Chapter xxxiii. application of integral transformations for solving problems of mathematical physics. Finite integral transformations
An integral equation is a functional equation containing an integral transformation over an unknown function. If the integral equation also contains derivatives of an unknown function, then one speaks of an integro differential equation.… … Wikipedia
Equations containing unknown functions under the integral sign. Numerous tasks physics and mathematical physics lead to I. at. various types. Let, for example, it is required using some optical device to obtain ... ... Great Soviet Encyclopedia
GOST 24736-81: Integrated digital-to-analog and analog-to-digital converters. main parameters- Terminology GOST 24736 81: Integrated digital-to-analog and analog-to-digital converters. The main parameters of the original document: Conversion time The time interval from the moment the signal changes at the input of analog-to-digital converters to ... ...
Conversion time- The 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, ms 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 transformation 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 solved ... ... Wikipedia
The Laplace transform is an integral transformation that relates a function of a complex variable (image) to a function of a real variable (original). With its help, the properties of dynamical systems are investigated and differential and ... Wikipedia
The Fourier transform is an operation that maps a function of a real variable to 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, akin to the Fourier transform. First introduced in the work of the Austrian mathematician Johann Radon in 1917. The most important property of the Radon transform is reversibility, that is, the possibility of ... ... Wikipedia
Books
- Integral transformations
- Integral transformations, Knyazev P.N. This book deals with the issues of the theory of integral transformations, closely related to the boundary value problems of the theory of analytic functions (Fourier transforms of analytic functions,…
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 indefinite integral there are rules that allow it to be transformed. 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 \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-hand side of equality (III). Formulas (I) and (II) are verified similarly. Example 1. ^ (n* - Applying the integration rule I and formulas 1 and 5 from the table of integrals, 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.
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, such as Jacobi polynomials. Formulas that allow you to restore the function f(t) from the 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, Mellin transform.
Examples of an 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 becomes 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
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The application of the integral transformation to the first group of data obviously comes down to replacing the functions of the variable Ay.
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 solution (16) of Sec.
The application of integral transformations over spatial coordinates on finite intervals and other rigorous analytical methods to boundary value problems for differential transport equations gives solutions in the form of infinite functional series. In this case, only the main part of this series is used for practical calculations from the obtained solution. Therefore, a simple method for determining an approximate solution that is equivalent to the main part of the exact solution must 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. The definition of the integral Fourier transform and the general scheme of application to the solution of boundary value problems are given in Chap.
The application of integral transformations provides a useful method for solving primarily plane and also spatial problems of elasticity theory. 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 differential equations with partial derivatives 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 // Mrtemetical analysis and its applications: P.
The use of integral transformations allows us to reduce the problem of integrating partial differential equations to integrating a system of ordinary differential equations to represent the desired functions. To illustrate this idea, we present here the solution of the elastic half-plane problem using the Fourier transform; for domains of a different type, other integral transformations turn out to be convenient. Thus, the half-plane problem can be reduced to the determination of a single function p(z) from given values of its real or imaginary part at the boundary. Restricting ourselves to those examples that were considered in § 10.4, we proceed to the presentation of the method of integral transformations.
After applying the integral transformations, the problem is reduced to paired integral equations, an approximate solution is constructed by expanding into a series in cosines, and the transformation is reversed in time by the trapezoidal method. Numerical results are presented illustrating the effect of Poisson's ratio on die settlements.
After applying the Hankel integral transformations in the coordinate and the Laplace transformations in time, an approximate solution of the problem is constructed by expanding it in a system of piecewise constant functions with the selection of a static singularity under the edge of the stamp. The inversion of the Laplace transform is done numerically. Some results of numerical calculations for a uniformly distributed load on the slab are presented, the influence of the permeability and stiffness of the slab and the Poisson's ratio of the soil on the degree of consolidation is studied.
The advantage of using integral transformations over other analytical methods for studying thermal processes associated with the integration of differential equations of energy transfer lies primarily in the standard nature and simplicity of finding solutions.
When applying the Mellin integral transformation to the general solutions of the equations of the plane theory of elasticity (6.1.1) - (6.1.5) in the Papkovich-Neuber form (6.5.34) and (6.5.35), questions of a general and particular nature arise.
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 condition for the application 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 region of integration, Fourier, Laplace, Mellin, Hankel, Meyer, Hilbert, etc. transformations are considered. With the help of these transformations, many problems of the theory of oscillations, thermal conductivity, diffusion and moderation of neutrons, hydrodynamics, theory of elasticity, physical kinetics can be solved .
Let us briefly outline the scheme for applying the indicated integral transformation.
