Sumudu transform
Encyclopedia
In mathematics
, the Sumudu transform, is an integral transform similar to the Laplace transform, introduced in the early 1990s by Gamage K. 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 Sinhala word, meaning “smooth”.
transform): NEWLINE
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 Fs(u) of f by means of the relation NEWLINE
transform NEWLINE
transform, NEWLINE
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Sumudu transform, is an integral transform similar to the Laplace transform, introduced in the early 1990s by Gamage K. Watugala to solve differential equations and control engineering
Control engineering
Control engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors...
problems. It is equivalent to the Laplace–Carson
John Renshaw Carson
John Renshaw Carson , who published as J. R. Carson, was a noted transmission theorist for early communications systems...
transform with the substitution p = 1/u. Sumudu is a Sinhala word, meaning “smooth”.
Formal definition
The Sumudu transform of a function f(t), defined for all real numbers t ≥ 0, is the function Fs(u), defined by: NEWLINE- NEWLINE
Properties and theorems
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- The transform of a Heaviside unit step functionHeaviside step functionThe Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
is a Heaviside unit step function in the transformed domain. NEWLINE - The transform of a Heaviside unit ramp functionRamp functionThe ramp function is an elementary unary real function, easily computable as the mean of its independent variable and its absolute value.This function is applied in engineering...
is a Heaviside unit ramp function in the transformed domain. NEWLINE - The transform of a monomial tn is the scaled monomial S{tn} = n!·un. NEWLINE
- If f(t) is a monotonically increasing function, so is F(u) and the converse is true for decreasing functions. NEWLINE
- 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 Cn continuous at the origin, so is the transformation F(u). NEWLINE
- 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. NEWLINE
- 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. NEWLINE
- 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. NEWLINE
- 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.
Relationship to other transforms
The Sumudu transform is a simple variant of the Laplace transform NEWLINE- NEWLINE
John Renshaw Carson
John Renshaw Carson , who published as J. R. Carson, was a noted transmission theorist for early communications systems...
transform): NEWLINE
- NEWLINE
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- L{tn}(s) = n!·s−(n+1) NEWLINE
- C{tn}(p) = n!·p−n NEWLINE
- S{tn}(u) = n!·un.
Mellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace 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 Fs(u) of f by means of the relation NEWLINE
- NEWLINE
- NEWLINE
Practical importance
In mechanical and material engineering, the Laplace–CarsonJohn Renshaw Carson
John Renshaw Carson , who published as J. R. Carson, was a noted transmission theorist for early communications systems...
transform NEWLINE
- NEWLINE
John Renshaw Carson
John Renshaw Carson , who published as J. R. Carson, was a noted transmission theorist for early communications systems...
transform, NEWLINE
- NEWLINE