Elements Of Partial Differential Equations By Ian Sneddon Pdf Download
Walberto Kennedy
In this chapter we shall discuss the properties of ordinary differential equations in more than two variables. Parts of the theory of these equations play important roles in the theory of partial differential equations, and it is essential that they should be understood thoroughly before the study of partial differential equations is begun. Collected in the first section are the basic concepts from solid geometry which are met with most frequently in the study of differential equations.
then to each pair of values of u, v there corresponds a set of numbers (x,y,z) and hence a point in space. Not every point in space corresponds to a pair of values of u and v, however. If we solve the first pair of equations
so that u and v are determined once x and y are known. The corresponding value of z is obtained by substituting these values for u and v into the third of the equations (2). In other words, the value of z is determined once those of x and y are known. Symbolically
so that there is a functional relation of the type (1) between the three coordinates x, y, and z. Now equation (1) expresses the fact that the point (x,y,z) lies on a surface. The equations (2) therefore express the fact that any point (x,y,z) determined from them always lies on a fixed surface. For that reason equations of this type are called parametric equations of the surface.
It should be observed that parametric equations of a surface are not unique; i.e., the same surface (1) can be reached from different forms of the functions F1, F2, F3 of the set (2). As an illustration of this fact we see that the set of parametric equations
in which t is a continuous variable, may be regarded as the parametric equations of a curve. For if P is any point whose coordinates are determined by the equations (5), we see that P lies on a curve whose equations are
which is characterized by the value s of the are length, then s is the distance P0P of P from some fixed point P0 measured along the curve (cf. Fig. 3). Similarly if Q is a point at a distance δs along the curve from P, the distance P0Q will be s + δs, and the coordinates of Q will be, as a consequence,
The curve C is arbitrary except that it passes through the point P and lies on the surface S. It follows that the line with direction ratios (11) is perpendicular to the tangent to every curve lying on S and passing through P. Hence the direction (11) is the direction of the normal to the surface S at the point P.
The expressions (8) give the direction cosines of the tangent to a curve whose equations are of the form (6). Similar expressions may be derived for the case of a curve whose equations are given in the form (4).
arise frequently in mathematical physics. The problem is to find n functions xi, which depend on t and the initial conditions (i.e., the values of x1, x2, . . . , xn when t = 0) and which satisfy the set of equations (1) identically in t.
A third example of the occurrence of systems of differential equations of the kind (1) arises in analytical mechanics. In Hamiltonian form the equations of motion of a dynamical system of n degrees of freedom assume the forms
In particular, if the dynamical system possesses only one degree of freedom, i.e., if its configuration at any time is uniquely specified by a single coordinate q (such as a particle constrained to move on a wire), then the equations of motion reduce to the simple form
Similarly if a heavy string is hanging from two points of support and if we take the y axis vertically upward through the lowest point O of the string, the equation of equilibrium may be written in the form
where P, Q, and R are given functions of x, y, and z. For that reason we study equations of this type now. In addition to their importance in theoretical investigations in physics they play an important role in the theory of differential equations, as will emerge later.
We shall not prove this theorem here but merely assume its validity. A proof of it in the special case in which the functions f1 and f2 are linear in y and z is given in M. Golomb and M. E. Shanks, Elements of Ordinary Differential Equations (McGraw-Hill, New York, 1950), Appendix B. For a proof of the theorem in the general case the reader is referred to textbooks on analysis.
This curve refers to a particular choice of initial conditions; i.e., it is the curve which not only satisfies the pair of differential equations but also passes through the point (a,b,c). Now the numbers a, b, and c are arbitrary, so that the general solution of the given pair of equations will consist of the curves formed by the intersection of a one-parameter system of cylinders of which y = y(x) is a particular member with another one-parameter system of cylinders containing z = z(x) as a member. In other words, the general solution of a set of equations of the type (7) will be a two-parameter family of curves.
In some instances it is a comparatively simple matter to derive one of the sets of surfaces of the solution (2) but not so easy to derive the second set. When that occurs, it is possible to use the first solution in the following way: Suppose, for example, that we are trying to determine the integral curves of the set of differential equations (6) and that we have derived the set of surfaces (8) but cannot find the second set necessary for the complete solution. If we write
and a system of curves on it, to find a system of curves each of which lies on the surface (1) and cuts every curve of the given system at right angles. The new system of curves is called the system of orthogonal trajectories on the surface of the given system of curves. The original system of curves may be thought of as the intersections of the surface (1) with the one-parameter family of surfaces
where c1 is a parameter. It is obvious on geometrical grounds that, in this case, the orthogonal trajectories are the generators shown dotted in Fig. 7. We shall prove this analytically at the end of this section (Example 5 below).
Existence and Uniqueness of Initial Value Problems: Picard's and Peano's Theorems, Gronwall's inequality, continuous dependence, maximal interval of existence. Linear Systems: Autonomous Systems and Phase Space Analysis, matrix exponential solution, critical points, proper and improper nodes, spiral points and saddle points.
First Order Partial Differential Equations: Classification, Method of characteristics for quasi-linear and nonlinear equations, Cauchy's problem, Cauchy-Kowalewski's Theorem. Second-Order Partial Differential Equations: Classification, normal forms and characteristics, Well-posed problem, Stability theory, energy conservation, and dispersion, Adjoint differential operators. Laplace Equation: Maximum and Minimum principle, Green's identity and uniqueness by energy methods, Fundamental solution, Poisson's integral formula, Mean value property, Green's function. Heat Equation: Maximum and Minimum Principle, Duhamel's principle. Wave equation: D'Alembert solution, method of spherical means and Duhamel's principle. The Method of separation of variables for for parabolic, hyperbolic and elliptic equations.
CO1Students will learn the basic principles and methods for the analysis of various partial differential equations. Able to solve the most common PDEs, recurrent in engineering using standard techniques.
CO2Apply some techniques/ methods to predict the behavior of certain phenomena. Identify real phenomena as models of PDEs.
CO3Apply logical/mathematical thinking: the analytic process.
CO4Understand the Fourier series for periodic functions and determine the Fourier coefficients and able to solve various PDEs using Fourier series.
CO5Students will learn the Integral transform method for parabolic, hyperbolic and elliptic equations.