Exact analytic solutions for some classes of partial differential equations

Abstract

Exact solutions of linear partial differential equations with variable coefficients and those of nonlinear partial differential equations are dilEcult to obtain. The present thesis attempts to extend the existing methods and devise new ones to tackle some of these equations. These methods include the classical similarity method using invajiance of partial differential equations under the group of infinitesimal transformations (Olver (1986), Bluman and Kumei (1989)), Bax;klund transformations (Rogers and Shadwick (1982)), and direct method for similarity solutions (Clarkson and Kruskal (1989)). Two generalizations of the direct similarity method may be mentioned: One applies to system of partial differential equations to obtain similarity solutions (Sachdev and Mayil Vaganan ( 1994a,b)), while the other generates new nonlinear partial differential equations from the given ones (Sachdev and Mayil Vaganan (1994c)). Each of these methods is applied to important physical equations with appropriate initial and boundary conditions. The physical phenomena that are covered in the present thesis include one-dimensional shock wave propagation, free surface flows in incompressible and compressible media under shallow water approximation, and damped waves governed by the generalized Burgers and Kortweg-de Vries equations. In each case, our solutions extend in a significant manner the existing class of solutions. Chapter I is introductory and explains succinctly how simple exact solutions are obtained by using product, similarity and direct similarity approaches. The heat equation serves a convenient simple model to elucidate these approaches. The direct similarity method is explained and reviewed. Backlund transformations are introduced and illustrated with the help of the Liouville equation. This preparatory material takes the reader smoothly to the original work reported in the succeeding Chapters. Chapter II is spilt into two Parts. Part 1 of Chapter II opens with an exact treatment of a second order linear partial diffarential equation with variable coefficients. Backlund transformations relating the solutions of linear partial differential equation with variable coefficients to those of partial differential equation with constant coefficients are found (Sachdev and Mayil Vaganan (1992)). Auto-B^klund transformations are also determined. To facilitate the generation of new solutions viaBacklund transformation, explicit solutions of both classes of the above mentioned partial differential equation are found using invariance properties of these equations and other methods. Some of these solutions are new. A few new results regarding the applicability of the Baxiklund transformations to nonlinear hyperbolic partial Sinnmary of the Thesis In Chapter III, a new second order nonlinear partial differential equation is derived from one-dimensional, unsteady, non-isentropic gaadynamic equations through the introduction of three ‘potential’ functions. Appropriate boundary conditions at the shock and at the piston in terms of the new functions are obtained. The nonlinear partial differential equation is analysed in great detail. Intermediate integrals and generalized Riemann invariants are discovered. Using the classical Lie group method, the direct similarity method, and equationsplitting, several families of new solutions are found. The direct similarity method is found to yield the most general results. Solutions with shocks (both finite and strong) are constructed to illustrate the applicability of the solutions. (Sachdev and Mayil Vaganan (1993)) Chapter IV is spilt into two Parts. Part 1 gives a comprehensive exact treatment of free surface flows, governed by shallow water equations, written in terms of sigma variables. Several new families of exact solutions of the governing partial differential equations are found and are shown to embed the well-known self-similar or travelling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the travelling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied. In Part 2, exact free surface flows with shear in a compressible barotropic medium are found, extending the work for the incompressible medium. The barotropic medium is of finite extent in the vertical direction, while it is infinite in the horizontal direction. The shallow water equations for a compressible barotropic medium, subject to boundary conditions at the free surface and at the bottom, are solved in terms of double series. Simple wave and time-dependent solutions are found; for the former the free surface is of arbitrary shape while for the latter it is a damping travelling wave in the horizontal direction. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed. In the case of an isothermal medium, simple wave and time-dependent solutions are found. differential equations are also derived in Part 2. Chapter V is spilt into two Parts. In Part 1, a new algorithm for relating different nonlinear partial differential equations is given. It generalizes the direct similarity method which reduces partial differential equations to ordinary differential equations. This algorithm is applied to several model equations-the Burgers, modified Burgers, non-plan.ar Burgers (including cylindrical and spherical Burgers), Korteweg-de Vries, modified Korteweg-de Vries and non-planar Korteweg-de Vries (including cylindrical Korteweg-de Vries) equations. Two new mappings relating physically important equations are derived. The first relates the Burgers equation with a linear damping to the cylindrical Burgers equation while the second maps the KdV equation with a linear damping to the non-planar KdV equation. These mappings are generalized to apply to corresponding equations in n-space derivatives. In Part. 2 of Chapter V, we obtain more general similarity solutions of a few of the generalized Burgers equations using the direct similarity method. The symmetry reductions of the Burgers equation found earlier are generalized. The solutions of the Burgers equation of the form a(x, t) + /3(t)U(z(x, t)) found earlier using the Hopf-Cole transformation are rederived. The solutions of the Burgers equation obtained here contain more arbitrary constants.