|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], the '''Sumudu transform''', is an [[integral transform]] similar to the [[Laplace transform]], introduced in the early 1990s by Gamage K. Watugala<ref name="Watugala"/> to solve [[differential equations]] and [[control engineering]] problems. It is equivalent to the [[Laplace–Carson transform]] with the substitution ''p'' = 1/''u''. Sumudu is a [[Sinhalese language|Sinhala]] word, meaning “smooth”.
| | Sobre los jugar a través de la red es muy facil Debatir de cuales son los 100 juegos mas exitosos juegos que existen en el mercado es muy difícil porque cada persona tiene su propia versión<br>A mi por ejemplo me gustan los de jugar sin pagar nada con pou favorito es el de de motos pero muchos días me distraigo games de juego preferido es el de de organizar ciudades<br>Los partidas mas populares son los de jugar sin descargas y cada dia son centenares de casados que se entretienen con los retos de forma online compitiendo con otros jugadores para ver quien es el vencedor y se disputan competiciones en muchos partes del mundo como Valencia<br>EL vencedor se lleva el enhorabuena de los demás games del reto<br>Si quieres conocer a que juegan debes visitar el Web de jugar a todos los juegos gratis para poder disfrutar de los mejores jugar gratuitamente online apou<br><br>Should you loved this article and you would love to receive more information relating to [https://descargarpouparapc.wordpress.com https://descargarpouparapc.wordpress.com/] kindly visit our own webpage. |
| | |
| ==Formal definition==
| |
| The Sumudu transform of a function ''f''(''t''), defined for all real numbers ''t'' ≥ 0, is the function ''F''<sub>''s''</sub>(''u''), defined by:
| |
| | |
| : <math> S\{f(t)\} = F_s(u)
| |
| = \int_0^\infty (1/u)e^{-t/u}f(t)\,dt.\qquad(1)</math>
| |
| | |
| Watugala<ref name="Watugala">Watugala, G. K., “Sumudu transform: a new integral transform to solve differential equations and control engineering problems.” International Journal of Mathematical Education in Science and Technology 24 (1993), 35–43.</ref> first advocated the transform as an alternative to the standard Laplace transform, and gave it the name Sumudu transform. It was early adopted by Weerakoon,<ref name="Weerakoon">Weerakoon, S., “Application of Sumudu transform to partial differential equations” International Journal of Mathematical Education in Science and Technology 25 (1994), 277–283.</ref> and later by others.<ref name="Hussain">Hussain, M. M., and Belgacem, F. M., "Transient solutions of Maxwell's equations based on Sumudu transform," Progress In Electromagnetics Research, PIER 74, 273–289, 2007.</ref>
| |
| | |
| == Properties and theorems ==
| |
| *The transform of a [[Heaviside step function|Heaviside unit step function]] is a Heaviside unit step function in the transformed domain.
| |
| *The transform of a Heaviside unit [[ramp function]] is a Heaviside unit ramp function in the transformed domain.
| |
| *The transform of a monomial ''t''<sup>''n''</sup> is the scaled monomial ''S''{''t<sup>n</sup>''} = ''n''!·''u<sup>n</sup>''.
| |
| *If ''f''(''t'') is a monotonically increasing function, so is ''F''(''u'') and the converse is true for decreasing functions.
| |
| *The Sumudu transform can be defined for functions which are discontinuous at the origin. In that case the two branches of the function should be transformed separately. If ''f''(''t'') is ''C<sup>n</sup>'' continuous at the origin, so is the transformation ''F''(''u'').
| |
| *The limit of ''f''(''t'') as ''t'' tends to zero is equal to the limit of ''F''(''u'') as ''u'' tends to zero provided both limits exist.
| |
| * The limit of ''f''(''t'') as ''t'' tends to infinity is equal to the limit of ''F''(''u'') as ''u'' tends to infinity provided both limits exist.
| |
| * Scaling of the function by a factor ''c'' > 0 to form the function ''f''(''ct'') gives a transform ''F''(''cu'') which is the result of scaling by the same factor.
| |
| * By taking the Sumudu transform of the output signal of a dynamic system when the input is a unit step, the transfer function of the dynamic system in the ''u''–domain can be defined. This is an easily comprehensible concept for the transfer function of a system.
| |
| | |
| All of these properties may be deduced from the corresponding properties of the [[Laplace transform]] using no more than simple high school algebra.
| |
| | |
| == Relationship to other transforms ==
| |
| The Sumudu transform is a simple variant of the [[Laplace transform]]
| |
| | |
| : <math> L\{f(t)\} = F(s)
| |
| = \int_0^\infty e^{-st}f(t)\,dt\qquad(2) </math>
| |
| | |
| which is also used in its so-called ''p''-multiplied form (sometimes known as the [[Laplace]]–[[John Renshaw Carson|Carson]] transform):
| |
| | |
| : <math> C\{f(t)\} = G(p)
| |
| = \int_0^\infty pe^{-pt}f(t)\,dt.\qquad(3) </math>
| |
| | |
| The three transforms can be compared by their action on common functions, such as the monomials ''t''<sup>''n''</sup>:
| |
| *''L''{''t<sup>n</sup>''}(''s'') = ''n''!·''s''<sup>−(''n''+1)</sup>
| |
| *''C''{''t<sup>n</sup>''}(''p'') = ''n''!·''p''<sup>−''n''</sup>
| |
| *''S''{''t<sup>n</sup>''}(''u'') = ''n''!·''u''<sup>''n''</sup>.
| |
| | |
| Equation (2) is employed in Western countries,<ref name="Oberhettinger">Oberhettinger, F. and Badii, L., Tables of Laplace transforms (Berlin: Springer, 1973).</ref> and the Laplace–Carson form remains the standard in Eastern Europe.<ref name="Ditkin">Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus (Oxford: Pergamon, 1965).</ref> The Sumudu transform is thus a minor variant of form (3) in which ''p'' is replaced by 1/''u'' and in this guise has been pressed into service for special purposes in the form shown in Equation (1).<ref name="Balser">Balser, W., From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics 1582 (Berlin: Springer, 1994), Section 2.1.</ref>
| |
| | |
| There are many interconnections between the various transforms. For example, the [[Mellin transform]] can by a change of variable be turned into a bilateral version of the Laplace. However, because the ranges of integration differ between the bilateral case and the standard one, the convergence and other properties of the Laplace and the Mellin transforms are also quite different. Similar distinctions apply to other connections between all the usual transforms.
| |
| | |
| In contrast, the Sumudu transform is essentially identical with the Laplace. Given an initial ''f''(''t''), its Laplace transform ''F''(''s'') can be translated into the Sumudu transform ''F<sub>s</sub>''(''u'') of ''f'' by means of the relation
| |
| | |
| : <math>F_s (u) = \frac{F\left(\frac{1}{u}\right)}{u} </math> | |
|
| |
| and its inverse,
| |
| | |
| : <math>F(s) = \frac{F_s\left( \frac{1}{s} \right)}{s}. \,</math>
| |
| | |
| It is thus possible to take a table of Laplace transforms<ref name="Oberhettinger"/> and rewrite it line by line as a table of Sumudu transforms (and vice versa). Similarly, every property proved of the Laplace transform may routinely be turned into a corresponding property of the Sumudu transform (and again vice versa). This proves the essential identity of the two transforms (Sumudu and Laplace).<ref name="Deakin">Deakin, M. A. B., “The ‘Sumudu transform’ and the Laplace transform.” International Journal of Mathematical Education in Science and Technology 28 (1997), 159.</ref><ref>Weerakoon, S., "The 'Sumudu transform' and the Laplace transform – Reply" International Journal of Mathematical Education in Science and Technology Vol 28 Issue 1 (1997), 160.</ref>
| |
| | |
| It is sometimes said that the Sumudu variant of the Laplace transform is more suitable for educational purposes than is the standard Laplace.<ref name="Watugala"/><ref name="Weerakoon"/> The argument for this viewpoint proceeds mostly from the somewhat simpler form for the transform of ''t<sup>n</sup>'' and the unit-preserving property of the Sumudu transform. However, even if this were so, the standard versions, Equations (2) and (3), are now so deeply entrenched that change is probably infeasible.
| |
| | |
| ==Practical importance==
| |
| | |
| In mechanical and material engineering, the [[Laplace]]–[[John Renshaw Carson|Carson]] transform
| |
| | |
| : <math> C\{f(t)\} = G(p)
| |
| = \int_0^\infty pe^{-pt}f(t)\,dt\qquad(3) </math>
| |
| | |
| is used in the study of the behavior of [[visco-elastic#Types_of_viscoelasticity|linear visco-elastic]] materials. When the linear visco-elastic [[constitutive law]] is transformed to the Laplace–Carson domain, its integral form reduces to the simple <math>\sigma (p)=E (p) \epsilon (p)</math>. This is not the case when using the Laplace transform itself. Some other constitutive laws are more appropriately described by the [[John Renshaw Carson|Carson]] transform,
| |
| | |
| : <math> Car\{f(t)\} = G_{Car}(p)
| |
| = p\int_0^\infty e^{-pt}f(t)\,dt\qquad(4) </math>
| |
| | |
| with the ''p'' in front of the integral.
| |
| | |
| ==See also==
| |
| * [[List of transforms]]
| |
| * [[Laplace transform]]
| |
| * [[N-transform]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Integral transforms]]
| |
Sobre los jugar a través de la red es muy facil Debatir de cuales son los 100 juegos mas exitosos juegos que existen en el mercado es muy difícil porque cada persona tiene su propia versión
A mi por ejemplo me gustan los de jugar sin pagar nada con pou favorito es el de de motos pero muchos días me distraigo games de juego preferido es el de de organizar ciudades
Los partidas mas populares son los de jugar sin descargas y cada dia son centenares de casados que se entretienen con los retos de forma online compitiendo con otros jugadores para ver quien es el vencedor y se disputan competiciones en muchos partes del mundo como Valencia
EL vencedor se lleva el enhorabuena de los demás games del reto
Si quieres conocer a que juegan debes visitar el Web de jugar a todos los juegos gratis para poder disfrutar de los mejores jugar gratuitamente online apou
Should you loved this article and you would love to receive more information relating to https://descargarpouparapc.wordpress.com/ kindly visit our own webpage.