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

Authors: M. P. Markakis
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

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.


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## Body

## 1. Introduction

First-order PDEs, which mostly appear in fluid mechanics, describe the motion of ideal as well as of real fluids [1–3] 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 (Section2), 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 (u1≪1) 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 Section4 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 versusx1(=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 fieldu¯=(u1,u2,u3) satisfying the following PDE:(2.1)(A0ij+Aκijuκ+Aκλijuκuλ)ui,j+(A033+A333u3)u3,3=0,i,j,κ,λ=1,2,
where summation convention has been adopted and(2.2)Aκλij=Aλκij,i,j,κ,λ=1,2,ui,j=∂ui∂xj,u3,3=∂u3∂x3,i,j=1,2,
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 where(2.3a)A12ii=Aκκij=0,i≠j,i,j,κ=1,2,
as well as the case where(2.3b)A2ii=A0ij=A1ij=0,i≠j,i,j=1,2.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,(2.4)ui≪1,i=1,2.
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 thead hocassumption that the components u2 and u3 are functions of the component u1, namely,(2.5)ui=fi(u1),i=1,2,3,
and thus by substituting (2.5), (2.1) (taking into account (2.3a) and (2.3b)) becomes(2.6)R1(u)u,1+R2(u)u,2+R3(u)u,3=0,
where u1 has been replaced by u and(2.7a)R1(u)=A011+A111u+A1111u2+A2211f22+(A221f2+2A1221uf2)f2′,(2.7b)R2(u)=A212f2+2A1212uf2+(A022+A122u+A1122u2+A2222f22)f2′,(2.7c)R3(u)=(A033+A333f3)f3′.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(2.8)∇×u̅=εkji∂ui∂xje̅k=0̅,i,j,k=1,2,3,
where εkji is the well-known Levi-Civita tensor and e̅k 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): (2.9a)f3′u,2-f2′u,3=0,(2.9b)u,3-f3′u,2=0,(2.9c)f2′u,1-u,2=0.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, i≠j). 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 thead 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:(2.10)(2.6)⟹u=G[x1-R1(u)R2(u)x2,x2-R2(u)R3(u)x3],(2.11a)(2.9a)⟹u=G1(x1,x2+f3′f2′x3),(2.11b)(2.9b)⟹u=G2(x2,x1+f3′x3),(2.11c)(2.9c)⟶u=G3(x3,x1+f2′x2),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:(2.12)x1=0,x2=0,x3=0,x1=x2,x2=x3,x1=x3,
we eventually obtain the following systems.Case 1 (G2≡G3).
We have(2.13)f2′=1-x1x2,f3′=1-x1x3.Case 2 (G≡G1).
We have(2.14)R1R2=x1x2-1-f3′f2′x3x2,R2R3=x2x3-x1x3.Case 3 (G≡G2).
We have(2.15)R1R2=x1x2-1,R2R3=x2x3-x1x3-f3′.Case 4 (G≡G3).
Subcase 1 (G≡G3).
We have(2.16)R1R2=x1x2-x3x2,R2R3=x2x3-x1x3-f2′x2x3.Subcase 2 (G≡G3).
We have(2.17)R1R2=-f2′,R2R3=x2x3-1.Subcases1 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:(2.18)R2(u)f2′(u)+R1(u)=0,(2.19)R3(u)f3′(u)-R3(u)f2′(u)+R1(u)+R2(u)=0.
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) becomes(2.20)yx′2+ρ22(x)y2yx′2+ρ11(x)yyx′+ρ20(x)y2=ω(x),
where yx′ denotes the derivative of y(x) with respect to x and(2.21)ρ22(x)=αP(x),ρ11(x)=A2+A3xP(x),ρ20(x)=βP(x),ω(x)=P1(x)P(x),
with(2.22a)α=A2222,β=A2211,(2.22b)P(x)=A022+A122x+A1122x2,P1(x)=-A011-A111x-A1111x2.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.


## 2.1. Transformation of the Governing Equations

Consider an irrotational fieldu¯=(u1,u2,u3) satisfying the following PDE:(2.1)(A0ij+Aκijuκ+Aκλijuκuλ)ui,j+(A033+A333u3)u3,3=0,i,j,κ,λ=1,2,
where summation convention has been adopted and(2.2)Aκλij=Aλκij,i,j,κ,λ=1,2,ui,j=∂ui∂xj,u3,3=∂u3∂x3,i,j=1,2,
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 where(2.3a)A12ii=Aκκij=0,i≠j,i,j,κ=1,2,
as well as the case where(2.3b)A2ii=A0ij=A1ij=0,i≠j,i,j=1,2.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,(2.4)ui≪1,i=1,2.
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 thead hocassumption that the components u2 and u3 are functions of the component u1, namely,(2.5)ui=fi(u1),i=1,2,3,
and thus by substituting (2.5), (2.1) (taking into account (2.3a) and (2.3b)) becomes(2.6)R1(u)u,1+R2(u)u,2+R3(u)u,3=0,
where u1 has been replaced by u and(2.7a)R1(u)=A011+A111u+A1111u2+A2211f22+(A221f2+2A1221uf2)f2′,(2.7b)R2(u)=A212f2+2A1212uf2+(A022+A122u+A1122u2+A2222f22)f2′,(2.7c)R3(u)=(A033+A333f3)f3′.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(2.8)∇×u̅=εkji∂ui∂xje̅k=0̅,i,j,k=1,2,3,
where εkji is the well-known Levi-Civita tensor and e̅k 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): (2.9a)f3′u,2-f2′u,3=0,(2.9b)u,3-f3′u,2=0,(2.9c)f2′u,1-u,2=0.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, i≠j). 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 thead 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:(2.10)(2.6)⟹u=G[x1-R1(u)R2(u)x2,x2-R2(u)R3(u)x3],(2.11a)(2.9a)⟹u=G1(x1,x2+f3′f2′x3),(2.11b)(2.9b)⟹u=G2(x2,x1+f3′x3),(2.11c)(2.9c)⟶u=G3(x3,x1+f2′x2),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:(2.12)x1=0,x2=0,x3=0,x1=x2,x2=x3,x1=x3,
we eventually obtain the following systems.Case 1 (G2≡G3).
We have(2.13)f2′=1-x1x2,f3′=1-x1x3.Case 2 (G≡G1).
We have(2.14)R1R2=x1x2-1-f3′f2′x3x2,R2R3=x2x3-x1x3.Case 3 (G≡G2).
We have(2.15)R1R2=x1x2-1,R2R3=x2x3-x1x3-f3′.Case 4 (G≡G3).
Subcase 1 (G≡G3).
We have(2.16)R1R2=x1x2-x3x2,R2R3=x2x3-x1x3-f2′x2x3.Subcase 2 (G≡G3).
We have(2.17)R1R2=-f2′,R2R3=x2x3-1.Subcases1 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:(2.18)R2(u)f2′(u)+R1(u)=0,(2.19)R3(u)f3′(u)-R3(u)f2′(u)+R1(u)+R2(u)=0.
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) becomes(2.20)yx′2+ρ22(x)y2yx′2+ρ11(x)yyx′+ρ20(x)y2=ω(x),
where yx′ denotes the derivative of y(x) with respect to x and(2.21)ρ22(x)=αP(x),ρ11(x)=A2+A3xP(x),ρ20(x)=βP(x),ω(x)=P1(x)P(x),
with(2.22a)α=A2222,β=A2211,(2.22b)P(x)=A022+A122x+A1122x2,P1(x)=-A011-A111x-A1111x2.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 transformation(3.1)y(x)=h[ξ(x)]f(x),
the left hand side of (2.20) results in a nonlinear expression involving h, hξ′, ξx′, f, and fx′. Thus, by taking into account this expression and setting(3.2)f(x)=exp⁡(-κ(x)2),κ(x)=∫ρ11(x)dx,(3.3)ξ(x)=∫ρ31/2(x)dx,ρ3(x)=ρ1124-ρ20,
with ρ11, ρ20 as in (2.21), (2.20) takes the form (3.4)hξ′2+ρ22f2h2hξ′2-ρ22ρ11f2ρ3-1/2h3hξ′+14ρ22ρ112f2ρ3-1h4-h2=ωf2ρ3.
Then, by substituting(3.5)h2(ξ)=s(ξ),
(3.4) becomes(3.6)sξ′2s+ρ22f2sξ′2-(2ρ22ρ11f2ρ3-1/2ssξ′-ρ22ρ112f2ρ3-1s2+4s+4ωf2ρ3)=0.
In addition, by substitution of (2.21) and (2.22b) into (3.3), we obtain(3.7)ρ3(x)=P2(x)4P2(x),P2(x)=A4+A5x+A6x2,
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: (3.8a)sξ′2s+ρ22f2sξ′2=F(ξ),(3.8b)2ρ22ρ11f2ρ3-1/2ssξ′-ρ22ρ112f2ρ3-1s2+4s+4ωf2ρ3=F(ξ),where F(ξ) is an unknown arbitrary function. Furthermore, after dividing (3.8b) by f2 and setting(3.9)F(ξ)=4ω(x)f2(x)ρ3(x)G̅(ξ),x=x(ξ),(3.8b) will be written as(3.10)2ρ22(x)ρ11(x)ρ3-1/2(x)ssξ′=ρ22(x)ρ112(x)ρ3-1(x)s2-4f2(x)s+4ω(x)f4(x)ρ3(x)[G̅(ξ)-1],x=x(ξ),
where G̅(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,(3.11)z̅z̅t′-z̅=ω(x)f(x)ρ3(x)[G̅(ξ)-1],x=x[ξ(t)],ξ=ξ(t)
with(3.12)s(ξ)=z̅[t(ξ)]f[x(ξ)],t(ξ)=-2∫ρ31/2(x)dξρ22(x)ρ11(x)f(x),
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′=ξx′-1) 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 in(3.13)[1f(x)z̅(t)+ρ22(x)][z̅t′-ρ22(x)ρ112(x)f(x)4ρ3(x)z̅(t)]2=ρ222(x)ρ112(x)ω(x)ρ32(x)G̅(ξ),
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 G̅ yields a nonlinear ODE, which represents the reduced form of (3.6). More precisely, after some algebra we extract(3.14)(3z̅+1ρ22f)z̅t′-ρ3ρ222ρ112f(1fz̅+ρ22)z̅t′2=ρ22ρ112f8ρ3z̅2+(2+ρ1128ρ3)z̅-2ωρ3f.
Furthermore, as far as the z̅t′2-term is concerned, combination of (3.12) with (3.1) and (3.5) yields z̅(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 z̅t′ is equal to (α̅+β̅u)(u2f2′+γ̅u22+δ̅uu22), where α̅,β̅,γ̅,δ̅ 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 (3.15)z̅t′≪1,
and hence we can neglect the z̅t′2 term in the left-hand side of (3.14) in comparison with the others, as it is of 𝒪[max{u24,u22f2′2}]. 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(u24/(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(3.16)t̅(x)=t[ξ(x)]=-2∫ρ3(x)dxρ22(x)ρ11(x)f(x).
Thus, by writing(3.17)z(x)=z̅[t̅(x)],
neglecting the z̅t′2 term and multiplying with t̅x′, then by using (3.16), (3.14) becomes(3.18)[3z+1ρ22(x)f(x)]zx′=-ρ11(x)4z2-16ρ3(x)+ρ112(x)4ρ22(x)ρ11(x)f(x)z+4ω(x)ρ22(x)ρ11(x)f2(x).
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 at(3.19)qqr′-q=-23αF00+F01x+O(x2)M0+M1x+O(x2),
where(3.20)r(x)=∫F10+F11x+O(x2)Q0+Q1x+O(x2)dx
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 obtain(3.21)qqr′-q=B2+B̅3x,(3.22)r(x)=∫(B0+B1x)dx=B0x+O(x2).
Finally, substitution of x=r/B0 into (3.21) yields(3.23)qqr′-q=B2+B3r.
Moreover, by the followed procedure (see [8]), we have that(3.24)z(x)=q[r(x)]E-P(x)3αf(x)
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)) as(3.25)r=CB3τΓ-1/2(τ)e-I(τ)/2-B2B3,q=CΓ-1/2(τ)e-I(τ)/2,
where(3.26)Γ(τ)=τ2+τ-B3,I(τ)=∫dτΓ(τ).
In the above relations τ represents the parameter while C is an arbitrary constant. Now, by substituting (3.27)Γ-1/2(τ)e-I(τ)/2=Ω(τ)
and taking into account (3.22), the above parametric solution takes the form(3.28)x=B4+B5CτΩ(τ),q=CΩ(τ).
All the coefficients appearing through the analysis are listed in Appendix E.


### 3.3. The Parametric Solution for the Field’s Componentsu1, u2

By combining (3.1), (3.5), (3.12), (3.17), and (3.24), we obtain(3.29)y2(x)=q[r(x)]f(x)E-P(x)3α.
Approximating linearly P(x), namely,(3.30)P(x)=A022+A122x+O(x2)
and substituting q from (3.28), as well as f and E from (D.10a), (D.10b) (Appendix D), then (3.29) yields(3.31)3αy2=[B6+CτΩ(B7+B8/τ)+C2τ2Ω2(B9+B10/τ)+C3τ3Ω3(B11+B12/τ)+B13C4τ4Ω4](c0+c1CτΩ+c2C2τ2Ω2).
Furthermore, by solving the first part of (3.28) for CτΩ(τ), we have(3.32)CτΩ(τ)=b0+b1x.
Now, approximating linearly the powers of CτΩ(τ) involved into (3.31), that is(3.33)C2τ2Ω2=b02+2b0b1x+O(x2),C3τ3Ω3=b03+3b02b1x+O(x2),C4τ4Ω4=b04+4b03b1x+O(x2)
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 that(3.34a)u(x1,x2,x3)=φ1(τ)=B4+B5CτΩ(τ),(3.34b)u22(x1,x2,x3)=φ22(τ,u)=13αb2+b3u+(b4+b5u)(1/τ)c3+c4u,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 in(3.35)R3f2′+R2(f2′-1)-R3f3′=0,u3=f3(u).
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)f3′2) by considering f3′≪1. Anyhow we will not investigate (3.35) in this work.Moreover, in order to evaluate the constantC involved into the parametric solution (3.34a), (3.34b), we need a boundary condition, that is, to locate to a point x̅0=(x10,x20,x30) where the field components u0=u(x̅0), u20=u2(x̅0) are known. Then, by solving (3.34b)) for τ, we extract the corresponding value of the parameter τ0=τ(x̅0), and finally, by using (3.34a), we arrive at(3.36)C=u0-B4B5τ0Ω(τ0).In the next section we apply the derived solution in the two-dimensional case of a flow past bodies with specific boundaries.


## 3.1. Transformation of (2.20)

Introducing transformation(3.1)y(x)=h[ξ(x)]f(x),
the left hand side of (2.20) results in a nonlinear expression involving h, hξ′, ξx′, f, and fx′. Thus, by taking into account this expression and setting(3.2)f(x)=exp⁡(-κ(x)2),κ(x)=∫ρ11(x)dx,(3.3)ξ(x)=∫ρ31/2(x)dx,ρ3(x)=ρ1124-ρ20,
with ρ11, ρ20 as in (2.21), (2.20) takes the form (3.4)hξ′2+ρ22f2h2hξ′2-ρ22ρ11f2ρ3-1/2h3hξ′+14ρ22ρ112f2ρ3-1h4-h2=ωf2ρ3.
Then, by substituting(3.5)h2(ξ)=s(ξ),
(3.4) becomes(3.6)sξ′2s+ρ22f2sξ′2-(2ρ22ρ11f2ρ3-1/2ssξ′-ρ22ρ112f2ρ3-1s2+4s+4ωf2ρ3)=0.
In addition, by substitution of (2.21) and (2.22b) into (3.3), we obtain(3.7)ρ3(x)=P2(x)4P2(x),P2(x)=A4+A5x+A6x2,
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: (3.8a)sξ′2s+ρ22f2sξ′2=F(ξ),(3.8b)2ρ22ρ11f2ρ3-1/2ssξ′-ρ22ρ112f2ρ3-1s2+4s+4ωf2ρ3=F(ξ),where F(ξ) is an unknown arbitrary function. Furthermore, after dividing (3.8b) by f2 and setting(3.9)F(ξ)=4ω(x)f2(x)ρ3(x)G̅(ξ),x=x(ξ),(3.8b) will be written as(3.10)2ρ22(x)ρ11(x)ρ3-1/2(x)ssξ′=ρ22(x)ρ112(x)ρ3-1(x)s2-4f2(x)s+4ω(x)f4(x)ρ3(x)[G̅(ξ)-1],x=x(ξ),
where G̅(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,(3.11)z̅z̅t′-z̅=ω(x)f(x)ρ3(x)[G̅(ξ)-1],x=x[ξ(t)],ξ=ξ(t)
with(3.12)s(ξ)=z̅[t(ξ)]f[x(ξ)],t(ξ)=-2∫ρ31/2(x)dξρ22(x)ρ11(x)f(x),
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′=ξx′-1) 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 in(3.13)[1f(x)z̅(t)+ρ22(x)][z̅t′-ρ22(x)ρ112(x)f(x)4ρ3(x)z̅(t)]2=ρ222(x)ρ112(x)ω(x)ρ32(x)G̅(ξ),
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 G̅ yields a nonlinear ODE, which represents the reduced form of (3.6). More precisely, after some algebra we extract(3.14)(3z̅+1ρ22f)z̅t′-ρ3ρ222ρ112f(1fz̅+ρ22)z̅t′2=ρ22ρ112f8ρ3z̅2+(2+ρ1128ρ3)z̅-2ωρ3f.
Furthermore, as far as the z̅t′2-term is concerned, combination of (3.12) with (3.1) and (3.5) yields z̅(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 z̅t′ is equal to (α̅+β̅u)(u2f2′+γ̅u22+δ̅uu22), where α̅,β̅,γ̅,δ̅ 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 (3.15)z̅t′≪1,
and hence we can neglect the z̅t′2 term in the left-hand side of (3.14) in comparison with the others, as it is of 𝒪[max{u24,u22f2′2}]. 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(u24/(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(3.16)t̅(x)=t[ξ(x)]=-2∫ρ3(x)dxρ22(x)ρ11(x)f(x).
Thus, by writing(3.17)z(x)=z̅[t̅(x)],
neglecting the z̅t′2 term and multiplying with t̅x′, then by using (3.16), (3.14) becomes(3.18)[3z+1ρ22(x)f(x)]zx′=-ρ11(x)4z2-16ρ3(x)+ρ112(x)4ρ22(x)ρ11(x)f(x)z+4ω(x)ρ22(x)ρ11(x)f2(x).
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 at(3.19)qqr′-q=-23αF00+F01x+O(x2)M0+M1x+O(x2),
where(3.20)r(x)=∫F10+F11x+O(x2)Q0+Q1x+O(x2)dx
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 obtain(3.21)qqr′-q=B2+B̅3x,(3.22)r(x)=∫(B0+B1x)dx=B0x+O(x2).
Finally, substitution of x=r/B0 into (3.21) yields(3.23)qqr′-q=B2+B3r.
Moreover, by the followed procedure (see [8]), we have that(3.24)z(x)=q[r(x)]E-P(x)3αf(x)
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)) as(3.25)r=CB3τΓ-1/2(τ)e-I(τ)/2-B2B3,q=CΓ-1/2(τ)e-I(τ)/2,
where(3.26)Γ(τ)=τ2+τ-B3,I(τ)=∫dτΓ(τ).
In the above relations τ represents the parameter while C is an arbitrary constant. Now, by substituting (3.27)Γ-1/2(τ)e-I(τ)/2=Ω(τ)
and taking into account (3.22), the above parametric solution takes the form(3.28)x=B4+B5CτΩ(τ),q=CΩ(τ).
All the coefficients appearing through the analysis are listed in Appendix E.


## 3.3. The Parametric Solution for the Field’s Componentsu1, u2

By combining (3.1), (3.5), (3.12), (3.17), and (3.24), we obtain(3.29)y2(x)=q[r(x)]f(x)E-P(x)3α.
Approximating linearly P(x), namely,(3.30)P(x)=A022+A122x+O(x2)
and substituting q from (3.28), as well as f and E from (D.10a), (D.10b) (Appendix D), then (3.29) yields(3.31)3αy2=[B6+CτΩ(B7+B8/τ)+C2τ2Ω2(B9+B10/τ)+C3τ3Ω3(B11+B12/τ)+B13C4τ4Ω4](c0+c1CτΩ+c2C2τ2Ω2).
Furthermore, by solving the first part of (3.28) for CτΩ(τ), we have(3.32)CτΩ(τ)=b0+b1x.
Now, approximating linearly the powers of CτΩ(τ) involved into (3.31), that is(3.33)C2τ2Ω2=b02+2b0b1x+O(x2),C3τ3Ω3=b03+3b02b1x+O(x2),C4τ4Ω4=b04+4b03b1x+O(x2)
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 that(3.34a)u(x1,x2,x3)=φ1(τ)=B4+B5CτΩ(τ),(3.34b)u22(x1,x2,x3)=φ22(τ,u)=13αb2+b3u+(b4+b5u)(1/τ)c3+c4u,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 in(3.35)R3f2′+R2(f2′-1)-R3f3′=0,u3=f3(u).
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)f3′2) by considering f3′≪1. Anyhow we will not investigate (3.35) in this work.Moreover, in order to evaluate the constantC involved into the parametric solution (3.34a), (3.34b), we need a boundary condition, that is, to locate to a point x̅0=(x10,x20,x30) where the field components u0=u(x̅0), u20=u2(x̅0) are known. Then, by solving (3.34b)) for τ, we extract the corresponding value of the parameter τ0=τ(x̅0), and finally, by using (3.34a), we arrive at(3.36)C=u0-B4B5τ0Ω(τ0).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,(4.1)[1-M2-(γ+1)M2u1-12(γ+1)M2u12-12(γ-1)M2u22]u1,1+[1-(γ-1)M2u1-12(γ-1)M2u12-12(γ+1)M2u22]u2,2-M2(u2+u1u2)(u1,2+u2,1)=0,
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 Aκλij of (2.1) (the u3,3 term vanishes) are given by(4.2)A011=1-M2,A111=-2.4M2,A1111=-1.2M2,A2211=-0.2M2,A212=A221=-M2,A1212=A1221=-M22,A022=1,A122=-0.4M2,A1122=-0.2M2,A2222=-1.2M2.
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 thex1x2 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) become(4.3)u=G[x1-R1(u)R2(u)x2],(4.4)u=G3(x1+f2′x2),
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 AppendixD), we extract that the discriminant Δ of P(u) (x has been replaced by u) is always positive (Δ>0), and, moreover, since A1122<0, by obtaining the roots of P(u), considering the respective to the Cases P-1 and P-2 intervals for u and assuming  u≤0.1(u≪1) as well, then a restriction to the domain of M is derived. More precisely, we find that formulae (3.34a), (3.34b) are valid for(4.5)M≤{0.71(u=10-1),0.74(u=5×10-2),0.78(u=10-2),0.79(u=10-4).
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 as(4.6)-I(τ)2=-1-δarctan1+2τ-δ.Now, in order to construct an appropriate procedure to obtainτ(x1,x2), we consider the well-known boundary condition (see [6, page 208])(4.7)u̅⋅∇φ=0,
where u̅=(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 at(4.8)(1+u(ξ1,ξ2))φ,ξ1+u2(ξ1,ξ2)φ,ξ2=0,
where by neglecting u(u≪1), we obtain(4.9)u2(ξ1,ξ2)=-φ,ξ1φ,ξ2=dξ2dξ1=g(ξ1,ξ2).
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,(4.10)φ22[τ,φ1(τ)]=g2(ξ1,ξ2).
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 have(4.11)ui(ξ1,x2)=ui(ξ1,ξ2)+∂ui∂x2(x2-ξ2)+12∂2ui∂x22(x2-ξ2)2+⋯,i=1,2.
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 (x2-ξ2)≪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(x2-ξ2)≤10-2. 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 that(4.12)τ(0,0)=τ0=b43αc3g2(0,0)-b2,C=-B4B5τ0Ω(τ0)
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 with(4.13a)φs(ξ1,ξ2)=ξ2-assin⁡(bξ1)=0,as,b∈(0,1),(4.13b)φp(ξ1,ξ2)=ap(ξ2+ξ20)2-ξ1-ξ10=0,ap,ξ10,ξ2>0,ξ20=(ξ10/ap)1/2,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.(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.
(a)(b)(c)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.
(a)(b)In Figure2(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)π-π/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 6×10-5, 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 1.5×10-6 (the error increasing with x1).


## 5. Summary and Conclusion

In this paper anad hocanalytical 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.



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