Islamia College and University Peshawar
Sir : Dr. Muhammad Idrees
Name : Muhammad Nawaz
Roll No : 181301
Assignment No : 01
Subject : Integral Equation
Semeter :
8th
Department : Mathematics
Batch : 2018 – 2022
Date : 8/06/2022
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Islamia College and University Peshawar
Sir : Dr. Muhammad Idrees
Name : Muhammad Nawaz
Roll No : 181301
Assignment No : 01
Subject : Integral Equation
Semeter :^8
th
Department : Mathematics
Batch : 2018 – 2022
Date : 8/06/
1. Oscillatory integrals :
(a). Background :
In Mathematical Analysis an Oscillatory integral is a type of distribution. Oscillatory integral make rigorous many arguments that, an naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equation as Oscillatory Integrals.
(b). Definition :
An Oscillatory Integral f^ (^ x)^ is written formally as f ( x )=∫ e iϕ ( x , y ) α ( x , y ) dy Where ϕ^ (^ x^ ,^ y^ )∧α^ (^ x^ ,^ y^ )^ are functions defined on Rx n × R (^) y N with the following properties;
- The function ϕ^ is real valued, Positive homogeneous of degree 1, and infinitely differentiable away from {y=0}.Also we assume that ϕ^ does not have any critical points ( a point where the gradient is undefined or the gradient is zero ) on the support of α. Such a function, ϕ^ is usually called a phase function. In some contexts more general functions are considered, and still referred to as phase function.
- The function α^ belongs to one of the symbol classes S1, m (Rx n × Ry N ) (^) for some m ∈ R (^). Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree m. As with the phase function ϕ, in some cases the function α^ is taken to be in more general, or just different, classes. When m←^ N^ the formal integral defining f^ (x)^ converges for all x^ and there is no need for any further discussion of the definition of f^ (x)^. However, when m ≥−N (^) the oscillatory integral is still defined as a distribution on Rn^ even though the integral may not converge. In this case the distribution f^ (^ x)^ is defined by using the fact that
measuring the volume of some standard geometric set, such as the intersection of two balls), and then sum (generally one ends up with summing a standard series such a geometric series or harmonic series). For non-negative integrands, this approach tends to give answers which only differ above and below from the truth by a constant (possibly depending on things such as the dimension d). Slightly more generally, this type of estimation works well in providing upper bounds for integrals which do not oscillate very much. With some more effort, one can often extract asymptotics rather than mere upper bounds, by performing some sort of expansion (e.g. Taylor expansion) of the integrand into a main term (which can be integrated exactly, e.g. by methods from undergraduate calculus), plus an error term which can be upper bounded by an expression smaller than the final value of the main term. However, there are many cases in which one has to deal with integration of highly oscillatory integrands, in which the naive approach of taking absolute values (thus destroying most of the oscillation and cancellation) will give very poor bounds. A typical such oscillatory integral takes the form ∫ R d α ( x ) e iλϕ ( x) dx (^) (1) where α^ is a bump function adapted to some reasonable set B (such as a ball), φ is a real-valued phase function (usually obeying some smoothness conditions), and λ ∈ R is a parameter to measure the extent of oscillation. One could consider more general integrals2 in which the amplitude function α^ is replaced by something a bit more singular, e.g. a power singularity |x| −α , but the aforementioned dyadic decomposition trick can usually decompose such a “singular oscillatory integral” into a dyadic sum of oscillatory integrals of the above type. Also, one can use linear changes of variables to rescale B to be a normalised set, such as the unit ball or unit cube. In one dimension, the definite integral ∫ J e iλϕ ( x) dx (^) (2)
is also of interest, where J is now an interval. While one can dyadically decompose around the endpoints of these intervals to reduce this integral to the previous smoother integral (1), in one dimension one can often compute the integrals (2) more directly. There are two modern tools to estimate (either as upper bounds or as asymptotics) such integrals. One is the principle of nonstationary phase, which roughly speaking asserts that (1) is rapidly decreasing in λ whenever φ is smooth and non-stationary (thus ∇φ does not vanish). This allows one to localise such integrals to the vicinity of the stationary points {x^ : ∇^ ϕ^ (^ x^ )=^0 }. If these stationary points are not isolated, then matters can become extremely complicated; however, in many important cases the stationary points are isolated, and then one can apply the principle of stationary phase, which roughly speaking asserts that the contribution of each stationary point x0 to an integral (1) is essentially equal to the amplitude α^ (x 0 )^ at that point, times the phase e iλϕ (^) ( x❑ 0 ) at that point, times the magnitude ¿^ {x^ ≈^ x 0 :ϕ^ (^ x^ )+O(^ 1 λ )}∨¿ (^) of the region where the phase is close to stationary. A more classical method is the method of steepest descent. This works for certain one-dimensional integrals, using the complex analysis method of contour shifting to shift the integral into a region where the phase acquires a large negative real part, and the integral can then be computed by taking absolute values and using cruder tools such as dyadic decomposition. For instance, one can use this method to show that
p. v .∫
R c P ( x ) e iλ x^2 dx=e iπ 4
R P(e iπ 4 x) e − λ x^2 dx (3) for all polynomials P and λ > 0, where the principal value on the left denotes the limit of the integral (^) ∫ −R R P( x ) e i x 2 dx (^) as x → ∞. This shows in particular that we expect this integral to be small when λ is large and P vanishes near the origin. However, the method of steepest descent requires analytic extension of all the phases involved (and in particular is incompatible with the use of bump functions), and is difficult to generalise to higher dimensions, and so this method has been largely abandoned as obsolete (though it still is applied for “non-commutative integrals”, which are of relevance, among other things, to
Intrinsic and Extrinsic High Oscillation
Example :