| dc.description.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. | |
