴0𝑖𝑗=𝐴1𝑖𝑗=0,𝑖𝑗,𝑖,𝑗=1,2.(2.3b)However, the proposed solution can also be applied to cases where the coefficients involved in (2.3a) and (2.3b) are sufficiently small, so that the respective terms of (2.1) can be neglected in comparison with the others. Moreover, the field is supposed to be weak in the 𝑥1𝑥2 plane, that is,𝑢𝑖1,𝑖=1,2.(2.4) In fact the approximations (see Appendix C), used in certain steps of the analytical procedure, are based on the weakness of the field under consideration.

As a first step, we make the ad hoc assumption that the components 𝑢2 and 𝑢3 are functions of the component 𝑢1, namely,𝑢𝑖=𝑓𝑖𝑢1,𝑖=1,2,3,(2.5) and thus by substituting (2.5), (2.1) (taking into account (2.3a) and (2.3b)) becomes𝑅1(𝑢)𝑢,1+𝑅2(𝑢)𝑢,2+𝑅3(𝑢)𝑢,3=0,(2.6) where 𝑢1 has been replaced by 𝑢 and𝑅1(𝑢)=𝐴011+𝐴111𝑢+𝐴1111𝑢2+𝐴1122𝑓22+𝐴221𝑓2+2𝐴2112𝑢𝑓2𝑓2,(2.7a)𝑅2(𝑢)=𝐴212𝑓2+2𝐴1212𝑢𝑓2+𝐴022+𝐴122𝑢+𝐴2211𝑢2+𝐴2222𝑓22𝑓2,(2.7b)𝑅3𝐴(𝑢)=033+𝐴333𝑓3𝑓3.(2.7c)Here, the prime “ ' ” denotes differentiation with respect to 𝑢(𝑓𝑖(𝑢),𝑖=2,3).

On the other hand, the irrotational condition of the field is written in the form×𝑢=𝜖𝑘𝑗𝑖𝜕𝑢𝑖𝜕𝑥𝑗𝑒𝑘=0,𝑖,𝑗,𝑘=1,2,3,(2.8) where 𝜖𝑘𝑗𝑖 is the well-known Levi-Civita tensor and 𝑒𝑘 represent the unit vectors corresponding to 𝑥𝑘, 𝑘=1,2,3, respectively. By substituting the assumption (2.5) into (2.8), we arrive at the following three equations (𝑢1 is replaced by 𝑢): 𝑓3𝑢,2𝑓2𝑢,3=0,(2.9a)𝑢,3𝑓3𝑢,2=0,(2.9b)𝑓2𝑢,1𝑢,2=0.(2.9c)

With respect to the physical relevance of (2.1), as well as of the constraints imposed above, we note the following. No “mixed” nonlinear terms involving the plane components 𝑢1, 𝑢2 together with 𝑢3 are included in (2.1). Furthermore the restrictions (2.3a) and (2.3b) focus on cases where specific nonlinear terms are involved into the governing equation. More precisely, the procedure developed in this paper confronts nonlinear equations where the partial derivatives of the field components appear in products together with specific combinations of these components, of the first and the second degree. Indeed by (2.3a) and (2.3b), it is obvious that two groups of nonlinear terms are formed with respect to the variations of the plane components 𝑢1, 𝑢2, along their own axes (𝑢𝑖,𝑖) and the other axis (𝑢𝑖,𝑗, 𝑖𝑗). This can be clearly observed in the two-dimensional steady small perturbation equation of fluid mechanics, treated in Section 4 (4.1) as an application of the present analysis.

All the above notations, as well as the ad hoc assumption (2.5), outline a normalized structure as regards the behavior of the field in phase space, due to a regulated physical setup. In fact the small perturbation (4.1) is representative of the imposed restrictions, since the origin of the field (the perturbed velocities due to slight “geometric perturbations” of the body’s surface) combined with the orientation of the uniform flow (with reference to the body—see Figure 1 in Section 4) can give rise to the specific nonlinear form of the governing equations (2.1), (2.3a), and (2.3b), as well as to the “weakness” and the ad hoc assumptions, (2.4) and (2.5), respectively.

2.2. Construction of Intermediate Integrals

Now, by integrating the correspondent to (2.6), (2.9a), (2.9b), and (2.9c) subsidiary Lagrange equations (see Appendix A), we, respectively, obtain the following general solutions:𝑥(2.6)𝑢=𝐺1𝑅1(𝑢)𝑅2𝑥(𝑢)2,𝑥2𝑅2(𝑢)𝑅3𝑥(𝑢)3,(2.10)(2.9a)𝑢=𝐺1𝑥1,𝑥2+𝑓3𝑓2𝑥3,(2.11a)(2.9b)𝑢=𝐺2𝑥2,𝑥1+𝑓3𝑥3,(2.11b)(2.9c)𝑢=𝐺3𝑥3,𝑥1+𝑓2𝑥2,(2.11c)where 𝐺, 𝐺1, 𝐺2, and 𝐺3 are arbitrary functions possessing continuous partial derivatives with respect to their arguments.

2.3. Reduction to a System of Nonlinear ODEs

In view of (2.10), (2.11a),(2.11b), and (2.11c), we construct a first set of relations by equating identically the functions 𝐺, 𝐺1, 𝐺2, and 𝐺3 as well as their arguments. Thus, excluding the cases where in the extracted equations:𝑥1=0,𝑥2=0,𝑥3𝑥=0,1=𝑥2,𝑥2=𝑥3,𝑥1=𝑥3,(2.12) we eventually obtain the following systems.

Case 1 (𝐺2𝐺3). We have 𝑓2𝑥=11𝑥2,𝑓3𝑥=11𝑥3.(2.13)

Case 2 (𝐺𝐺1). We have 𝑅1𝑅2=𝑥1𝑥2𝑓13𝑓2𝑥3𝑥2,𝑅2𝑅3=𝑥2𝑥3𝑥1𝑥3.(2.14)

Case 3 (𝐺𝐺2). We have 𝑅1𝑅2=𝑥1𝑥2𝑅1,2𝑅3=𝑥2𝑥3𝑥1𝑥3𝑓3.(2.15)

Case 4 (𝐺𝐺3). Subcase 1 (𝐺𝐺3). We have 𝑅1𝑅2=𝑥1𝑥2𝑥3𝑥2,𝑅2𝑅3=𝑥2𝑥3𝑥1𝑥3𝑓2𝑥2𝑥3.(2.16)Subcase 2 (𝐺𝐺3). We have 𝑅1𝑅2=𝑓2𝑅,2𝑅3=𝑥2𝑥31.(2.17)

Subcases 1 and 2 are, respectively, derived by equating the arguments of 𝐺 and 𝐺3 in two possible combinations. Then, in order to obtain a system of equations not containing explicitly 𝑥1, 𝑥2, and 𝑥3, we find that Cases 1, 3 and Subcase 2 are compatible to each other. Thus by combining their respective equations, we derive the following ODEs:𝑅2(𝑢)𝑓2(𝑢)+𝑅1(𝑢)=0,(2.18)𝑅3(𝑢)𝑓3(𝑢)𝑅3(𝑢)𝑓2(𝑢)+𝑅1(𝑢)+𝑅2(𝑢)=0.(2.19) Taking into account (2.7a), (2.7b), and (2.7c), we note that (2.18) contains only 𝑓2 and 𝑓2, and thus it constitutes the main equation, the manipulation of which is presented in the next section.

Therefore, the ordinary differential equations (2.18) and (2.19) represent the reduced forms of the partial differential equations (2.6), (2.9a), (2.9b), and (2.9c), via assumption (2.5). Then by substituting (2.7a), (2.7b)) and replacing 𝑓2 with 𝑦 and 𝑢 with 𝑥, (2.18) becomes𝑦𝑥2+𝜌22(𝑥)𝑦2𝑦𝑥2+𝜌11(𝑥)𝑦𝑦𝑥+𝜌20(𝑥)𝑦2=𝜔(𝑥),(2.20) where 𝑦𝑥 denotes the derivative of 𝑦(𝑥) with respect to 𝑥 and𝜌22𝛼(𝑥)=𝑃(𝑥),𝜌11𝐴(𝑥)=2+𝐴3𝑥𝑃(𝑥),𝜌20𝛽(𝑥)=𝑃𝑃(𝑥),𝜔(𝑥)=1(𝑥),𝑃(𝑥)(2.21) with𝛼=𝐴2222,𝛽=𝐴1122,(2.22a)𝑃(𝑥)=𝐴022+𝐴122𝑥+𝐴2211𝑥2,𝑃1(𝑥)=𝐴011𝐴111𝑥𝐴1111𝑥2.(2.22b)We note that 𝐴2 and 𝐴3 as well as all the other coefficients appearing in the next sessions are listed in Appendix E. Henceforth, the prime will denote differentiation with respect to the corresponding suffix.

3. Integration of (2.20)

3.1. Transformation of (2.20)

Introducing transformation𝑦[𝜉]𝑓(𝑥)=(𝑥)(𝑥),(3.1) the left hand side of (2.20) results in a nonlinear expression involving , 𝜉, 𝜉𝑥, 𝑓, and 𝑓𝑥. Thus, by taking into account this expression and setting𝑓(𝑥)=exp𝜅(𝑥)2𝜌,𝜅(𝑥)=11(𝑥)𝑑𝑥,(3.2)𝜌𝜉(𝑥)=31/2(𝑥)𝑑𝑥,𝜌3𝜌(𝑥)=2114𝜌20,(3.3) with 𝜌11, 𝜌20 as in (2.21), (2.20) takes the form 𝜉2+𝜌22𝑓22𝜉2𝜌22𝜌11𝑓2𝜌31/23𝜉1+4𝜌22𝜌211𝑓2𝜌3142=𝜔𝑓2𝜌3.(3.4) Then, by substituting2(𝜉)=𝑠(𝜉),(3.5) (3.4) becomes𝑠𝜉2𝑠+𝜌22𝑓2𝑠𝜉22𝜌22𝜌11𝑓2𝜌31/2𝑠𝑠𝜉𝜌22𝜌211𝑓2𝜌31𝑠2+4𝑠+4𝜔𝑓2𝜌3=0.(3.6) In addition, by substitution of (2.21) and (2.22b) into (3.3), we obtain𝜌3𝑃(𝑥)=2(𝑥)4𝑃2(𝑥),𝑃2(𝑥)=𝐴4+𝐴5𝑥+𝐴6𝑥2,(3.7) with 𝑃(𝑥) as in (2.22b).

3.2. The Split of (3.6)

We now split (3.6) into the following system of equations: 𝑠𝜉2𝑠+𝜌22𝑓2𝑠𝜉2=𝐹(𝜉),(3.8a)2𝜌22𝜌11𝑓2𝜌31/2𝑠𝑠𝜉𝜌22𝜌211𝑓2𝜌31𝑠2+4𝑠+4𝜔𝑓2𝜌3=𝐹(𝜉),(3.8b)where 𝐹(𝜉) is an unknown arbitrary function. Furthermore, after dividing (3.8b) by 𝑓2 and setting𝐹(𝜉)=4𝜔(𝑥)𝑓2(𝑥)𝜌3(𝑥)𝐺(𝜉),𝑥=𝑥(𝜉),(3.9)(3.8b) will be written as2𝜌22(𝑥)𝜌11(𝑥)𝜌31/2(𝑥)𝑠𝑠𝜉=𝜌22(𝑥)𝜌211(𝑥)𝜌31(𝑥)𝑠24𝑓2(𝑥)𝑠+4𝜔(𝑥)𝑓4(𝑥)𝜌3(𝑥)𝐺(𝜉)1,𝑥=𝑥(𝜉),(3.10) where 𝐺(𝜉) represents now the unknown arbitrary function. We see that (3.10) is an Abel equation of the second kind, and thus by following the analysis presented in [8, Chapter 1, Section 3.4] and taking into account (3.2) and (3.3), it is reduced to a simpler Abel equation, namely,𝑧𝑧𝑡𝑧=𝜔(𝑥)𝑓(𝑥)𝜌3(𝑥)[]𝐺(𝜉)1,𝑥=𝑥𝜉(𝑡),𝜉=𝜉(𝑡)(3.11) with𝑠(𝜉)=𝑧[]𝑡(𝜉)𝑓[]𝜌𝑥(𝜉),𝑡(𝜉)=231/2(𝑥)𝑑𝜉𝜌22(𝑥)𝜌11,(𝑥)𝑓(𝑥)(3.12) now, by differentiating 𝑠, given by (3.12), with respect to 𝜉 and using (3.2), (3.3) (we consider the appropriate domains where 𝜉(𝑥) is invertible and hence 𝑥𝜉=𝜉𝑥1) as well as the expression of 𝑡 provided by (3.12), we obtain 𝑠𝜉, substitution of which into the left-hand side of (3.8a) results in1𝑓(𝑥)𝑧(𝑡)+𝜌22(𝑥)𝑧𝑡𝜌22(𝑥)𝜌211(𝑥)𝑓(𝑥)4𝜌3(𝑥)𝑧(𝑡)2=𝜌222(𝑥)𝜌211(𝑥)𝜔(𝑥)𝜌23(𝑥)𝐺(𝜉),(3.13) where (3.9) has been substituted for 𝐹(𝜉).

Equations (3.11) and (3.13) form a new system equivalent to that of (3.8a), (3.8b), obtained by splitting (3.6). The elimination of the arbitrary function 𝐺 yields a nonlinear ODE, which represents the reduced form of (3.6). More precisely, after some algebra we extract31𝑧+𝜌22𝑓𝑧𝑡𝜌3𝜌222𝜌211𝑓1𝑓𝑧+𝜌22𝑧𝑡2=𝜌22𝜌211𝑓8𝜌3𝑧2+𝜌2+2118𝜌3𝑧2𝜔𝜌3𝑓.(3.14) Furthermore, as far as the 𝑧𝑡2-term is concerned, combination of (3.12) with (3.1) and (3.5) yields 𝑧(𝑡)=𝑦2(𝑥)/𝑓(𝑥), 𝑥=𝑥[𝜉(𝑡)]. By differentiating with respect to 𝑡 and taking into account certain relations obtained above, as well as that 𝑥, 𝑦, and 𝑦𝑥 represent 𝑢,𝑢2=𝑓2(𝑢) and 𝑓2(𝑢), respectively, we conclude that 𝑧𝑡 is equal to (𝛼+𝛽𝑢)(𝑢2𝑓2+𝛾𝑢22+𝛿𝑢𝑢22), where 𝛼,𝛽,𝛾,𝛿 represent expressions of the equation’s coefficients. Therefore, when the plane field’s components, as well as the variation of 𝑢2 with respect to 𝑢, are very small compared with the unit (e.g., if they denote perturbed components in a small perturbation theory), we can perfectly consider 𝑧𝑡1,(3.15) and hence we can neglect the 𝑧𝑡2 term in the left-hand side of (3.14) in comparison with the others, as it is of 𝒪[max{𝑢42,𝑢22𝑓22}]. We should note here that in our previous work [5], after following a different analysis concerning two simplified forms of the full equation, an analogous to (3.15), but weaker approximation, has been applied, since the neglected term was of 𝒪(𝑢42/(4𝑢2)), yielding less accurate results compared to the obtained herein solution of (3.14) especially when 𝑢 takes smaller values than 𝑢2. Moreover by means of (3.3) and (3.12), we have[]𝜌𝑡(𝑥)=𝑡𝜉(𝑥)=23(𝑥)𝑑𝑥𝜌22(𝑥)𝜌11.(𝑥)𝑓(𝑥)(3.16) Thus, by writing𝑧(𝑥)=𝑧,𝑡(𝑥)(3.17) neglecting the 𝑧𝑡2 term and multiplying with 𝑡𝑥, then by using (3.16), (3.14) becomes13𝑧+𝜌22𝑧(𝑥)𝑓(𝑥)𝑥𝜌=11(𝑥)4𝑧216𝜌3(𝑥)+𝜌211(𝑥)4𝜌22(𝑥)𝜌11(𝑥)𝑓(𝑥)𝑧+4𝜔(𝑥)𝜌22(𝑥)𝜌11(𝑥)𝑓2.(𝑥)(3.18) The above equation is also an Abel equation of the second kind and thus we proceed as in [8, Chapter 1, Section 3.4]. More precisely, by using the formulas (D.10a), (D.10b) (see Appendix D), after some algebra, we arrive at𝑞𝑞𝑟2𝑞=𝐹3𝛼00+𝐹01𝑥𝑥+𝒪2𝑀0+𝑀1𝑥𝑥+𝒪2,(3.19) where𝐹𝑟(𝑥)=10+𝐹11𝑥𝑥+𝒪2𝑄0+𝑄1𝑥𝑥+𝒪2𝑑𝑥(3.20) and 𝛼 being as in (2.22a). Now, by applying (C.4) (Appendix C) to both rational functions in the right-hand sides of (3.19) and (3.20), we obtain𝑞𝑞𝑟𝑞=𝐵2+𝐵3𝑥,(3.21)𝐵𝑟(𝑥)=0+𝐵1𝑥𝑑𝑥=𝐵0𝑥𝑥+𝒪2.(3.22) Finally, substitution of 𝑥=𝑟/𝐵0 into (3.21) yields𝑞𝑞𝑟𝑞=𝐵2+𝐵3𝑟.(3.23) Moreover, by the followed procedure (see [8]), we have that𝑞[]𝑧(𝑥)=𝑟(𝑥)𝐸𝑃(𝑥)3𝛼𝑓(𝑥)(3.24) with 𝑃(𝑥) as in (2.22b) and 𝐸 given by (D.10b) (see Appendix D). Finally, the Abel equation (3.23) is solved parametrically (Appendix B, formulas (B.7)) as𝐶𝑟=𝐵3𝜏Γ1/2(𝜏)𝑒𝐼(𝜏)/2𝐵2𝐵3,𝑞=𝐶Γ1/2(𝜏)𝑒𝐼(𝜏)/2,(3.25) whereΓ(𝜏)=𝜏2+𝜏𝐵3,𝐼(𝜏)=𝑑𝜏.Γ(𝜏)(3.26) In the above relations 𝜏 represents the parameter while 𝐶 is an arbitrary constant. Now, by substituting Γ1/2(𝜏)𝑒𝐼(𝜏)/2=Ω(𝜏)(3.27) and taking into account (3.22), the above parametric solution takes the form𝑥=𝐵4+𝐵5𝐶𝜏Ω(𝜏),𝑞=𝐶Ω(𝜏).(3.28) All the coefficients appearing through the analysis are listed in Appendix E.

3.3. The Parametric Solution for the Field’s Components 𝑢1, 𝑢2

By combining (3.1), (3.5), (3.12), (3.17), and (3.24), we obtain𝑦2𝑞[](𝑥)=𝑟(𝑥)𝑓(𝑥)𝐸𝑃(𝑥).3𝛼(3.29) Approximating linearly 𝑃(𝑥), namely,𝑃(𝑥)=𝐴022+𝐴122𝑥𝑥+𝒪2(3.30) and substituting 𝑞 from (3.28), as well as 𝑓 and 𝐸 from (D.10a), (D.10b) (Appendix D), then (3.29) yields3𝛼𝑦2=𝐵6𝐵+𝐶𝜏Ω7+𝐵8/𝜏+𝐶2𝜏2Ω2𝐵9+𝐵10/𝜏+𝐶3𝜏3Ω3𝐵11+𝐵12/𝜏+𝐵13𝐶4𝜏4Ω4𝑐0+𝑐1𝐶𝜏Ω+𝑐2𝐶2𝜏2Ω2.(3.31) Furthermore, by solving the first part of (3.28) for 𝐶𝜏Ω(𝜏), we have𝐶𝜏Ω(𝜏)=𝑏0+𝑏1𝑥.(3.32) Now, approximating linearly the powers of 𝐶𝜏Ω(𝜏) involved into (3.31), that is𝐶2𝜏2Ω2=𝑏20+2𝑏0𝑏1𝑥𝑥+𝒪2,𝐶3𝜏3Ω3=𝑏30+3𝑏20𝑏1𝑥𝑥+𝒪2,𝐶4𝜏4Ω4=𝑏40+4𝑏30𝑏1𝑥𝑥+𝒪2(3.33) and substituting (3.32) and (3.33) into (3.31), then by replacing 𝑥 with 𝑢=𝑢1 and 𝑦 with 𝑢2=𝑓2(𝑢) and taking also into account (3.28), we conclude that𝑢𝑥1,𝑥2,𝑥3=𝜙1(𝜏)=𝐵4+𝐵5𝐶𝜏Ω(𝜏),(3.34a)𝑢22𝑥1,𝑥2,𝑥3=𝜙221(𝜏,𝑢)=𝑏3𝛼2+𝑏3𝑏𝑢+4+𝑏5𝑢(1/𝜏)𝑐3+𝑐4𝑢,(3.34b)with Ω as in (3.27) and 𝛼 given by (2.22a). Equations (3.34a), (3.34b) constitute the approximate analytical parametric solution of the problem for 𝑢1, 𝑢2. As far as the component 𝑢3 is concerned, combination of (2.18) and (2.19) results in𝑅3𝑓2+𝑅2𝑓21𝑅3𝑓3=0,𝑢3=𝑓3(𝑢).(3.35) The above equation can be simplified a little if we neglect the last term in the left-hand side (it is of the form (𝑎+𝑏𝑓3)𝑓32) by considering 𝑓31. Anyhow we will not investigate (3.35) in this work.

Moreover, in order to evaluate the constant 𝐶 involved into the parametric solution (3.34a), (3.34b), we need a boundary condition, that is, to locate to a point 𝑥0=(𝑥10,𝑥20,𝑥30) where the field components 𝑢0=𝑢(𝑥0), 𝑢20=𝑢2(𝑥0) are known. Then, by solving (3.34b)) for 𝜏, we extract the corresponding value of the parameter 𝜏0=𝜏(𝑥0), and finally, by using (3.34a), we arrive at𝑢𝐶=0𝐵4𝐵5𝜏0Ω𝜏0.(3.36)

In the next section we apply the derived solution in the two-dimensional case of a flow past bodies with specific boundaries.

4. Parametric Solution for a 2-D Flow

As an application of the parametric solution obtained above for the plane case of (2.1), we consider the full nonlinear PDE governing the two-dimensional (𝑢3=0, 𝑥3=0) steady small perturbation frictionless flow past a solid body surface [6], namely,1𝑀2(𝛾+1)𝑀2𝑢112(𝛾+1)𝑀2𝑢2112(𝛾1)𝑀2𝑢22𝑢1,1+1(𝛾1)𝑀2𝑢112(𝛾1)𝑀2𝑢2112(𝛾+1)𝑀2𝑢22𝑢2,2𝑀2𝑢2+𝑢1𝑢2𝑢1,2+𝑢2,1=0,(4.1) where 𝑀 is the correspondent to the uniform flow Mach number, which stands for the physical parameter of the problem, and 𝛾 is the ratio of the specific heats usually taken equal to 1.4; hence, the respective (dimensionless) coefficients 𝐴0𝑖𝑗, 𝐴𝜅𝑖𝑗 and 𝐴𝑖𝑗𝜅𝜆 of (2.1) (the 𝑢3,3 term vanishes) are given by𝐴011=1𝑀2,𝐴111=2.4𝑀2,𝐴1111=1.2𝑀2,𝐴1122=0.2𝑀2,𝐴212=𝐴221=𝑀2,𝐴1212=𝐴2112𝑀=22,𝐴022=1,𝐴122=0.4𝑀2,𝐴2211=0.2𝑀2,𝐴2222=1.2𝑀2.(4.2) Relations (2.3a), (2.3b) also hold true. As mentioned in Section 2, the above equation represents a highly appropriate case, where the physical relevance of the imposed constraints (2.3a)-(2.3b)–(2.5) can be explained by a normalized physical background like the one generated by a uniform flow passing over a slightly “perturbed” surface, according to a specific geometry (see Figure 1, and the applications at the end of this section). Here, 𝑢1, 𝑢2 represent the dimensionless perturbation velocity components along the 𝑥1, 𝑥2 axes (see Figure 1), normalized by the uniform velocity of the steady flow, which is parallel to the 𝑥1 direction in the physical plane.

A wavy surface (projection in the 𝑥1𝑥2 plane) is shown in Figure 1, as a representative case able to produce small plane perturbations in the velocity field (the surface is supposed to have very small amplitude). Moreover, the irrotationality condition (2.9c) holds true and (2.10) and (2.11c) become𝑥𝑢=𝐺1𝑅1(𝑢)𝑅2𝑥(𝑢)2,(4.3)𝑢=𝐺3𝑥1+𝑓2𝑥2,(4.4) respectively, where 𝐺, 𝐺3 denote arbitrary functions and 𝑅1, 𝑅2 are as in (2.7a),(2.7b)) (we mention that 𝑢=𝑢1). Obviously in the two-dimensional case, (2.9a),(2.9b)) and (2.11a),(2.11b)) become identities. Comparison between (4.3) and (4.4) results in (2.18).

If we refer now to the proper conditions restricted by the analysis (see Appendix D), we extract that the discriminant Δ of 𝑃(𝑢) (𝑥 has been replaced by 𝑢) is always positive (Δ>0), and, moreover, since 𝐴2211<0, by obtaining the roots of 𝑃(𝑢), considering the respective to the Cases P-1 and P-2 intervals for 𝑢 and assuming 𝑢0.1(𝑢1) as well, then a restriction to the domain of 𝑀 is derived. More precisely, we find that formulae (3.34a), (3.34b) are valid for𝑀0.71𝑢=101,0.74𝑢=5×102,0.78𝑢=102,0.79𝑢=104.(4.5) Thus, for the specific 2-D steady flow field, the obtained approximate solution can be applied only to subsonic flows. Moreover, as far as the integral 𝐼(𝜏)=𝑑𝜏/Γ(𝜏), involved into the function Ω(𝜏), is concerned (see (3.26), (3.27)), the discriminant 𝛿 of Γ(𝜏) is evaluated negative (𝛿<0), and therefore the integral 𝐼 is obtained as𝐼(𝜏)21=𝛿arctan1+2𝜏.𝛿(4.6)

Now, in order to construct an appropriate procedure to obtain 𝜏(𝑥1,𝑥2), we consider the well-known boundary condition (see [6, page 208])𝑢𝜙=0,(4.7) where 𝑢=(1+𝑢(𝜉1,𝜉2),𝑢2(𝜉1,𝜉2)) is the total dimensionless velocity vector of the flow at the solid surface, while 𝜙(𝜉1,𝜉2)=0 represents the equation of the “surface line”, that is, the section of the body’s surface with the 𝑥1𝑥2 plane. Here, 𝜉1, 𝜉2 denote the plane coordinates on this line with 𝜉1[0,𝐿], 𝐿 being the body’s length, and |𝜉2|1. Condition (4.7) states that at the surface of the body the direction of the flow must be tangential to the surface line. Developing (4.7), we arrive at𝜉1+𝑢1,𝜉2𝜙,𝜉1+𝑢2𝜉1,𝜉2𝜙,𝜉2=0,(4.8) where by neglecting 𝑢(𝑢1), we obtain𝑢2𝜉1,𝜉2𝜙=,𝜉1𝜙,𝜉2=𝑑𝜉2𝑑𝜉1𝜉=𝑔1,𝜉2.(4.9) By squaring (4.9) and substituting 𝑢2 and 𝑢 by their parametric expressions (3.34b),(3.34a), we derive a transcendental equation for 𝜏, namely,𝜙22𝜏,𝜙1(𝜏)=𝑔2𝜉1,𝜉2.(4.10) Thus, for a given pair (𝜉1,𝜉2) on the surface line, the solution of (4.10) results in 𝜏(𝜉1,𝜉2), substitution of which into (3.34a), (3.34b) yields the perturbed velocity vector (𝑢1,𝑢2) of the flow at (𝜉1,𝜉2). In fact, in the case of the flow under consideration, only the perturbed velocity 𝑢 is evaluated by, use of the extracted parametric solution, since due to (4.9) 𝑢2 simply expresses approximately the slope of the surface line. Furthermore, assuming that the functions 𝑢𝑖(𝑥1,𝑥2), 𝑖=1,2 are analytic inside a domain located on any line 𝑥1(=𝜉1)= constant with 𝑥1[0,𝐿] (𝐿 represents the body’s length) and 𝑥2, slightly different from 𝜉2(𝑥2>𝜉2), by developing in Taylor series around (𝜉1,𝜉2), we have𝑢𝑖𝜉1,𝑥2=𝑢𝑖𝜉1,𝜉2+𝜕𝑢𝑖𝜕𝑥2𝑥2𝜉2+12𝜕2𝑢𝑖𝜕𝑥22𝑥2𝜉22+,𝑖=1,2.(4.11) Taking into account the small perturbation theory (the derivatives involved into the series (4.11), as well as 𝜉2, are very small compared to unity) and also that 𝑥2 lies close enough to 𝜉2, so that (𝑥2𝜉2)1, all the terms after the first in the right-hand side of (4.11) can be neglected. Thus, we can approximately evaluate the perturbed plane flow field inside a thin zone over the body’s surface. Obviously, the thickness of this zone depends on the order of magnitude of 𝜉2. For example, if the boundary has a sinusoidal shape (one of the cases considered below), that is, 𝜉2=𝑎sin(𝑏𝜉1), 𝜉1[0,𝐿], 𝑎,𝑏(0,1) and if we take 𝑎=0.05, then within the domain {(𝑥1,𝑥2)𝑥1[0,𝐿],𝑥2(𝜉2,𝜉2+2𝑎)} (a plane zone of thickness 2𝑎 (measured in the 𝑥2-direction) with parallel sinusoidal boundaries), the error in (4.11) is 𝒪(𝑥2𝜉2)102. Therefore, the above approximation is valid inside a zone over the solid surface of thickness less or equal to 2𝑎(=0.1). In addition, in order to obtain 𝜏0=𝜏(𝑥10,𝑥20) and 𝐶 (see the end of Section 3), the axes origin is used which is located at the point where the flow arrives at the body surface and consequently (𝑥10,𝑥20)=(0,0), (𝑢0,𝑢20)=(0,𝑔(0,0))(𝑢0=𝑢(0,0), 𝑢20=𝑢2(0,0)), where 𝑔 is given by (4.9). Therefore, by means of (3.34b)) and (3.36) we conclude that𝜏(0,0)=𝜏0=𝑏43𝛼𝑐3𝑔2(0,0)𝑏2𝐵,𝐶=4𝐵5𝜏0Ω𝜏0(4.12) with Ω provided by (3.26) and (3.27), where the integral 𝐼 is evaluated by (4.6).

The derived solution, constructed by relations (3.34a), (3.34b), and (4.10), is applied to the two-dimensional steady frictionless flow past a boundary of sinusoidal (wavy wall), as well as of a parabolic shape. The problem is governed by (4.1). Especially for the “sinusoidal” boundary problem, implicit solutions in the form of transcendental equations have been extracted in [5, Section 5, Equation (72) and (74), (77)], for the more simplified case of (2.1), where the only nonzero coefficients were 𝐴011, 𝐴111, and 𝐴022 [5, Equation (9)]. Here, boundary condition (4.7) holds with𝜙𝑠𝜉1,𝜉2=𝜉2𝑎𝑠sin𝑏𝜉1=0,𝑎𝑠,𝑏(0,1),(4.13a)𝜙𝑝𝜉1,𝜉2=𝑎𝑝𝜉2+𝜉202𝜉1𝜉10=0,𝑎𝑝,𝜉10,𝜉2>0,𝜉20=𝜉10/𝑎𝑝1/2,(4.13b)where 𝜙𝑠 and 𝜙𝑝 describe the sinusoidal and parabolic form of the surface, respectively, while 𝑎𝑠 and 𝑎𝑝 denote the amplitude and the curvature of the surface line in the cases under consideration. The low magnitude of 𝑎𝑠 and the large magnitude of 𝑎𝑝 allow the small perturbation theory to be applied. Additionally, in both (4.13a), (4.13b), we have 𝜉1(=𝑥1)[0,𝐿], where 𝐿 stands for the assumed body’s length, while in the “sinusoidal” case, the wavelength of the wavy surface is equal to 2𝜋/𝑏.

As far as the graphs exhibited below are concerned, the “dashed” line represents the sinusoidal or the parabolic boundary, with geometries: 𝑎𝑠=0.05, 𝑏=0.5 (Figures 2(a), 2(c)—(4.13a)) and 𝑎𝑝=5×103, 𝜉10=10 (Figures 3(a), 3(b), (4.13b)). Moreover, the solid blue line in Figures 2(b) and 3(a) has been obtained as the solution for 𝑢1 of the linearized form of (4.1), where the slope of the solid surface has been substituted for the component 𝑢2. In both geometries, the body’s length 𝐿 is taken equal to 12𝜋 (three wavelengths in the wavy case) and the correspondent to the uniform unperturbed flow Mach number is set equal to 0.7. We note that by changing the values of the geometric parameters involved in (4.13a) and (4.13b), as well as the value of the Mach number, the perturbed field presents qualitatively similar graphs to those obtained here. Finally, as mentioned above the perturbed velocity 𝑢2 is obtained as the slope of the surface.

Figure 2: (a) 𝑢1(×102) versus 𝑥1, 𝑀=0.7, sinusoidal shape: 𝑎𝑠=0.05, 𝑏=0.5; (b) 𝑢1(×102) versus 𝑥1, 𝑀=0.7, sinusoidal shape: 𝑎𝑠=0.05, 𝑏=0.5, Ad hoc parametric solution (black line)—linearized equation (blue line); (c) 𝑢2 versus 𝑥1, 𝑀=0.7, sinusoidal shape: 𝑎𝑠=0.05, 𝑏=0.5.
Figure 3: Solid lines: ad hoc parametric solution (black)—linearized equation (blue). (a) 𝑢1(×104) versus 𝑥1, 𝑀=0.7, parabolic shape: 𝑎𝑝=5×103, 𝜉10=10; (b) 𝑢2(×10) versus 𝑥1, 𝑀=0.7, parabolic shape: 𝑎𝑝=5×103, 𝜉10=10.

In Figure 2(b), we note that the linear approximation is excellent throughout the body’s length except in small intervals centered at the picks of the sinusoidal surface with radius approximately equal to 𝜋/6(((2𝑘+1)𝜋𝜋/6,(2𝑘+1)𝜋+𝜋/6),𝑘=0,1,). Outside these locations the maximum error of the linear approximation (with respect to the ad hoc solution) is approximately equal to 6×105, while inside these intervals the difference between the two solutions increases with 𝑥1 moving towards the pick. Furthermore, concerning the comparison of the solutions in the case of the parabolic surface (Figure 3(a)), we find that for the considered body’s length, the maximum error of the linear approximation is approximately equal to 1.5×106 (the error increasing with 𝑥1).

5. Summary and Conclusion

In this paper an ad hoc analytical parametric solution has been obtained, concerning a nonlinear PDE governing a two-dimensional steady irrotational vector field. However, in Section 2 of this work the three-dimensional case is treated. As a result, we obtain a system of two (nonlinear) ODEs being equivalent to that of the original PDEs (including the irrotationality conditions). The analytical tools have been used in order to integrate the first ODE (concerning the two-dimensional case), in combination with linear approximations of certain polynomial and rational expressions, succeeded in transforming the above equation to a parametrically solvable Abel form. In particular, as established in Section 3, the “splitting” technique proved excellent in manipulating and transforming strongly nonlinear ODEs to integrable equations, and hence it may be considered representative of the general pattern of the analysis. Thus, we believe that the developed methodology, possibly modified, extended and enriched with more analytical techniques, can be a powerful tool of research on nonlinear problems in mechanics and physics.


A. Lagrange Method for Quasilinear PDEs of First Order

According to this method, a general solution of the quasilinear equation𝐻1𝑥1,𝑥2,𝑥3,𝑢𝑢,1+𝐻2𝑥1,𝑥2,𝑥3,𝑢𝑢,2+𝐻3𝑥1,𝑥2,𝑥3,𝑢𝑢,3𝑥=𝑅1,𝑥2,𝑥3,𝑥,𝑢𝑢=𝑢1,𝑥2,𝑥3,𝑢,𝑖=𝜕𝑢𝜕𝑥𝑖,𝑖=1,2,3(A.1) has the form𝐺𝑤1,𝑤2,𝑤3=0,(A.2) where𝑤1𝑥1,𝑥2,𝑥3,𝑢=𝑎,𝑤2𝑥1,𝑥2,𝑥3,𝑢=𝑏,𝑤3𝑥1,𝑥2,𝑥3,𝑢=𝑐,(A.3) with 𝑎,𝑏,𝑐 being constants, are solutions of the subsidiary Lagrange equations𝑑𝑥1𝐻1=𝑑𝑥2𝐻2=𝑑𝑥3𝐻3=𝑑𝑢𝑅(A.4) and 𝐺 is an arbitrary function possessing continuous partial derivatives with respect to its arguments.

B. Analytical Parametric Solution of the Equation 𝑦𝑦𝑥𝑦=𝐴𝑥+𝐵

It is well known that the general ODE of the first order𝐹𝑥,𝑦,𝑦𝑥=0(B.1) can accept a parametric solution of the form𝑥=𝑥(𝑡),𝑦=𝑦(𝑡),(B.2) in case where the following system can be integrated, namely,𝑑𝑥𝐹𝑑𝑡=,𝑡𝐹,𝑥+𝑡𝐹,𝑦,(B.3a)𝑑𝑦𝑑𝑡=𝑡𝑑𝑥𝑑𝑡=𝑡𝐹,𝑡𝐹,𝑥+𝑡𝐹,𝑦,(B.3b)where the notation 𝐹𝑥=𝑑𝐹/𝑑𝑥, 𝐹,𝑥=𝜕𝐹/𝜕𝑥 has been adopted. The above system is obtained by the substitution of 𝑦𝑥=𝑡 and differentiation of (B.1) with respect to 𝑡. In particular, if 𝑡 can be eliminated from (B.2), then a closed-form solution of (B.1) is extracted.

Therefore, as far as the Abel equation 𝑦𝑦𝑥𝑦=𝐴𝑥+𝐵 is concerned, since it is solvable for 𝑥, that is,𝑥=𝑡1𝐴𝐵𝑦𝐴,(B.4) then (B.3b) is considered, namely, (𝐹(𝑥,𝑦,𝑡)=𝑦𝑡𝑦𝐴𝑥𝐵=0)𝑑𝑦𝑑𝑡=𝑡𝑦𝑡2.𝑡𝐴(B.5) Integration of (B.5) in combination with (B.4) results in 𝐶𝑥=𝐴(𝑡1)exp𝑡𝑑𝑡𝑡2𝐵𝑡𝐴𝐴,𝑦=𝐶exp𝑡𝑑𝑡𝑡2𝑡𝐴(B.6) with 𝐶 being an arbitrary constant. Moreover, by substituting 𝜏=𝑡1 and taking into account [19, Integral 2.175.1], the parametric solution (B.6) takes the form 𝐶𝑥=𝐴𝜏𝜏2+𝜏𝐴1/21exp2𝑑𝜏𝜏2𝐵+𝜏𝐴𝐴,𝜏𝑦=𝐶2+𝜏𝐴1/21exp2𝑑𝜏𝜏2.+𝜏𝐴(B.7)

C. Approximations due to the Weakness of the Field

The weakness of the field under consideration, especially of the 𝑢1(=𝑢) coordinate, that is, 𝑢1, allows us to establish the following approximations. (1)We linearly approximate all the polynomials 𝑝(𝑥) (𝑥 represents 𝑢) of degree greater or equal than two, namely, 𝑥𝑝(𝑥)=𝑎+𝑏𝑥+𝒪2.(C.1)(2)Considering the ratio of binomials 𝑝1(𝑥)=𝛼+𝛽𝑥,𝛾+𝛿𝑥(C.2) we evaluate 𝑝1(𝑥)=(𝛼+𝛽𝑥)(𝛾𝛿𝑥)𝛾2𝛿2𝑥2=𝑥𝛼𝛾+(𝛽𝛾𝛼𝛿)𝑥+𝒪2𝛾2𝑥+𝒪2,(C.3) and therefore we obtain 𝑝1𝛼(𝑥)𝛾+1𝛾𝛽𝛼𝛿𝛾𝑥.(C.4)

D. Expressions for 𝑓(𝑥) and 𝐸(𝑥)

In this appendix, we extract appropriate formulas for the function 𝑓(𝑥), appearing in (3.2), as well as for the function 𝐸(𝑥)=exp(𝜅(𝑥)/12), involved into the reduction procedure of the Abel equation (3.18) [8, Chapter 1, Section 3.4]. Thus, by considering the function 𝜌𝜅(𝑥)=11(𝑥)𝑑𝑥, given from (3.2) and substituting 𝜌11 from (2.21), by means of [9, Expression 2.175.1], we arrive at𝜅(𝑥)=𝐴7[]ln𝑃(𝑥)+𝐴8𝑑𝑥𝑃(𝑥),𝑃(𝑥)=𝐴022+𝐴122𝑥+𝐴2211𝑥2.(D.1) The coefficients involved in various expressions appearing in this appendix are listed in Appendix E. We mention that all these coefficients (appeared through the analytical procedure in this work) are functions of the physical parameter(s), of the problem. Therefore, for this (these) parameter(s) taking values such that the discriminant Δ of 𝑃(𝑥) becomes positive (Δ=(𝐴122)24𝐴022𝐴2211) if 𝜌1, 𝜌2 represent the roots of 𝑃(𝜌1,2=(𝐴122±Δ2𝐴2211) and considering the following cases:

Case P-1

Case P-2
Δ>0,𝐴2211<0𝑥<𝜌1or𝑥>𝜌2+𝜖,Δ>0,𝐴2211>0𝑥<𝜌2𝜖or𝑥>𝜌1(D.3) with 𝜖=Δ/|𝐴2211|, then by elementary algebra (using [9, Expression 2.172]) we can easily prove the following Lemma.

Lemma D.1. If Case P-1 or P-2 is valid, then the integral 𝑑𝑥/𝑃 can be written in the form: 𝑑𝑥=1𝑃(𝑥)Δ[],||||ln1𝜆(𝑥)𝜆(𝑥)<1(D.4) with 𝜆𝜆(𝑥)=0+𝜆1𝑥𝜇0+𝜇1𝑥.(D.5)

The coefficients 𝜆0, 𝜆1 are different as regards these two cases, while 𝜇0, 𝜇1 are common (see Appendix E). Thus, substituting (D.4) into (D.1), we obtain𝑓(𝑥)=exp𝜅(𝑥)2=𝑃𝐴9[](𝑥)1𝜆(𝑥)𝐴10,𝐸=exp𝜅(𝑥)12=𝑃𝑎9[](𝑥)1𝜆(𝑥)𝑎10.(D.6) Then by writing 𝑃 in the form 𝑃(𝑥)=𝐴022(1+𝐴11𝑥+𝐴11𝑥2) and developing in power series (assuming that |𝐴11𝑥+𝐴11𝑥2|<1) up to the first order, we take𝑃𝐴9(𝑥)=𝐴12+𝐴13𝑥𝑥+𝒪2,𝑃𝑎9(𝑥)𝑎12+𝑎13𝑥𝑥+𝒪2.(D.7) Furthermore, by applying (C.4) to (D.5), we arrive at𝜆(𝑥)=𝜆2+𝜆3𝑥,𝜆2(𝑥)=𝜆22+2𝜆2𝜆3𝑥𝑥+𝒪2.(D.8) Then developing [1𝜆(𝑥)]𝐴10 and [1𝜆(𝑥)]𝑎10 up to the second order and substituting (D.8), we conclude that[]1𝜆(𝑥)𝐴10=𝜆4+𝜆5𝑥𝑥+𝒪2,[]1𝜆(𝑥)𝑎10=𝜇4+𝜇5𝑥𝑥+𝒪2.(D.9) Finally, substitution of (D.7) and (D.9) into (D.6) results in𝐴𝑓(𝑥)=12+𝐴13𝑥𝜆4+𝜆5𝑥,(D.10a)𝑎𝐸=12+𝑎13𝑥𝜇4+𝜇5𝑥.(D.10b)

E. List of Coefficients



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