Acta Polytechnica DOI:10.14311/AP.2020.60.0098 Acta Polytechnica 60(2):98–110, 2020 © Czech Technical University in Prague, 2020 available online at https://ojs.cvut.cz/ojs/index.php/ap SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY Amlan Kanti Haldera, ∗, Andronikos Paliathanasisb, c, Peter Gavin Lawrence Leachc, d a Pondicherry University, Department of Mathematics, 605014 Kalapet, India b Universidad Austral de Chile, Instituto de Ciencias Físicas y Matemáticas, Valdivia, Chile c Durban University of Technology, Institute for Systems Science, Durban 4000, Republic of South Africa d University of KwaZulu-Natal, School of Mathematics, Statistics and Computer Science, Durban, South Africa ∗ corresponding author: amlan91.res@pondiuni.edu.in Abstract. We study the similarity solutions and we determine the conservation laws of various forms of beam equations, such as Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of second and third orders. The reduced second-order ode is a perturbed form of Painlevé-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the abovementioned beam forms. The Lie Algebra is mentioned explicitly for all the cases. Keywords: Symmetry analysis, singularity analysis, conservation laws, beam equation. 1. Introduction There are basically two types of beams. One type is supported at both ends and the other type is supported at only one end. It is a cantilever. The latter is of greater mathematical and physical interest for the free end can vibrate. This causes stresses in the beam. The first mathematical description was made by Leonhard Euler and Daniel Bernoulli around 1750, but there were some earlier attempts by Leonardo da Vinci and Galileo Galilei who were more than a little hampered by no knowledge of differential equations. Jacob Bernoulli laid the groundwork for the development of Leonhard Euler and Daniel Bernoulli. In 1894, the polymath Lord Rayleigh proposed an improvement to the Euler-Bernoulli model by including a term related to rotational stress. In 1921, Timoshenko introduced considerable improvements in what is now termed the Timoshenko-Prescott model. There has been a considerable experimental and numerical work devoted to the comparison of predictions of the theories and experimental results. It should be emphasised that the infinitesimal theory of elasticity is three-dimensional and that the three models mentioned above are linear models. They make for easier mathematics, but there is a price to pay. Curiously, the simplest model, that of Euler-Bernoulli, still finds favour amongst some practitioners. Some of the experimental work [1] undertaken to compare reality with theoretical prediction tries to make the experiment as close to a one-dimensional model as possible. One of the more interesting studies is the propagation of shock waves along the beam. This involves firing a bullet into the fixed end of the cantilever which is of a small diameter – 25 mm – to emulate a uniform boundary condition at the fixed end of the beam. The literature devoted to the theory and practice of beams is extensive both in time and space. A fairly recent paper by Labuschagne [2] is very good in its historical aspects as well as being clearly written. Earlier papers in addition to that of Davies cited above are by Hudson [3] and Bancroft [4]. An interesting feature is that the beams are taken to be cylindrical in shape even though the beams one sees in buildings are anything but cylindrical with some exceptions to be found in beamed structures of the nineteenth century. One assumes that this makes the analysis simpler due to the radial symmetry. Even a square cross-section would significantly complicate the mathematics. 98 https://doi.org/10.14311/AP.2020.60.0098 https://ojs.cvut.cz/ojs/index.php/ap vol. 60 no. 2/2020 Similarity solutions and conservation laws for the Beam Equations. . . In this work, we study the algebraic properties of the Euler-Bernoulli, the Rayleigh and of the Timoshenko- Prescott according to the admitted Lie point symmetries, for the source-free equation as also in the case where a homogeneous source term exists. The application of the symmetry analysis for the Euler-Bernoulli equation is not new, there are various studies in the literature [5–9], however in this paper, we obtained some new results, as the reduction of the Euler-Bernoulli form to a perturbed form of Painlevé-Ince [10] equation, which is integrable and the third-order ode, which falls into the category of equations studied by Chazy, Bureau and Cosgrove. Also, we show that the three beam equations of our study admit the same travelling-wave solution. Certain phenomenal works were recently done for the static Euler-Bernoulli equation by Ruiz [11] and Da Silva PL [12]. In Ruiz [11], the Euler-Bernoulli equation with an external agent is studied with respect to the joint invariants of the algebra and complete solutions are specified whereas in [12], for the static Euler-Bernoulli equation with specific nonlinear term, it was found that the algebraic structure of Lie point symmetries is similar to that of the Noether symmetries. It is worthwhile to mention the paper of Freire IL [13] where the Lane-Emden system is reduced to the Emden-Fowler equations and, correspondingly, the solutions of the system have been studied with the aid of the point symmetries. To elaborate the above mentioned works, we focus on the more general Euler-Bernoulli equation with and without the external forcing term and compute its solution using the point symmetries. It is also our intuition that the point symmetries of the form of Euler-Bernoulli under consideration do possess some similarities with the Noether symmetries to follow the results of the above mentioned work. This paper is structured in the following way: In Section 2, we mention the Lie point symmetries and the corresponding algebra. In Section 3, we discuss the travelling-wave solutions for all the beam forms and further reductions of Euler-Bernoulli form using the scaling symmetries. In Section 4, we study the forced forms of the beam equations. Section 5 is devoted to the singularity analysis of a third-order equation, which is obtained by the reduction of Euler-Bernoulli equation by using the scaling symmetry. Conservation laws for the three beam equations are derived in Section 6. The conclusion and appropriate references are mentioned henceforth. 2. Lie symmetry analysis For the convenience of the reader, we give a briefly discussion in the theory of Lie point symmetries. In particular, we present the basic definitions and main steps for the determination of Lie point symmetries for a give differential equation. Consider HA (t,x,u,u,i) = 0 to be a set of differential equations and u,i = ∂u∂yi in which y i = (t,x). Then, under the action of the infinitesimal one-parameter point transformation t′ = t (t,x,u; ε) , (1) x′ = x (t,x,u; ε) , (2) u ′A = uA (t,x,u; ε) , (3) in which ε is an infinitesimal parameter, the set of differential equations HA is invariant if and only if, HA (t′,x′,u′) = HA (t,x,u) , (4) or equivalently lim ε→0 HA (t′,x′,u′; ε) −HA (t,x,u) ε = 0. (5) The latter expression is the definition of the Lie derivative L of HA along the direction Γ = ∂t′ ∂ε ∂t + ∂x′ ∂ε ∂x + ∂u′ ∂ε ∂u. (6) Hence, we shall say that the vector field Γ will be a Lie point symmetry for the set of differential equations HA if and only if the following condition is true LΓ ( HA ) = 0. (7) In other words, the operator Γ can be considered to be symmetry provided Γ[n]HA = 0, 99 A. K. Halder, A. Paliathanasis, P. G. L. Leach Acta Polytechnica whenever HA(t,x,u,u,i ) = 0 and Γ[n] denotes the n-th prolongation of the specified operator in its defined space. The set of all such operators can be denoted by G, which can be regarded as the symmetry group for the set of differential equations HA(t,x,u,u,i ) = 0 [14–16]. 2.1. The Euler-Bernoulli equation. The Euler-Bernoulli form of the beam equation is [1, 17], αβuxxxx + utt = 0. (8) The Lie point symmetries are Γ1a = ∂x , Γ2a = ∂t, Γ3a = u∂u , Γ4a = 2t∂t + x∂x, Γ5a = a(t,x)∂u, where a(t,x) satisfies the Euler-Bernoulli form of the beam equation. The Lie Algebra is (A3,3 ⊕A1) ⊕s ∞A1, according to the Morozov-Mubarakzyanov classificatin scheme [18–21]. 1 2.2. The Rayleigh equation. The Rayleigh form of the beam equation is [1, 17], αβuxxxx + utt −βuxxtt = 0. (9) The Lie point symmetries are Γ1b = ∂t , Γ2b = ∂x, Γ3b = u∂u , Γ4b = b(t,x)∂u, where b(t,x) satisfies the Rayleigh form of the beam equation. Consequently, the admitted Lie algebra is A3 ⊕s ∞A1. 2.3. The Timoshenko-Prescott equation. The Timoshenko and Prescott form of the beam equation is [1, 24], αβuxxxx + utt −β(1 + �)uxxtt + �βutttt α = 0. (10) The Lie point symmetries are Γ1c = ∂t , Γ2c = ∂x , Γ3c = u∂u, Γ4c = c(t,x)∂u, where c(t,x) satisfies the Timoshenko and Prescott form of the beam equation. Hence, the admitted Lie Algebra is A3 ⊕s ∞A1. Therefore, we can say that the Timoshenko-Prescott equation and the Rayleigh equation are algebraic equivalent, but different with the Euler-Bernoulli equation as it admits a higher-dimensional Lie algebra. We proceed our analysis by applying the Lie point symmetries to determine similarity solutions for the three equations of our study. 1We use the SYM package developed by Prof.Stelios Dimas [22, 23]. 100 vol. 60 no. 2/2020 Similarity solutions and conservation laws for the Beam Equations. . . 3. The travelling-wave solution The travelling-wave solution for eq (8), with respect to Γ2a + cΓ1a, where c, denotes the frequency, leads to the fourth order equation, c2v′′(s) + αβv′′′′(s) = 0, (11) where s = x− ct, v(s) = u(x,t). The Lie point symmetries of equation (11) are Γ1d = ∂s, Γ2d = ∂v, Γ3d = s∂v, Γ4d = v∂v, Γ5d = sin cs √ αβ ∂v, Γ6d = cos cs √ αβ ∂v. The reduced form has six symmetries and hence linearisable, the solution for the fourth-order equation can be given as, v(s) = C0 + C1s + C2 sin cs √ αβ + C3 cos cs √ αβ , where Ci, i = 0, 1, 2, 3 are arbitrary constants. Corresponding, to which the solution for Euler-Bernoulli form of beam can be given as u(x,t) = C0 + C1(x− ct) + C2 sin c(x− ct) √ αβ + C3 cos c(x− ct) √ αβ . 3.1. Further reduction of the Euler-Bernoulli equation. The reduction with respect to Γ3a and Γ4a, leads to fourth-order odes. For the similarity variables with respect to 2Γ3a+Γ4a, s = t x2 , u(t,x) = tv(s), the reduced ode is ( 2 αβ + 120s2 ) v′ + ( s + 300s3 ) v′′ + 144s4v′′′ + 16s5v′′′′ = 0. The latter equation is solvable. We continue by considering the similarity variable u(t,x) = xv(s), with respect to Γ3a+Γ4a for s being same as above leads to the fourth-order ode, 24sv′ + ( 1 αβ + 156s2 ) v′′ + 16s3 (7v′′′ + sv′′′′) = 0. (12) The equation has a total of five Lie point symmetries with ∂v, v∂v are the two simpler symmetries, the other three are in terms of Hypergeometric functions, which is complicated enough to be mentioned here. We apply ∂v to perform the reduction. The new invariant functions are s = h and g(h) = v′(s), hence, the reduced equation is a third-order ode, g′′′(h) = − 7g′′(h) h − ( 39 4h2 + 1 16h4αβ ) g′(h) − 3g(h) 2h3 . (13) 101 A. K. Halder, A. Paliathanasis, P. G. L. Leach Acta Polytechnica The latter equation admits four Lie point symmetries. The simpler one being g∂g, the other three are hyperbolic functions of sin h,cos h and Hypergeometric function respectively. We consider g∂g, to do the reduction. The subsequent second-order equation is, m′′(n) = ( −3m(n) − 7 n ) m′(n) −m(n)3 − 7m(n)2 n − ( 39 4n2 + 1 16n4αβ ) m(n) − 3 2n3 , (14) where n = h and m(n) = g ′(h) g(h) . This is the perturbed form of Painlevé-Ince equation and the singularity analysis of this equation shows that it is integrable. In our subsequent paper, we look at the analysis and discuss it elaborately. The reduction of (12) with respect to v∂v, leads to a third-order equation with zero symmetries, g′′′(h) = ( −4g(h) − 7 h ) g′′(h) − 3g′2 − ( 6g(h)2 + 21 g(h) h + 39 4h2 + 1 16h4αβ ) g′(h) −g(h)4 − 7g(h)3 h − ( 39 4h2 + 1 16h4αβ ) g(h)2 − 3g(h) 2h3 , (15) where h = s and g(h) = v ′(s) v(s) . This equation is integrable as ascertained by the singularity analysis, the calculations of which are mentioned in a following section. 3.2. The Travelling wave solution for the Rayleigh equation. The reduction using Γ1b + cΓ2b leads to the fourth-order equation which is maximally symmetric, where c is the frequency. The definition of similarity variables s and v(s) is similar to that of the previous case.( 1 −βc2 ) v′′′′(s) + c2v′′(s) = 0, (16) which is in the form of equation (11). 3.3. The travelling-wave solution for the Timoshenko-Prescott equation. In a similar way, the application of the generic symmetry vector, Γ1c + cΓ2c, in (10) provides the fourth-order ode, ( α2β −βc2a−αβc2ε + c4εβ ) v′′′′(s) + ac2v′′ (s) = 0, (17) which is again in the form of (11). Consequently, we conclude that the three different beam equations provide the same travel-wave solutions. We continue our analysis by assuming the existence of a source term f (u) in the beam equations. 4. Symmetry analysis with a source term In this section, we study the impact of the forcing-source term f(u) in the rhs of the Euler-Bernoulli, Rayleigh and Timoshenko-Prescott beam equations. 4.1. Euler-Bernoulli. The Lie symmetry analysis for the Euler-Bernoulli equation (8) with the forced term f(u), leads to the following possible cases for the forcing term f1 (u) = au + b, (18) f2 (u) = (au + b)n, (19) f3 (u) = eau+b, (20) f4 (u) = arbitrary. (21) For f1(u), the admitted Lie point symmetries for the Euler-Bernoulli equation are, Γf11 = ∂t , Γ f1 2 = ∂x , Γ f1 3 = u∂u , Γ f1 ∞ = b (t,x) ∂u, 102 vol. 60 no. 2/2020 Similarity solutions and conservation laws for the Beam Equations. . . where they form the 3A1 Lie Algebra and b (t,x) is a solution of the original equation. For the source f2 (u) , the admitted Lie point symmetries are, Γf21 = ∂t , Γ f2 2 = ∂x , Γ f2 3 = 2 (n− 1) t∂t + (n− 1) x∂x − 4 ( u + b a ) ∂u, which they form the 2A1 ⊕s A1 Lie Algebra. For f3 (u) the admitted Lie point symmetries are, Γf31 = ∂t , Γ f3 2 = ∂x , Γ f3 3 = 2t∂t + x∂x − 4 a ∂u, where the corresponding Lie Algebra is the 2A1 ⊕s A1. Finally, for the arbitrary functional form of f (u), the admitted Lie point symmetries are the only two symmetry vectors, Γf41 = ∂t , Γ f4 2 = ∂x, which form the 2A1 Lie Algebra and provide the travelling-wave solution. 4.2. Rayleigh and Timoshenko-Prescott equations. For the other two beam equations, namely the Rayleigh and Timoshenko-Prescott equations with a source term, we find that, for a linear function f = f1 (u), the two equations admit the same Lie point symmetries with the force-free case, while for arbitrary function f (u) = f4 (u), admit only two Lie point symmetries, the vector fields Γf41 , Γ f4 2 which provide travelling-wave solutions. 4.3. Symmetry classification of ODE. We show the reduction with the Lie point symmetries, Γf41 + cΓ f4 2 , because the three beam equations of our consideration provide the same fourth-order ODE, which now, with a source term, takes the following form, v′′′′ + c2v′′ = f (v) , (22) We perform the symmetry classification of the latter differential equation and we find that, for the arbitrary function f (v), the latter equation admits only the autonomous symmetry vector ∂v. However, for a constant source f (v) = a0, the Lie point symmetries are, ∂s ,∂v , s∂v , ( 2c2v −a0s2 ) ∂v , cos (cs) ∂v , sin (cs) ∂v, where the generic solution of equation (22) is, v (s) = v1 sin (cs) + v2 cos (cs) + v3s + v4 + a0 2c2 s2. (23) On the other hand, for f (v) = a1v + a0, equation (22) admits the six Lie point symmetries, ∂s , (a1v + a0) ∂v , exp ( ±i √ 2c2 + 2 √ c4 + 4a1 2 s ) ∂v , exp ( ±i √ 2c2 − 2 √ c4 + 4a1 2 s ) ∂v, where the generic solution of (22) is, v (s) = v1 exp ( i √ 2c2 + 2 √ c4 + 4a1 2 s ) + v2 exp ( −i √ 2c2 + 2 √ c4 + 4a1 2 s ) + +v3 exp ( +i √ 2c2 − 2 √ c4 + 4a1 2 s ) + v4 exp ( −i √ 2c2 − 2 √ c4 + 4a1 2 s ) . (24) 103 A. K. Halder, A. Paliathanasis, P. G. L. Leach Acta Polytechnica 4.4. Scaling solutions for the forced Euler-Bernoulli equation. We continue by presenting the reduction with the scaling symmetries for the Euler-Bernoulli equation for the power-law and the exponential sources f2 (u) and f3 (u). For simplicity and without a loss of generality, we select b = 0. For the power-law source f2 (u) = aun, the application of the Lie point symmetry Γ f2 3 provides the reduced fourth-order ode, 4αβv′′′′ + s2v′′ + 3n + 5 n− 1 sv′ + 8(1 + n)v (n− 1)2 − 4avn = 0, (25) in which s = x√ t and u (t,x) = v (s) t 2 n−1 . For the exponential source f3 (u) = eau, the reduced equation given by the scaling symmetry is, 4αβv′′′′ + s2v′′ + 3sav′ + 4aeav + 8 = 0, (26) where now s = x√ t and u (t,x) = −2 a ln (t) + v (s). 5. Singularity analysis The third-order ode that we apply the singularity analysis to is (15) or 24νxy(x) + y(x)2 + 156νx2y(x)2 + 112νx3y(x)3 + 16νx4y(x)4 + y′(x) + 156νx2y′(x) +336νx3y(x)y′(x) + +96νx4y(x)2y′(x) + 48νx4y′2(x) + 112νx3y′′(x) +64νx4y(x)y′′(x) + 16νx4y′′′(x) = 0, (27) where g (h) = y (x) , h = x and αβ = v. We apply the ARS algorithm [25–27] and we make the usual substitution to obtain the leading-order behaviour [28], y → a(x−x0)p, (28) which provides 32aνpx4(x−x0)−3+p − 48aνp2x4(x−x0)−3+p + 16aνp3x4(x−x0)−3+p − 112aνpx3(x−x0)−2+p +112aνp2x3(x−x0)−2+p + ap(x−x0)−1+p + 156aνpx2(x−x0)−1+p + 24aνx(x−x0)p +a2(x−x0)2p + 156a2νx2(x−x0)2p + 112a3νx3(x−x0)3p + 16a4νx4(x−x0)4p −64a2νpx4(x−x0)−2+2p + 112a2νp2x4(x−x0)−2+2p + 336a2νpx3(x−x0)−1+2p +96a3νpx4(x−x0)−1+3p = 0. (29) From the latter, it is evident that p →−1. Hence, − 96aνx4 (x−x0)4 + 176a2νx4 (x−x0)4 − 96a3νx4 (x−x0)4 + 16a4νx4 (x−x0)4 + 224aνx3 (x−x0)3 − 336a2νx3 (x−x0)3 + 112a3νx3 (x−x0)3 − a (x−x0)2 (30) + a2 (x−x0)2 − 156aνx2 (x−x0)2 + 156a2νx2 (x−x0)2 + 24aνx x−x0 . We extract the obvious dominant terms, − 96aνx4 (x−x0)4 + 176a2νx4 (x−x0)4 − 96a3νx4 (x−x0)4 + 16a4νx4 (x−x0)4 , (31) and solve for a, (−3 + a)(−2 + a)(−1 + a)a = 0, (32) 104 vol. 60 no. 2/2020 Similarity solutions and conservation laws for the Beam Equations. . . to obtain, a → 0 ,a → 1 ,a → 2 ,a → 3. (33) In order to find the resonances, we substitute, y → a(x−x0)−1 + m(x−x0)−1+s, (34) and we linearize around m. Then, the usual x → z + x0, simplify the calculations and provide the dominant terms factor as, 16ν(−3 + 2a + s) ( 2 − 6a + 2a2 − 3s + 2as + s2 ) x40z −4+s, (35) and for the three values of a, we obtain the three sets of resonances, for each value of the coefficient term a : a = 1 : s →−1, s → 1, s → 2, a = 2 : s →−2, s →−1, s → 1, and a = 3 : s →−3, s →−2, s →−1. For a = 1, the solution is expressed in terms of a Right Painlevé Series, for a = 3, in terms of a Left Painlevé Series and for a = 2, in terms of a Mixed Painlevé Series. We commence the consistency test. For the Right Painlevé Series, we write y → (x−x0)−1 + F0 + F1(x−x0) + F2(x−x0)2 + F3(x−x0)3 + ... (36) where F0 and F1 are the second and third constants of integration. The output is enormous and hence omitted. The substitution x → z + x0, just makes it easier to collect as powers. The terms in z−1 are, 2F0 z + 24νx0 z + 312F0νx20 z + 336F 20 νx30 z + 336F1νx30 z + + 64F 30 νx40 z + 192F0F1νx40 z + 128F2νx40 z + ...... = 0. (37) This is solved to give, 64νx40F2 → −F0 − 12νx0 − 156F0νx 2 0 − 168F 2 0 νx 3 0 + −168F1νx30 − 32a 3 0νx 4 0 − 96F0F1νx 4 0. (38) The expression for F2 is substituted into the major output, −7F 20 + 3F1 − 240ν − 22F0 x0 − 2880F0νx0 − 3780F 20 νx 2 0 − 2220F1νx 2 0 + −1680F 30 νx 3 0 − 1680F0F1νx 3 0 − 240F 4 0 νx 4 0 + −480F 20 F1νx 4 0 + 240F 2 1 νx 4 0 + 480F3νx 4 0 = 0, (39) and the coefficient of the constant term is solved to give F3 as, 480νx50F3 = 22F0 + 7F 2 0 x0 − 3F1x0 + 240νx0 + 2880F0νx 2 0 + 3780F 2 0 νx 3 0 +2220F1νx30 + 1680F 3 0 νx 4 0 + 1680F0F1νx 4 0 + 240F 4 0 νx 5 0 + 480F 2 0 F1νx 5 0 − 240F 2 1 νx 5 0. (40) Thus, there is not a problem with the determination of the coefficients of the terms in the Right Painlevé Series. As certain terms in the third-order ode are less dominant, there cannot be a Left Painlevé Series. The possibility of the existence of a Mixed Painlevé Series is moot due to the practical difficulty of calculating coefficients. Consequently, equation (15) is integrable according to the Painlevé test 105 A. K. Halder, A. Paliathanasis, P. G. L. Leach Acta Polytechnica 6. Conservation laws The Ibragimov’s theory of nonlinear self-adjointness details the construction of conservation laws for a scalar pde[29–31]. Our first step is to verify the self-adjointness condition on the various forms of beams and later on, compute the conservation laws. The preliminaries can be easily accessed from[29]. The main motivation behind using the Ibragimov’s approach is to obtain the conservation laws to deduce certain special solutions for the Beam equations following the methodology specified by Cimpoiasu [32], where the author have used the nonlinear self-adjointness method to compute solutions for the Rossby waves. The Noether’s theorem can be easily applied to obtain the conserved terms but it is our intuition that the non-local conserved terms as obtained using the Ibragimov’s method can contribute in obtaining new solutions in a different subspace of the complex plane. The main objective is to deduce new solutions using the point symmetries, singularities and conservation laws. For instance, the Euler-Bernoulli equation does imply the existence of series type solutions through the method of Singularity analysis as mentioned in Section 5 for equation (15). Let the scalar PDE admit the following generators of the infinitesimal transformation, V = ξi(x,u,ui, ..) ∂ ∂xi + η(x,u,ui, ...) ∂ ∂u . (41) Then the scalar PDE and its adjoint equation, as defined above, admits the conservation law Ci = ξiL + W ( ∂L ∂ui −Dj ( ∂L ∂uij ) + DjDk ( ∂L ∂uijk ) − ... ) + Dj (W) ( ∂L ∂uij −Dk ( ∂L ∂uij ) + ... ) +DjDk(W) ( ∂L ∂uijk − ... ) , where W = η − ξiui and L denotes the Lagarangian of the corresponding form of the beam equation. For the Euler-Bernoulli, Rayleigh and Timshenko-Prescott forms the Lagarangians are as follows L = q(t,x)(utt + αβuxxxx), L = q(t,x)(αβuxxxx + utt −βuxxtt), L = q(t,x)(αβuxxxx + utt −β(1 + �)uxxtt + �βutttt α ), where q(t,x) is the new dependent variable. To verify the non-linear self adjointness, the substitution of q(t,x) = φ(t,x,u), to the adjoint equation of (8), (9) and (10) must satisfy for all solutions u of those equations. The possible values of φ(t,x,u) is a constant term, let us say, A0 and A1u(t,x) + A2, where A1 and A2 are arbitrary constants. A complete description of this method can be obtained from [29]. 6.1. Conservation laws for the various form of beam. With respect to each of the symmetries in (8), we compute the nonzero conservation laws. For Γ1a, the conservation components are ct = uxφt −uxtφ(t,x,u), cx = φ(t,x,u)(utt + αβuxxxx) + αβuxφxxxx −αβuxxφxx + αβφxuxxx −uxxxxφ(t,x,u). (42) For Γ2a, ct = φ(t,x,u)(utt + αβuxxxx) + utφt −uttφ(t,x,u), cx = αβ(utφxxx −uxtφxx −uxxtφx −uxxxtφ(t,x,u)). (43) 106 vol. 60 no. 2/2020 Similarity solutions and conservation laws for the Beam Equations. . . For Γ3a, ct = −uφt + utφ(t,x,u), cx = αβ(uxφxx −uφxxx + uxxφx + uxxxx). (44) For Γ4a, ct = 2t(φ(t,x,u)(utt + αβuxxxx)) + φt(2tut + xux) −φ(t,x,u)(2tutt + 2ut + xuxt), cx = x(φ(t,x,u)(utt + αβuxxxx)) + αβφxxx(2tut + xux) −αβφxx(2tuxt + xuxx + ux) + αβφx(2tuxxt + xuxxx + 2uxx) −αβφ(t,x,u)(2tuxxxt + xuxxxx + 3uxxx). (45) F0r Γ5a, ct = −a(t,x)φt + atφ(t,x,u), cx = −αβa(t,x)φxxx + αβaxφxx −αβaxxφx + φ(t,x,u)axxx. (46) Next, we compute the conservation laws of equation (9), with respect to its symmetries. Γ1b leads to the following conserved components, ct = φ(t,x,u)(αβuxxxx + utt −βuxxtt) + utφt −φ(t,x,u)utt, cx = αβ(utφxxx −uxtφxx + uxxtφx −uxxxtφ(t,x,u)). (47) For Γ2b, ct = uxφt −uxtφ(t,x,u), cx = φ(t,x,u)(αβuxxxx + utt −βuxxtt) + αβ(uxφxxx −uxxφxx + uxxxφx −φ(t,x,u)uxxxx). (48) For Γ3b, ct = utφ(t,x,u) −uφt, cx = αβ(uxφxx −uφxxx −uxxφx + uxxxφ(t,x,u)). (49) For Γ4b, ct = btφ(t,x,u) − b(t,x)φt, cx = αβ(bxφxx − b(t,x)φxxx − bxxφx + bxxxφ(t,x,u)). (50) For the Timoshenko-Prescott form of the beam (10), the conserved components are as follows: For Γ1c, ct = φ(t,x,u) ( αβuxxxx + utt −β(1 + �)uxxtt + �βutttt α ) + ut ( φt + �βφttt α ) −utt ( φ(t,x,u) + �βφtt α ) + uttt �βφt α −utttt �βφ(t,x,u) α , cx = αβ(utφxxx −uxtφxx + uxxtφx −uxxxtφ(t,x,u)). (51) 107 A. K. Halder, A. Paliathanasis, P. G. L. Leach Acta Polytechnica For Γ2c, ct = ux ( φt + φttt�β α ) −uxt ( φ(t,x,u) + φtt�β α ) + uxtt φt�β α −uxttt �β α , cx = φ(t,x,u) ( αβuxxxx + utt −β(1 + �)uxxtt + �βutttt α ) + αβ(uxφxxx −uxxφxx + uxxxφx −φ(t,x,u)uxxxx). (52) For Γ3c, ct = −u ( φt + �βφttt α ) + ut ( φ(t,x,u) + �βφtt α ) −utt �βφt α + uttt �βφ(t,x,u) α , cx = αβ(uxφxx −uφxxx −uxxφx + uxxxφ(t,x,u)). (53) For Γ4c, ct = − c(t,x) ( φt + �βφttt α ) + ct ( φ(t,x,u) + �βφtt α ) − ctt �βφt α + cttt �βφ(t,x,u) α , cx = αβ(cxφxx − c(t,x)φxxx − cxxφx + cxxxφ(t,x,u)). (54) 7. Conclusion In this work, we focused on the algebraic properties for three different forms of the beam equations with or without a source. For the source free equations, we found that the Euler-Bernoulli equation is invariant under the Lie algebra (A3,3 ⊕A1)⊕s ∞A1, while, the Rayleigh and Timoshenko-Prescott equations are invariant under the Lie algebra A3 ⊕s ∞A1. In the case of an isotropic source f (u), we found that the Euler-Bernoulli, Rayleigh and Timoshenko-Prescott equations are invariant under the Lie algebra 2A1 for an arbitrary source f (u). Moreover, for the Euler-Bernoulli beam equations, the admitted Lie algebras are A3 ⊕s ∞A1 for Linear f (u) = au + b, 2A1 ⊕s A1 for exponential or power-law functional form. For the other two beam equations, there are no specific functional forms of f (u), where the equations admit different algebras. Therefore, for the source-free equations, we derived the conservation laws by applying the Ibragimov’s method. We applied the Lie point symmetries to reduce the pdes and we proved that the three beam equations provide exactly the same travelling-wave solutions. The most important result of our paper is the reduction of Euler-Bernoulli equation to a second-order equation, of the form of perturbed Painlevé-Ince equation and to a third-order equation, which was studied by Chazy, Bureau and Cosgrove. One of our subsequent papers will be on the singularity analysis of the perturbed Painlevé-Ince equation. Moreover, our future work also includes deriving further solutions of the different forms of beams using conservation laws. 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Central European Journal of Physics 12:81 – 89, 2014. doi:10.2478/s11534-014-0430-6. 110 http://dx.doi.org/10.1088/1751-8113/44/43/432002 http://dx.doi.org/10.1016/j.jmaa.2006.10.078 http://dx.doi.org/10.1016/j.cnsns.2013.12.005 http://dx.doi.org/10.2478/s11534-014-0430-6 Acta Polytechnica 60(2):1–13, 2020 1 Introduction 2 Lie symmetry analysis 2.1 The Euler-Bernoulli equation. 2.2 The Rayleigh equation. 2.3 The Timoshenko-Prescott equation. 3 The travelling-wave solution 3.1 Further reduction of the Euler-Bernoulli equation. 3.2 The Travelling wave solution for the Rayleigh equation. 3.3 The travelling-wave solution for the Timoshenko-Prescott equation. 4 Symmetry analysis with a source term 4.1 Euler-Bernoulli. 4.2 Rayleigh and Timoshenko-Prescott equations. 4.3 Symmetry classification of ODE. 4.4 Scaling solutions for the forced Euler-Bernoulli equation. 5 Singularity analysis 6 Conservation laws 6.1 Conservation laws for the various form of beam. 7 Conclusion Acknowledgements References