Integral transformation. Chapter xxxiii

Operational methods.

For many problems of heat conduction, the use of classical methods turns out to be inefficient, for example, the use of the method of separation of variables for problems with internal heat sources.

The basic rules and theorems of operational calculus were obtained by M. Vishchenko-Zakharchenko and Hevisvide. They are most widely used in electrical engineering thanks to the work of Heaviside.

The Heaviswide operational method is equivalent to the Laplace integral transform method.

The Laplace transform method consists in the fact that it is not the function itself (original) that is studied, but its modification (image).

Integral transformation of a function
is determined by the formula

(40)

Here S can be a complex number; but the thing-th part is greater than 0.

- original;
- function image. For the image to exist, the integral (51) must converge.

If the problem is solved in images, then the original is determined from the image (example conversion) using the inversion formula

(41)

Instead of formula (52), to determine the original function from its image, you can use the following inversion formula

(41.a)

This formula makes it possible to obtain the original function only with the help of the operation of differentiation and passage to the limit.

If the image is a feature

(42)

which is a partial case of two entire transcendental functions, then by the decomposition theorem we have

(43)

where - simple function roots
; the denominator does not contain free terms and

2. If the image
is the ratio of two denominations (fractional-rational function), and the degree of denomination
less than the degree of face value
, and denomination
has roots of multiplicity K at points , then

where the sum is taken over all roots
. If all roots are simple, i.e. all K are equal to one, then formula (5) goes into (43)

The integral Laplace transform has its drawbacks. In particular, difficulties arise when solving problems where the conditions are given as a function of spatial coordinates, or when solving multidimensional problems.

In this regard, a number of methods for integral transformations in spatial coordinates were proposed in accordance with the geometric shape of the body.

If the transformation is taken along the spatial coordinate x, then the integral transformation of the function
can be represented like this:

(44)

If the transformation kernel K(p,x) is taken in the form
or
, then this integral transformation is called the sine or cosine Fourier transform, respectively.

If the Bessel function is chosen as the transformation kernel
, then it is called the Hankel transform.

The complex Fourier transform is convenient to use for bodies of unlimited length, the sine Fourier transform should be used when a value is given on the surface of the body by formulas, i.e. at GU!, and the cosine is the Fourier transform when the differential is solved. transport equations for BC2. Hankel transformations are applicable when the body is axially symmetrical. The practical application of these integral transformations in the presence of detailed image tables does not cause any particular difficulties.

The transition from images to originals can be carried out using the conversion formulas for:

Complex Fourier Transform

(45)

Sine Fourier Transform

(46)

Cosine Fourier Transform

(47)

Hankel transform

(48)

The considered integral transformations are applicable for bodies of semi-limited extent.

Finite integral transformations

The limitations of the integral transformations of Fourier, Hankel, and partly of Laplace, on the one hand, and the urgent need to solve problems with a finite range of variables, on the other hand, led to the creation of methods for finite integral transformations. They are more preferable even for problems solved by classical methods.

The idea of ​​the method of finite integral transformations was proposed by N.S. Kommekov

(49)

Further study of the issues of the method of finite integral transformations was reflected in the works of Griabarg G.A., Sleddon, Tranter, Degas (Deig) and others.

If the integration boundary is between 0 and e, the kernel of the finite sine - and cosine - Fourier transforms, as well as the Hankel transforms, respectively, have the form:

(50)

(51)

With GU1 and GU2
, and at GU3 is the roots of the equation

(52)

Transformations of indefinite integrals Just as in algebra rules are given that allow one to transform algebraic expressions in order to simplify them, so for an indefinite integral there are rules that allow one to perform its transformations. I. The integral of the algebraic sum of functions is equal to the algebraic sum of the integrals of each member separately, i.e. S dx=lf(x)dx+l (i)="" ii.="" the constant factor can be "taken out="" for \u003d "" sign \u003d"" of the integral, e. \u003d"" (c-constant value, the formula for integration by parts, namely: We prove formula (III). We take the differential from the right side of equality (III) Applying formula 4 from the table § 2 chapter IX, we get x. We transform the term according to formula 5 of the same table: and the term d J / "(d:) f (l;) dx according to the formula (B) § 1 of this chapter is equal to d \ f (*) f = \u003d / (x) f "(l:) dx + f (x) /" (x) dx - / "(x) f (*) dx \u003d \u003d f (x) y "(x) dx, i.e. we have obtained what is obtained by differentiating the left side of equality (III). Formulas (I) and (II) are verified in a similar way. we get J(x1--sin x:) dx= ^ xr dx-^ sin xdx = x* x9 = (-cosx) + C= y + cos x + C. EXAMPLE 2. I ^ dx Applying rule II and the formula J COS X 6 from the table of integrals, we get J cos2* J COS2* to 1 Example 3. ^ Inx dx. There is no such integral in the table of integrals given in § 1. We calculate it by integrating by parts; To do this, we rewrite this integral as follows: J In xdx= ^ In l: 1 dx. Putting /(x) = In l: and<р"(д;)=п1, применим правило интегрирования по частям: J 1 п лг tf* = 1 п л: ср (л;) - J (In х)" ф (х) dx. Но так как ф (л:) = J ф" (л:) dx = ^ 1. = j х0 dx, то, применяя формулу 1 таблицы интегралов (п = 0), получим Ф = *. Окончательно получаем Inxdx = x In л:- = л: In х- J dx - x In jc - x + C. Пример 4. Рассмотрим ^ л; sfn л; rfx. Положим f(x) - x и ф" (л:) = sinx. Тогда ф(лг) = - cosjc, так как (-cos*)" = = sin*. Применяя интегрирование по частям, будем иметь J х sin х dx = - х cos *- J (*)" (- cos x) dx = = - x cos * + ^ cos x dx = - x cos x + sin x + C. Пример 5. Рассмотрим ^ хгехdx. Положим /(x) = xг и ф"(лг) = е*. Тогда ф(лг) = е*, так как (ех)" = ех. Применяя интегрирование по частям, будем иметь J хгех dx = x*ex- J (л:1)" dx = = хгех - 2 ^ хех dx. (*) Таким образом, заданный интеграл выражен при помощи более простого интеграла J хех dx. Применим к последнему интегралу еще раз формулу интегрирования по частям, для этого положим f(x) = x и ф/(лг) = ех. Преобразования неопределенных интегралов Отсюда ^ хех dx = хех - ^ (х)" ех dx = ~хе*-J ех dx = xe* - ех Соединяя равенства (*) и (**), получим окончательно ^ х2е* dx = x2ex - 2 [хех - ех + С] = = х2ех - 2хех + 2ех - 2 С = = хгех - 2хех + 2ех + С, где Ct = - 2С, так что С, есть произвольное постоянное интегрирования.

The transformation to which a function of real variables is assigned a function

Real variables, and the variable 7, generally speaking complex, is called the integral transformation with respect to the variable. The variable is called the transformation variable. For the sake of greater clarity, below the transformation variable will be denoted by the symbol Integral transformation (1) is determined by the limits of the transformation , the kernel and the weight function The limits can be infinite; function properties will be set below. The function is called the integral transformation, and also the integral transformation, image or image of the function. Below, the first of these equivalent terms will be used mainly. A function is often called the original or prototype of a function.

Integral transformations with respect to several or all variables at once are possible. Generalization for this case given above

definitions are obvious. Below we will consider transformations with respect to one variable only. The successive application of such transformations, however, is equivalent to some transformation in several variables.

The transformed functions will be denoted by the same symbols as before the transformation, but with some sign above the symbol: a dash, a wavy line, and By which variable the transformation was carried out, it will be clear from what arguments the transformed function depends on. For example, we will not explicitly write out the integral transformation of a function with respect to the variable Arguments in those cases where this cannot lead to misunderstandings.

The transformation by which the function is again transformed into a function is called the inverse integral transformation (1) or simply the inverse transformation. In this case, the transformation (1) itself is called direct.

An integral transformation is defined when the integral on the right side of (1) exists. For the practical application of integral transformations, however, it is important that there also exist inverse transformations, which, together with (1), would establish a one-to-one correspondence between two classes of functions: the original class of functions and the class of functions that are their integral transformations. Under this condition, it is also possible to establish a correspondence between the operations on both classes of functions and the solution of the problem given for the functions of one class, lead to a problem for the functions of another class, which may be simpler. Having solved this last one, with the help of an inverse transformation, a solution to the original problem is found. An example well known to the reader is the operational calculus based on the use of the Laplace integral transform. Here, differentiation of functions of the original class of functions corresponds to multiplication by an independent variable of functions that are Laplace transforms. Due to this, problems for ordinary differential equations with constant coefficients are reduced to algebraic problems for transformed functions.

The idea of ​​using integral transformations in problems for partial differential equations is similar: they try to choose an integral transformation that would allow differential operations with respect to one of the variables to be replaced by algebraic operations. When this succeeds, the transformed problem is usually simpler than the original one. Having found the solution of the transformed problem, with the help of the inverse transformation, the solution of the original one is also found. The main difference from operational calculus is the application of integral transformations to equations with

partial derivatives is the use of a wider set of integral transformations, which is important when the coefficients of the equations are variable.