Sumudu transform

The Sumudu transform is an integral transform introduced in 1990 by G K Watagala.<ref>Watugala, G. K. (January 1993). "Sumudu transform: a new integral transform to solve differential equations and control engineering problems". International Journal of Mathematical Education in Science and Technology. 24 (1): 35–43. doi:10.1080/0020739930240105. ISSN 0020-739X.</ref><ref>scholar.google.co.in https://scholar.google.co.in/scholar_url?url=https://downloads.hindawi.com/archive/2006/091083.pdf&hl=en&sa=X&ei=BGirZdzlKfKc6rQPgIKU6As&scisig=AFWwaeYEO-3HYLsYwW2gbhf7ehJl&oi=scholarr. Retrieved 2024-01-20. {{cite web}}: Missing or empty |title= (help)</ref><ref>Asiru, Muniru Aderemi (May 2002). "Further properties of the Sumudu transform and its applications". International Journal of Mathematical Education in Science and Technology. 33 (3): 441–449. doi:10.1080/002073902760047940. ISSN 0020-739X. S2CID 123671820.</ref><ref>Zhang, Jun; Shen, Fangyang (2018). "Sumudu Transform for Automatic Mathematical Proof and Identity Generation". In Latifi, Shahram (ed.). Information Technology - New Generations. Advances in Intelligent Systems and Computing. Vol. 558. Cham: Springer International Publishing. pp. 677–683. doi:10.1007/978-3-319-54978-1_85. ISBN 978-3-319-54978-1.</ref> It is defined as the set as <ref>Kılıcman, A.; Gadain, H. E. (2010-06-01). "On the applications of Laplace and Sumudu transforms". Journal of the Franklin Institute. 347 (5): 848–862. doi:10.1016/j.jfranklin.2010.03.008. ISSN 0016-0032.</ref><ref>Watugala, G. K. (January 1993). "Sumudu transform: a new integral transform to solve differential equations and control engineering problems". International Journal of Mathematical Education in Science and Technology. 24 (1): 35–43. doi:10.1080/0020739930240105. ISSN 0020-739X.</ref><ref>Kapoor, Mamta (June 15, 2023). "Sumudu Transform for Time Fractional Physical Models an Analytical Aspect". Journal of Applied Analysis & Computation. 13 (3): 1255–1273. doi:10.11948/20220096. ISSN 2156-907X.</ref>

<math display=block>S = \{f(t) : \ni M, p ,q> 0 , |f(t)|= M \exp(1/u) \}</math>

where <math> p \leq u \leq q </math>

and the Sumudu transform is defined as

<math display=block>A[f(t)]= \frac 1 u \int_0^\infty f(t) \exp\left(\frac1u\right) \, dt. </math>

Relationship with other transforms

Sumudu transform is 1/u Laplace transform <math>S[f(t)](u)=\frac{1}{u} L[f(t)](\frac{1}{u})</math> And with u² Elzaki transform <math>S[f(t)](u)= u^2 E[f(t)](u)</math>