Some Remarks on the Properties of Double Laplace Transforms (original) (raw)

Abstract

In this note, we have discussed and proved the different properties of double Laplace transforms like linearity property, change of scale, shifting property, double Laplace transform of partial derivatives, double Laplace transform of integral, multiplication by xt and division by xt.

Key takeaways

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  1. The note establishes and proves multiple properties of double Laplace transforms.
  2. Key properties include linearity, change of scale, and shifting, among others.
  3. Double Laplace transforms simplify solving partial differential equations.
  4. Constants α, β, a, and b play a crucial role in the properties discussed.
  5. The work connects double Laplace transforms with applications in engineering and mathematics.

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References (8)

  1. D. G. Duff, "Transform methods for solving partial differential equations," Chapman and Hall/CRC, Boca Raton, F. L., 2004.
  2. A. Estrin and T. J. Higgins, "The solution of boundary value problems by multiple laplace transformation," Journal of the Franklin Institute, vol. 252, no. 2, pp. 153-167, 1951.
  3. H. Eltayeb and A. Kilicman, "On double sumudu transform and double laplace transform," Malaysian Journal of Mathematical Sciences, vol. 4, no. 1, pp. 17-30, 2010.
  4. A. Kilicman and H. Eltayeb, "Some remarks on the sumudu and laplace transforms and applications to differential equations," ISRN Applied Mathematics, doi: 10.5402/2012/591517, 2012.
  5. Y. Singh and H. K. Mandia, "Relationship between double laplace transform and double mellin transform in terms of generalized hypergeometric function with applications," International Journal of Scientific and Engineering Research, vol. 3, no. 5, 2012.
  6. H. Eltayeb and A. Kilicman, "A note on double laplace transform and telegraphic equations," Abstract and Applied Analysis, vol. 2013.
  7. A. Kilicman and H. E. Gadain, "On the applications of laplace and sumudu transform," Journal of the Franklin Institute, vol. 347, pp. 848 -862, 2010.
  8. A. Kilicman and H. Eltayeb, "A note on integral transforms and partial differential equations," App. Math. Sc., vol. 4, no. 3, pp. 109-118, 2010.

FAQs

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What are the key properties of double Laplace transforms discussed in the paper?add

The paper discusses properties such as linearity, change of scale, shifting, and derivatives. Each property is proven mathematically to enhance the utility of double Laplace transforms.

How does the double Laplace transform relate to other transform methods?add

It shows a significant relationship with the double Sumudu and double Mellin transforms as detailed by Eltayeb and Kilicman. The connections enhance the applicability of these transforms in solving differential equations.

In which specific equations has the double Laplace transform been recently applied?add

Recently, Eltayeb and Kilicman applied it to solve general linear telegraph and partial integro-differential equations. This application showcases its utility in complex mathematical problems.

What advantages do double Laplace transforms offer in engineering and mathematics?add

They simplify calculations, thus effectively solving various partial differential equations. This capability makes them a valuable tool across multiple engineering disciplines.

What method was used to prove the change of scale property?add

The change of scale property was proven by substituting ax and bt into the double Laplace transform integral. This substitution helped in deriving the generalized form involving constants a and b.