Elzaki and Sumudu transforms


The Sumudu transform was proposed by G.K. Watugala in 1993:
\[ S\left[ f(t) \right] (\nu ) = \frac{1}{\nu} \int_0^{\infty} f(t)\,e^{-t/\nu} \,{\text d}t . \]
A sufficient condition for the existence of the Sumudu transform of a function f is its exponential order: \( \left\vert f(t) \right\vert \le M\,e^{k\,t} , \) for some positive constants M and k. The Sumudu transformation is related to the Laplace transform via formula:
\[ S\left[ f(t) \right] (\nu ) = \frac{1}{\nu}\,{\cal L} \left[ f(t) \right] \left( \frac{1}{\nu} \right) = \int_0^{\infty} f(t\nu )\,e^{-t} \,{\text d}t, \]
and to Elzaki transform
\[ S\left[ f(t) \right] (\nu ) = \nu^2 E\left[ f(t) \right] (\nu ) , \]
where the Elzaki transformation is
\[ E\left[ f(t) \right] (\nu ) = \nu\,\int_0^{\infty} f(t)\,e^{-t/\nu} \,{\text d}t = \nu^2 \int_0^{\infty} f(t\nu )\,e^{-t} \,{\text d}t . \]
The inverse Elzaki transform:
\[ E^{-1} \left[ F(\nu ) \right] (t ) = \frac{1}{2\pi{\bf j}}\,\int_{a-{\bf j}\infty}^{a+{\bf j}\infty} F\left( \frac{1}{\nu} \right)\,e^{t\nu} \,\nu \,{\text d}\nu = \sum \,\mbox{residues of } \left[ F\left( \frac{1}{\nu} \right)\,e^{t\nu} \,\nu \right] . \]
The inverse Sumudu transform:
\[ S^{-1} \left[ F(\nu ) \right] (t ) = \frac{1}{2\pi{\bf j}}\,\int_{a-{\bf j}\infty}^{a+{\bf j}\infty} \,{\text d}t . \]

The ELzaki and Laplace transforms exhibit a duality relation expressed as follows

\[ E \left[ f(t) \right] (\nu ) = \nu\,{\cal L}\left[ f(t) \right] \left( \frac{1}{\nu} \right) \qquad\mbox{and} \qquad {\cal L}\left[ f(t) \right] \left( \lambda \right) = \lambda \, E \left[ f(t) \right] \left( \frac{1}{\lambda} \right) . \]
The main properties of these transformations are
  1. Convolution property:
    \begin{align*} S \left[ f(t) * g(t) \right] (\nu ) &= \nu\,S \left[ f(t) \right] (\nu ) \, \nu\,S \left[ g(t) \right] (\nu ) = \nu^2 S \left[ f(t) \right] S \left[ g(t) \right] ; \\ E \left[ f(t) * g(t) \right] (\nu ) &= \frac{1}{\nu}\,E \left[ f(t) \right] E \left[ g(t) \right] \end{align*}
    where \( (f*g)(t) = \int_0^t f(\tau )\,g(t-\tau )\,{\text d}\tau = (g*f)(t) \) is the convolution of two functions.
  2. Differentiation property:
    \begin{align*} S \left[ f' (t) \right] (\nu ) &= \frac{1}{\nu}\,S \left[ f (t) \right] (\nu ) - \frac{1}{\nu}\,f(+0) , \\ S \left[ f'' (t) \right] (\nu ) &= \frac{1}{\nu^2}\,S \left[ f (t) \right] (\nu ) - \frac{1}{\nu^2}\,f(+0) - \frac{1}{\nu}\,f'(+0) , \\ E \left[ f' (t) \right] (\nu ) &= \frac{1}{\nu} \,E \left[ f (t) \right] (\nu ) - \nu\,f(+0) , \\ E \left[ f'' (t) \right] (\nu ) &= \frac{1}{\nu^2} \,E \left[ f (t) \right] (\nu ) - \nu\,f'(+0) -f(+0) , \\ S \left[ t\,f' (t) \right] (\nu ) &= \\ S \left[ t^2 f' (t) \right] (\nu ) &= \\ E \left[ t\,f' (t) \right] (\nu ) &= \nu^2 \frac{\text d}{{\text d}\nu} \left[ \frac{1}{\nu}\, E \left[ f (t) \right] (\nu ) -\nu\,f(0) \right] - E \left[ f (t) \right] (\nu ) + \nu^2 f(0) , \\ E \left[ t^2 f' (t) \right] (\nu ) &= \nu^4 \frac{{\text d}^2}{{\text d}\nu^2} \left[ \frac{1}{\nu}\, E \left[ f (t) \right] (\nu ) - \nu\,f(0) \right] , \\ S \left[ t\,f'' (t) \right] (\nu ) &= \\ S \left[ t^2 f'' (t) \right] (\nu ) &= \\ E \left[ t\,f'' (t) \right] (\nu ) &= \nu^2 \frac{\text d}{{\text d}\nu} \left[ \frac{1}{\nu^2}\, E \left[ f (t) \right] (\nu ) -\nu\,f'(0) - f(0) \right] - \frac{1}{\nu}\,E \left[ f (t) \right] (\nu ) + \nu\, f(0) + \nu^2 f' (0) , \\ E \left[ t^2 f'' (t) \right] (\nu ) &= \nu^4 \frac{{\text d}^2}{{\text d}\nu^2} \left[ \frac{1}{\nu^2}\, E \left[ f (t) \right] (\nu ) - f(0) - \nu\,f' (0) \right] . \end{align*}

 

Example: The following formulas follow from the corresponding Laplace transformations:

\begin{align*} S \left[ H(t) \right] (\nu ) &= 1, \\ E \left[ H(t-a) \right] (\nu ) &= \nu^2 e^{-a/\nu} , \\ S \left[ \delta (t) \right] (\nu ) &= \frac{1}{\nu} , \\ E \left[ \delta (t-a) \right] (\nu ) &= \nu\, e^{-a/\nu} , \\ S \left[ \sin t\,H(t) \right] (\nu ) &= \frac{\nu}{1 + \nu^2} , \\ E \left[ \sin (at)\,H(t) \right] (\nu ) &= \frac{a\,\nu^3}{1+ a^2 \nu^2} , \\ S \left[ \cos t\,H(t) \right] (\nu ) &= \\ E \left[ \cos (at)\,H(t) \right] (\nu ) &= \frac{\nu^2}{1+ a^2 \nu^2} , \\ S \left[ \sinh t\,H(t) \right] (\nu ) &= \frac{\nu}{1 - \nu^2} , \\ E \left[ \sinh (at)\,H(t) \right] (\nu ) &= \frac{a\,\nu^3}{1- a^2 \nu^2} , \\ S \left[ \cos t\,H(t) \right] (\nu ) &= \\ E \left[ \cosh (at)\,H(t) \right] (\nu ) &= \frac{\nu^2}{1- a^2 \nu^2} , \\ S \left[ t^n \,H(t) \right] (\nu ) &= \\ E \left[ t^n \,H(t) \right] (\nu ) &= n! \nu^{n+2} , \\ S \left[ J_0 (at) \,H(t) \right] (\nu ) &= \\ E \left[ J_0 (at) \,H(t) \right] (\nu ) &= \frac{\nu^2}{\sqrt{1 + a^2 \nu^2}} . \end{align*}
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Example: Consider the Bessel equation subject to the initial condition

\[ t\,y'' + y' + a^2 t\,y =0, \qquad y(0) =1. \]
Application of the Elzaki transform yields
\[ \nu^2 \frac{\text d}{{\text d}\nu} \left[ \frac{1}{\nu^2}\,E[y] -1 -\nu y'(0) \right] + \frac{1}{\nu}\,E[y] -\nu + a^2 \left[ \nu^2 \frac{\text d}{{\text d}\nu}\,E[y] - \nu\,E[y] \right] =0 . \]
Let us denote by T(ν) the Elzaki transform of the unknown function y(t). The above equation can be rewritten as
\[ T' (\nu ) - \frac{2}{\nu}\, T(\nu ) + a^2 \nu^2 T' (\nu ) - a^2 \nu\,T(\nu ) =0 \]
or separating of variables,
\[ \frac{T' (\nu )}{T(\nu )} = \frac{2}{\nu} - \frac{a^2 \nu}{1 + a^2 \nu^2} . \]
Integrating both sides, we get
\[ T(\nu ) = \frac{C\,\nu^2}{\sqrt{1 + a^2 \nu^2}} , \]
where C is a constant of integration. Inventing the Elzaki transform, we obtain
\[ y(t) = E^{-} \left[ T(\nu ) \right] (t) = C\,J_0 \left( at \right) . \]
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Example: Consider the initial value problem for the Euler equation

\[ t^2 y'' + 4t\,y' + 2\,y = 6\,t^2 , \qquad y(0) = y' (0) =0. \]
First we apply the Sumudu transform to this equation to find
\[ \nu^2 S'' + 4\nu\,S' + 2\,S = 12\,\nu^2 , \]
which is the same differential equation. We conclude that the Sumudu transform does not provide any advantage.

Now if the Elzaki transform to the given equation is applied, it yields

\[ \nu^ \frac{{\text d}^2}{{\text d} \nu^2} \left[ \frac{E[y](\nu )}{\nu^2} \right] + 4\nu^2\, \frac{{\text d}}{{\text d} \nu} \left[ \frac{E[y](\nu )}{\nu} \right] + 2\,E[y] = 12\,\nu^2 , \]
The general solution to the latter is
\[ E[y] = \nu^4 + c_1 \nu + c_2 . \]
Using the initial conditions, we find the values of constant c1 and c2 to be both zeroes. By using the inverse Elzaki transform we find the solution in the form: \( y(t) = t^2 /2 . \)    ■

 

  1. Ahmed, S.A., A Comparison between Modified Sumudu Decomposition Method and Homotopy Perturbation Method, Scientific Research, 2018, Vol. 9, No. 3, doi: 10.4236/am.2018.93014
  2. Asiru, M.A., Further properties of the Sumudu transform and its applications, International Journal of Mathematical Education in Science and Technology, 2002, Vol. 33, No. 3, pp. 441--449.
  3. Belgacem F.B.M. and Karaballi, A.A., “Sumudu transform fundamental properties investigations and applications,” Journal of Applied Mathematics and Stochastic Analysis, pp. 1–23, 2006.
  4. Eltayeb, H., Kılıçman, A., and Fisher, B., A new integral transform and associateddistributions,Integral Transforms Spec. Funct.21(2010) 367-379.
  5. Elzaki, T.M. and Biazar, J., Homotopy Perturbation Method and Elzaki Transform for Solving System of Nonlinear Partial Differential Equations, World Applied Sciences Journal, 2013, Vol. 24 (7): 944-948; doi: 10.5829/idosi.wasj.2013.24.07.1041
  6. Elzaki, T.M. and Kim, H., The Solution of Burger’s Equation by Elzaki Homotopy Perturbation Method, Applied Mathematical Sciences,2014, Vol. 8, No. 59, 2931 - 2940; http://dx.doi.org/10.12988/ams.2014.44314
  7. Hussain M.G.M. and Belgacem, F.B.M., Transient solutions of Maxwell’s equations based on Sumudu transform, Progress in Electromagnetics Research, 2007, Vol. 74, 273-289.
  8. V. G. Gupta and B. Sharma, “Application of Sumudu transform in reaction-diffusion systems and nonlinear waves,” Applied Mathematical Sciences, vol. 4, no. 9, pp. 435–446, 2010.
  9. Kadem, A., Solving the one-dimensional neutron transport equation using Cheby-shev polynomials and the Sumudu transform, Analele Universitatii din Oradea. Fascicola Matematica, 2005, Vol.12, pp. 153-171.
  10. Kılıçman A. and Eltayeb, H., On the applications of Laplace and Sumudu transforms, Journal of the Franklin Institute, 2010, vol. 347, no. 5, pp. 848–862, 2010.
  11. Kılıçman A. and Eltayeb, H., On the second order linear partial differential equations with variable coefficients andd ouble Laplace transform, Far East J. Math.Sci. (FJMS)34(2009) 257-270.
  12. Kılıçman A. and Eltayeb, H., A note on integral transform and partial differentialequations, Appl. Math. Sci. (Ruse)4(2010) 109-118.
  13. Kılıçman A. and Eltayeb, H., On a new integral transform and differential equations, Mathematical Problems in Engineering, 2010, Volume 2010, Article ID 463579, 13 pages http://dx.doi.org/10.1155/2010/463579
  14. Kılıçman A., Eltayeb, H., Atan, A.M., A note on the comparison between Laplace and Sumudu transforms, Bulletin of the Iranian Mathematical Society, 2011, Vol.37, No. 1, pp 131-141.
  15. Kılıçman A. and Gadain, H.E., An application of double Laplace transform anddouble Sumudu transform,Lobachevskii J. Math.30(2009) 214-223.
  16. Pandey, R.K. and Mishea, H.K., Numerical simulation of time-fractional fourth order differential equations via homotopy analysis fractional Sumudu transform method, American Journal of Numerical Analysis, 2015, Vol. 3 No. 3, 52-64 DOI: 10.12691/ajna-3-3-1
  17. Rathore, S., Kumar, D., Singh, J., Gupta, S., Homotopy analysis Sumudu transform method for nonlinear equations, Int. J. Industrial Mathematics, 2012, Vol. 4, No. 4, Article ID IJIM-00204, 13 pages
  18. Singh, J., Kumar, D., and Sushila, Homotopy perturbation Sumudu transform method for nonlinear equations, Adv. Theor. Appl. Mech., Vol. 4, 2011, no. 4, 165 - 175
  19. 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, 1993, Vol. 24, No. 1, pp. 35-43
  20. Watugala, G.K., The Sumudu transform for functions of two variables, Mathematical Engineering in Industry, 2002, vol. 8, no. 4, pp. 293–302.
  21. Weerakoon, S., “Complex inversion formula for Sumudu transform,” International Journal of Mathematical Education in Science and Technology, vol. 29, no. 4, pp. 618–621, 1998.
  22. Weerakoon, S., Application of Sumudu transform to partial differential equations, International Journal of Mathematical Education in Science and Technology, vol. 25, no. 2, pp. 277–283, 1994.
  23. Zhang, J., A Sumudu based algorithm for solving differential equations, Computer Science Journal of Moldova, 2007, Vol. 15(3), pp. 303-313