Transformations of indefinite integrals.

An integral equation is a functional equation containing an integral transformation over an unknown function. If an integral equation also contains derivatives of an unknown function, then they speak of an integro-differential equation.... ... Wikipedia

Equations containing unknown functions under the integral sign. Numerous problems in physics and mathematical physics lead to artificial intelligence. various types. Let, for example, you need to use some optical device to obtain... ... Great Soviet Encyclopedia

GOST 24736-81: Integrated digital-to-analog and analog-to-digital converters. Main settings- Terminology GOST 24736 81: Integrated digital-to-analog and analog-to-digital converters. Main parameters of the original document: Conversion time The time interval from the moment the signal changes at the input of the analog-to-digital converters to... ...

Conversion time- Time interval from the moment the signal changes at the input of analog-to-digital converters until the corresponding stable code appears at the output, µs Source: original document See also related terms: 13 conversion time... ... Dictionary-reference book of terms of normative and technical documentation

The Laplace transform is an integral transform that relates a function of a complex variable (image) to a function of a real variable (original). It is used to study the properties dynamic systems and decide... ...Wikipedia

The Laplace transform is an integral transform that relates a function of a complex variable (image) to a function of a real variable (original). With its help, the properties of dynamic systems are studied and differential and ... Wikipedia are solved

The Fourier transform is an operation that associates a function of a real variable with another function of a real variable. This new feature describes the coefficients (“amplitudes”) when decomposing the original function into elementary components ... ... Wikipedia

Integral transformation of a function of several variables, related to the Fourier transform. It was first introduced in the work of the Austrian mathematician Johann Radon in 1917. The most important property Radon transformation reversibility, that is, the possibility ... ... Wikipedia

Books

• Integral transformations
• Integral transformations, Knyazev P.N.. This book presents issues of theory integral transformations, closely related to boundary value problems of the theory analytical functions(Fourier transforms of analytical functions,...

Page 1

The application of the integral transformation to the first group of data obviously comes down to replacing the functions of the variable Ау.  

The application of integral transformations (4) reduces the solution of the viscoelastic problem (3) to the solution of the purely elastic problem (5) in images. Taking into account the previously given solution (16) section.  

Application of integral transformations over spatial coordinates on finite intervals and other rigorous analytical methods to boundary value problems for differential equations transfer gives solutions in the form of infinite functional series. In this case, from the obtained solution only main part this row. Therefore, a simple method for determining an approximate solution equivalent to the main part of the exact solution should undoubtedly be of great practical importance.  

Application of the integral Fourier transform to problems on the line and half-line.  

Application of the integral Fourier transform to problems on the line and half-line. Definition of the Fourier integral transform and general scheme applications to the solution of boundary value problems are given in Chap.  

Application of integral transformations gives useful method solutions primarily planar, as well as spatial problems theory of elasticity. It is essential that the number of independent variables in partial differential equations can be reduced. The role of the corresponding independent variables passes to the parameters, and thus it is possible to reduce partial differential equations with respect to many variables to ordinary differential equations.  

Application of integral transformations to the construction of exact solutions to filtration problems in fractured-porous media // Mechanical analysis and its applications: S.  

The use of integral transformations allows us to reduce the problem of integrating partial differential equations to the integration of a system of ordinary differential equations to represent the required functions. To illustrate this idea, we present here a solution to the elastic half-plane problem using the Fourier transform; For domains of other types, other integral transformations turn out to be convenient. Thus, the half-plane problem can be reduced to determining one single function p (z) from given values its real or imaginary part on the boundary. Limiting ourselves to those examples that were considered in § 10.4, we move on to presenting the method of integral transformations.  

After applying integral transformations, the problem is reduced to paired integral equations, an approximate solution is constructed by expanding in a series in cosines, and the inversion of the transformation in time is performed by the trapezoidal method. Numerical results are presented to illustrate the influence of Poisson's ratio on die settlement.  

After applying Hankel integral transformations in coordinates and Laplace transformations in time, an approximate solution to the problem is constructed by expanding it into a system of piecewise constant functions, highlighting the static feature under the edge of the die. The inversion of the Laplace transform is performed numerically. Some results of numerical calculations for uniformly distributed load on the slab, the influence of the permeability and stiffness of the slab and the Poisson's ratio of the soil on the degree of consolidation was investigated.  

The advantage of using integral transformations over others analytical methods The study of thermal processes associated with the integration of differential equations of energy transfer consists primarily of standardization and ease of finding solutions.  

When applying the Mellin integral transform to general solutions equations of the plane theory of elasticity (6.1.1) - (6.1.5) in the Papkovich-Neiber form (6.5.34) and (6.5.35) questions of a general and specific nature arise.  

The idea of ​​using integral transformations in problems for partial differential equations is similar: one strives to choose an integral transformation that would allow differential operations on one of the variables to be replaced by algebraic operations. When this is successful, the transformed problem is usually simpler than the original one. Having found a solution to the transformed problem, using inverse conversion find a solution to the original one.  

The main condition for the use of integral transformations is the presence of an inversion theorem, which allows one to find the original function knowing its image. Depending on the weight function and the domain of integration, the transformations of Fourier, Laplace, Mellin, Hankel, Meyer, Hilbert, etc. are considered. With the help of these transformations, many problems in the theory of oscillations, thermal conductivity, diffusion and moderation of neutrons, hydrodynamics, elasticity theory, and physical kinetics can be solved .  

Let us briefly outline the application scheme of the indicated integral transformation.  

Transformations of indefinite integrals Just as in algebra rules are given that allow you to transform algebraic expressions in order to simplify them, and for indefinite integral there are rules that allow its transformation. I. The integral of the algebraic sum of functions is equal to algebraic sum integrals from each term separately, i.e. S dx=lf(x)dx+l (i)="" ii.="" the constant factor can be "extracted="" beyond the="" sign="" of the integral, e.="" ( c-constant value formula for integration by parts, namely: Let us prove formula (III). Let us take the differential from the right side of equality (III) Applying formula 4 from the table in § 2 of Ch. 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 formula (B) § 1 of this chapter is equal to d\f (*) f = =/ (x) f" (l:) dx + f (x) /" (x) dx -/" (x) f (*) dx = =f(x)y"(x)dx, i.e. we got what we get when differentiating the left side of equality (III). Formulas (I) and (II) are checked similarly. Example 1. ^ (l* - Applying the integration rule I and formulas 1 and 5 from the table of integrals, we obtain J (x1 - sin). l:) dx= ^ xg dx-^ sin xdx = x* x9 = (-cosх) + 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 obtain J cos2* J COS2* to 1 Example 3. ^ Inx dx. There is no such integral in the table of integrals given in § 1. Let us calculate it by integrating by parts; to do this, we will 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.

Operating methods.

For many problems of thermal conductivity, the use of classical methods is ineffective, 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 Heaviswid. They became most widespread in electrical engineering thanks to the work of Heaviside.

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

The Laplace transform method consists in studying not the function itself (original), but its modification (image).

Integral transformation of a function
is determined by the formula

(40)

Here S can be a complex number; but at the same time the thing part is greater than 0.

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

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

(41)

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

(41.a)

This formula makes it possible to obtain the original function only using the operation of differentiation and passing to the limit.

If the image is a function

(42)

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

(43)

Where - simple roots of a function
; in this case the denominator does not contain free terms and

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

where the sum is taken over all roots
. If all the 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 specified as a function of spatial coordinates, or solving multidimensional problems.

In this regard, a number of methods for integral transformations along 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 transform 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 extent; the sine Fourier transform should be used when the value on the surface of the body is specified by formulas, i.e. at GU!, and cosine is the Fourier transform when the differential is solved. transport equations at GI2. 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 inversion formulas for:

Complex Fourier transform

(45)

Sine Fourier Transform

(46)

Cosine Fourier transform

(47)

Hankel transformation

(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 Laplace, on the one hand, and the urgent need to solve problems with a finite range of changes in variables, on the other, 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 finite integral transformation method was proposed by N.S. Kommekov

(49)

Further elaboration of the issues of the method of finite integral transformations was reflected in the works of Griabarga G.A., Sleddon, Tranter, Deug (Deig) and others.

If the integration boundary lies 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 are the roots of the equation

(52)

INTEGRAL TRANSFORMATION, functional transformation of the form

where C is a finite or infinite contour in the complex plane, K(x, t) is the kernel of the integral transformation. Integral transformations are most often considered, for which K(x, t) = K(xt) and C is the real axis or its part (a, b). If - ∞< а, b < ∞, то интегральное преобразование называется конечным. При К(х, t) = К(х - t) интегральное преобразование называется интегральным преобразованием типа свёртки. Если х и t - точки n-мерного пространства, а интегрирование ведётся по области этого пространства, то интегральное преобразование называется многомерным. Используются также дискретные интегральные преобразования вида

where n = 0, 1, 2,..., and (Gn(t)) is some system of functions, for example Jacobi polynomials. Formulas that allow you to restore the function f(t) from a known function F(x) are called inversion formulas. Integral transformations are also defined for generalized functions (distributions).

Integral transformations are widely used in mathematics and its applications, in particular in solving differential and integral equations of mathematical physics. The most important for theory and applications are the Fourier transform, Laplace transform, and Mellin transform.

Examples of integral transform are the Stieltjes transform

where c v (α, β) = J ν (α) Y v (ß) - Υ ν (α)J ν (β), J v (x), Y v (x) are cylindrical functions of the 1st and 2nd cities. The inversion formula for the Weber transform is

As a → 0, the Weber transform transforms into the Hankel transform

For v = ± 1/2, this transformation reduces to the sine and cosine Fourier transforms.

An example of a convolution transformation is the Weierstrass transformation