الفهرس 
Bessel potentialsOnly 14 pages are availabe for public view
 |  | from | 120 |  |  |
| | | from | 120 | | |
Abstract
in this section we develop some of the theory of frontier transform, which provides extremely powerful tool for converting certain linear partial differential equations into either aldebraic equations or else differential equations involving fewer variables.
In this thesis, we estimate higher order and fractional of ultra –
hyperbolic kernel related to the spectrum, as follow:Chapter 1:
In this chapter, we refer to the fundamental concepts which are used in
In this section we develop some of the theory of Frontier transform, which
provides extremely powerful tool for converting certain linear partial differential
equations into either algebraic equations or else differential equations involving fewer
variables.Then, we see that the Fourier transform is an especially powerful technique for
studying linear, constant-coefficient partial differential equations. For examples this means that for a partial differential equation involving time, it may be useful to
perform in holding the space variables fixed the solution of the resolvent eq. (13) with the right hand side is the Laplace
transform of the solution of the heat equation (11) with initial data . (If
and , we could now represent in terms of the fundamental solution, to
rederive formula (7).)Turning now to more general (in particular, non-continuous) functions, we note that
the largest class for which the integral defining ̂ converges (absolutely) is the
space . Beyond these particular facts, what we would like here is to
reestablish in the general context the symmetry between and ̂ that holds for .
This is where the special role of the Hilbert space enters.Of special significance will be the negative power in the range, .For
these there will be generalization of the formal operator (18) as an integral operator
.That is, with a slight change of notation we shall have
(For the purpose it is convenient to use the class of functions, which are
indefinitely differentiable on and all of whose derivatives remain bounded when
multiplied by polynomials.The Rises potentials lead to very elegant and useful formulae, as we have already
seen .nevertheless the present formalism suffers from a shortcoming which may be
explained as follows .The importance of the potentials lies above all in their role
as “smoothing operators.”While the local behavior | | , of the kernels
| |
is suited to this purpose, the global behavior | | is less favorable and leads to
increasing awkwardness the greater isA way out of this dilemma is by a modification of the Riesz potentials which
maintains the essential local behavior but eliminates the irrelevant problems of
infinity .There are several roughly equivalent ways of doing this , but the simplest and
most natural approach consists in replacing the “non-negative” operator , by the
“strictly positive” operator , ( Identity) and defining the Bessel potentials
byTo put matters in logical order, we must begin by deriving the kernel of the Bessel
potential, that is the presumed function , with the property thatA close examination of eq. (37) suggests a method of constructing a sequence of
approximations to the exact solution let be a first approximation to the
solution. If were the exact solution, then equals . If is not equal to
the exact solution, then will not equal either or . However, we might
expect closer to than is. Therefore, we set the second approximation
equal to then is a better approximation than . Repeating this operation,
we get at the beginning of the st iteration,The question we ask is whether the sequence { } converges to the exact solution.
We must find conditions under which the iterative procedure in eq. (38) is convergent.
Two such conditions are apparent from the iterative procedure: the space from which
the sequence { } is taken should be complete, and the operator should be such that when applied to two elements of the space it transforms the elements into new
elements which are closer together than the original pair. That is,Thus, a contractive mapping “contracts distances”: the distance between the images
and is smaller, by scale factor of . Then the distance between the elements
and . When takes any positive value (i. e. not necessarily between zero and
one), the mapping is called Lipschtiz continuous. It is clear that a Lipschtiz
continuous (and hence, a contractive) mapping is continuous.If in eq. (38) is a contractive mapping on a subset of a normed space , then
the iteration sequence { } is a Cauchy sequence. If is complete, the Cauchy
sequence converges to the limit point in. However, the limit in general need not lie in
the domain of the mapping , and therefore is not a candidate for a solution of
the equation . If is a closed set, that is, contains all its limit points, then
the limit of sequence { } must lie in , and we can expect the iterative procedure
to converge to the true solution. These ideas are expressed in the following Banach
Fixed Point Theorem, which gives the sufficient conditions for the existence and
uniqueness of solutions.The theorem not only presents an existence and uniqueness result but also gives
an algorithm to obtain the solution by an iterative procedure. For this reason eq.
(38) is known as the method of successive approximationsThe contraction mapping theorem (or the method of successive approximations)
can be used in proofs of existence and in obtaining approximate solutions of
algebraic, differential, and integral equations