Nonlocal Symmetry and Generating Solutions for the Inhomogeneous Burgers Equation

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1 Proceedings of Institute of Mathematics of NAS of Ukraine 004, Vol. 50, Part, 77 8 Nonlocal Symmetry and Generating Solutions for the Inhomogeneous Burgers Equation Valentyn TYCHYNIN and Olexandr RASIN Prydniprovsk State Academy of Civil Engineering and Architecture, 4a Chernyshevsky Str., Dnipropetrovsk, Ukraine tychynin@pgasa.dp.ua, tychynin@ukr.net Dnipropetrovsk National University, 3 Naukovy Per., Dnipropetrovsk, Ukraine sasha rasin@rambler.ru In the present paper we consider a class of inhomogeneous Burgers equations. Nonlocal transformations of a dependent variable that establish relations between various equations of this class were constructed. We identified the subclass of the equations, invariant under the appropriate substitution. The formula of non-local superposition for the inhomogeneous Burgers equation was constructed. We also present examples of generation of solutions. Non-local invariance of the inhomogeneous Burgers equation Let us consider an inhomogeneous Burgers equation: u t + uu x u xx = c, u = u(x, t, ( where c = c(x, t is an arbitrary smooth function, and u t = u t, u x = u x, u xx = u Let us make the first order non-local substitution of the dependent variable: x. u = f(t, x, v, v x, where v = v(x, t is the new dependent variable. We seek the transformation of the equation ( into another inhomogeneous Burgers equation: v t + vv x v xx = g, with arbitrary smooth function g = g(x, t. To obtain this transformation we substitute and its differential prolongations into equation (. For differential prolongations of the equation we obtain the determining relation: f t f v v x v + f v g f vx v xx v f vx v x + f vx g x + ff x + ff v v x + ff vx v xx f xx f vx v x f vxxv xx f vv v x v x f vvx v xx f vxv x v xx c =0. Having split these expressions with respect to the derivative v xx, we obtain the system of equations: f vxvx =0, f vx v + ff vx f vvx v x f vxx =0, (4 f t f v v x v + f v g + ff v v x f vx v x + f vx g x + ff x c f vv v x f xx f vx v x =0. The nontrivial solution of the system ( is f = v x +4F x + v vf v F, (5

2 78 V. Tychynin and O. Rasin c =F xx +F t +4F x F, (6 g = F xx +F t +4F x F. (7 Here F = F (x, t is an arbitrary smooth function. Let us consider a non-local invariance transformation, namely c = g. The equations (6, (7 give us the condition F xx = 0. Integration yields F = A(tx + B(t, where A(t andb(t are arbitrary smooth functions. So we have found the non-local invariant transformation for the inhomogeneous Burgers equation in a general form: f = v x 4A v +vax +vb, v +Ax +B (8 g = c =A t x +B t +4A x +4AB. (9 Substituting c = g = 0 into expressions (6, (7, we find the system: F xx +F t +4F x F =0, F xx +F t +4F x F =0. The general and trivial solutions of this system are: F = x + c, F =0. (0 t +c Here c, c are an arbitrary constants. Having substituted (0 into (5, we get the following non-local invariant transformations of the homogeneous Burgers equation for the general and trivial solutions respectively: u = v xt +v x c v t v c + vx + vc, u = v x + v. vt vc + x + c v Example. Using (5, (6, (7 we transform the inhomogeneous Burgers equation: u t + uu x u xx =8 sin x cos 3 x, ( into homogeneous one. We obtain the system of the equations for F by substituting c =8 sin x cos 3 x and g = 0 in (6, (7: F t +F x F F xx =0, F t +F x F + F xx =4 sin x cos 3 x. There is the general solution of the system F =tgx. It gives us the following substitution: u = v xcos x 4 v cos x +vsin x cos x. ( cos x( v cos x +sinx This expression can be applied to generation of solutions of the equation (. Thus, the partial solution of the homogeneous Burgers equation: v = 4x t + x, generates the following solution of the equation (: u = cos x +t + x +x sin x cos x cos x(x cos x +t sin x + x sin x. (4 (3

3 Nonlocal Symmetry and Solutions for the Burgers Equation 79 Example. It is an example of application of the non-local transformation (8 for a partial case of equation (: A =,B = t. Then we have the equation: u t + uu x u xx =4x +4t +, (5 and corresponding invariant transformation: u = ( u x 4 ( u + ( ux+ ( ut. (6 ( u +x +t One of the similarity solutions of the equation (5 (see Appendix is ( u = x t + e t tg (x ++t et. 4 Having substituted it into (6 we find a new solution of the equation (5: (6t +6x +4tg u = ( e t Using this algorithm we get a chain of solutions: 4 (x ++t e t 8x x t 8t 6xt (. 4x 4t + e t tg e t 4 (x ++t x t 3+x +4xt +3x +t +3t x +t + 4x3 +8x +x t +xt +6xt +5x +5t +4t t 3+4x +8xt +4x +4t +4t. Linearization of the inhomogeneous Burgers equation We are looking for a non-local transformation u = f(v, v x, (7 of equation (, where v = v(x, t is a smooth function, which yields equation: v t v xx + ϕ =0. (8 Here ϕ = ϕ(x, t, v is an arbitrary smooth function. To obtain this transformation we substitute (7 and its differential prolongations into equation (. For differential prolongations of the equation (8 we obtain the determining correlation: f v ϕ + f vx ϕ x + f vx ϕ v v x + ff v v x + ff vx v xx f vv v x v x f vvx v xx f vxv x v xx + c =0. Having split these expressions with respect to the derivative v xx, we obtain the system of equations: f vxvx =0, f vvx v x ff vx =0, f v ϕ + f vx ϕ x + f vx ϕ v v x + ff v v x + c f vv v x =0. The nontrivial solution is v x f =, v + c (9

4 80 V. Tychynin and O. Rasin c = F x, ϕ = F (v + c. (0 ( Here c is an arbitrary constant and F = F (x, t is an arbitrary smooth function. Thus we obtain the Cole Hopf substitution with the parameter c. v x u =. ( v + c So the inhomogeneous Burgers equation may be transformed into the linear equation with variable coefficients [, 4]: v t F (v + c v xx =0. Here the function F is obtained from the equation (0. transformation into the homogeneous equation: v t Fv v xx =0. In the case c = 0, we obtain the (3 Theorem. Formula of a nonlinear superposition for inhomogeneous Burgers equation can be written in the following way: u = x ln (( τ + τ, x ln (k τ = (k u, k=,, (4 t ln (k τ = (k u x (k u + ψ, ψ x = c(x, t Here ( u, u are known solutions, and u is the new one. Proof. Let ( τ, τ be solutions of equation (3. Then τ = ( τ + τ equation (3. By using the substitution u = x ln(τ we can find u : u = x ln ( τ = x ln (( τ + τ. On the other side (k τ, k =, are connected with (k u, k =, in the following way: is a new solution of x ln (k τ = (k u, t ln (k τ = (k u x (k u + ψ, ψ x = c(x, t, k =,. So superposition formula (4 is obtained. Example 3. We can use superposition formula for equation (5. There are two solutions of equation (5: ( ( u = x t + e t tg (x ++tet, 4 u = x t, The formula (4 gives us a third one: u = x t + tg (( x t e t e t +e e4t 6 sec (( x t e t.

5 Nonlocal Symmetry and Solutions for the Burgers Equation 8 3 Lie symmetries for the inhomogeneous Burgers equation To apply the classical method [3] to ( we require the infinitesimal operator to be of this form: X = ξ 0 (x, t, u t + ξ (x, t, u x + η(x, t, u u. The invariance condition for equation ( yields an overdetermined system of differential equations for the coordinates of X. Having solved the system of equations we obtain the following expressions for infinitesimals and function c(x, t: ξ 0 = F, ξ = F x + F, η = uf + F x + F, ( c = F F F F 3/ dtdt F F dt F F 3/ dt + F / F dt + F 3 (ω F 3/ + x F F dtf, ( ω = x F / F F 3/ dt F /. Here F = F (t, F = F (t, F 3 = F 3 (ω are arbitrary smooth functions. If F =0wehave: c = F x F + F 3, Obviously, the connection between the right hand side of the equation ( and its Lie symmetries exists. [] Whitham F.R.S., Linear and nonlinear waves, John Wiley and Sons, 974. [] Tychynin V.A., Non-local linearization and group properties of hyperbolic and parabolic type equations, Ph.D.Thesis, Kyiv, Institute of Mathematics, 983. [3] Fushchych W.I., Shtelen W.M. and Serov N.I., Symmetry analysis and exact solutions of equations of nonlinear mathematical physics, Dordrecht, Kluwer, 993. [4] Miškinis P., New exact solutions of one-dimensional inhomogeneous Burgers equation, Rep. Math. Phys., 00, V.48, N, 75 8.