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Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics



Javid Ahmad Ganie, Altaf Ahmad and Renu Jain
 
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ABSTRACT

In this study basic analogue of double Sumudu transform of functions expressible as polynomials or convergent series are derived. The applicability of this relatively new transform is demonstrated using some special functions, which arise in the solution of evolution equations of population dynamics as well as partial differential equations.

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  How to cite this article:

Javid Ahmad Ganie, Altaf Ahmad and Renu Jain, 2018. Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics. Asian Journal of Mathematics & Statistics, 11: 12-17.

DOI: 10.3923/ajms.2018.12.17

URL: https://scialert.net/abstract/?doi=ajms.2018.12.17
 
Received: July 18, 2018; Accepted: August 07, 2018; Published: November 05, 2018


Copyright: © 2018. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

INTRODUCTION

The theory of Sumudu transform, meant for functions of exponential order is applicable for many applications in mathematics (ordinary and partial differential equations) and control engineering problems. Watugala1 extended the transform to functions of two variables with emphasis on solutions to partial differential equations, which is slightly different from ours. The aim of this paper was to derive, the basic analogue of the double Sumudu transform. Thus, this new transform has very special and useful properties, which can help to intricate applications in sciences and engineering as believed its double transform will also be a natural choice in solving problems with scale and units preserving requirements. Therefore, our aim is to apply the basic analogue of the double Sumudu transform to the age and physiology-dependent population dynamic problem2.

Integral transforms in the classical analysis are the most widely used to solve differential equations and integral equations. A lot of study has been done on the theory and application of integral transforms3,4. Most popular integral transforms are due to Laplace, Fourier, Mellin and Hankel. Most popular integral transforms are due to Laplace, Fourier, Mellin and Hankel. Originally, the Sumudu transform was proposed by Watugala5 as follow:

Let:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

over the set of functions:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

It is applied to the solution of ordinary differential equations in control engineering problems. Subsequently, Weerakoon6 gave the Sumudu transform of partial derivatives and the complex inversion transform who has applied it to the solution of partial differential equations. Basically, the Sumudu transform is not a new integral transform but simply s-multiplied Laplace transform, providing the relation between them7,8. The Sumudu transform is itself linear and preserves linear properties9. In recent past the theory of q-analysis, have been applied in the many areas of mathematics and physics like ordinary fractional calculus, optimal control problems, q-transform analysis, geometric functional theory in finding solutions of the q-difference and q-integral equations 10-13. Albayrak et al.14 introduced the q-analogues of the Sumudu transform and established several theorems related to q-Sumudu transform of some functions. The convolution theorem for q-Sumudu transform has been introduced by Albayrak et al.14. The reader is expected to be familiar with notations of q-calculus. It start with basic definitions and facts from the q-calculus which is necessary for understanding of this study. In this sequel, It assumed that q satisfies the condition 0<|q|<1. q-exponentials have the properties:

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The subject deals with the investigations of q-integrals and q-derivatives of arbitrary order and has gained importance due to its various applications in the areas like ordinary calculus, solutions of the q-differential and q-integral equations, q-transform analysis15-18. The q-integrals are defined as Jackson19:

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The q-analogues of Sumudu transform are defined as follows20:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

over the set of functions:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

and:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

over the set:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

Double sumudu transform: Let f (t, x), t, x ∈ R+ be a function which can be expressed as a convergent infinite series, then its double sumudu transform is given by Tchuenche and Mbare2:

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Definition 1: The q-analogue of the double Sumudu transform is defined as:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

over the set:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

where, u and v are the transform variables for t and x, respectively.

RESULTS

Theorem 1: Let f (t, x), t, x ∈ R+ be a real valued function, then:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics
(1)

The case f (x−y) is more interesting from the biological point of view where such functions are frequently used in mathematical biology with f representing the population density, x the age and y the time or vice-versa. The proof for the case x>y simple and sound enough but with a tedious manipulation. We limit ourselves to the first quadrant as negative populations are biologically irrelevant. Thus, geometrically if the line separating the first quadrant into two equal parts represents the η-axis (the lower part being represented by Q1 and the upper part Q2, while that separating both the second and fourth quadrants represents the ζ-axis (arrow pointing upwards) and -axis (arrow from origin into the fourth quadrant) respectively, then the proof is as follows:

Let f be an even function, then:

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(2)

changing variables and applying Fubini’s theorem let:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

then we have:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

Similarly:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

hence (3) becomes:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

and for odd functions:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics
(3)

from Eq. 1 and 3, it is obvious that if f is even function. Then:

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Lemma 1.1: Let f and g be two real valued functions satisfying3, then:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics
Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

where, a and b are positive constants:

Corollary 1.2:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics
Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

The proof is simple by rewriting the left hand side of the equations as:

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and performing the integrations, bearing in mind that H satisfies Fubini’s Theorem. The application of the basic analogue to double Sumudu transform to partial derivatives is as follows:

Let:

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(4)

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

the inner integral gives:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics

also:

Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics
(5)

alternatively:

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(6)

where, Fq(u, 0) = 0Fq(u) and Fq(0,v) = 0Fq(v). It is obvious from Eq. 5 and 6 that:

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If u and v are equal, we obtain a special case of the Basic analogue of double Sumudu transform known as iterated sumudu transform. Thus, the basic analogue of iterated Sumudu transform of any given function of two variables is defined by:

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APPLICATIONS

In this phase, the validity of the basic analogue of the double Sumudu transform is applied to an evolution equation of population dynamics, namely the famous Kermack-Mackendrick Von Fo- erster type model. Let f be the population density of individuals aged a at time t, λ the death modulus. Then population evolves according to the following system:

ft+fa+λ(a)f

where:

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(7)

taking the q-double Sumudu transform of Eq. 7 with u, v as the transform variables for t, a, respectively after some little arrangements, we get:

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(8)

In order to find the inverse of basic analogue of double Sumudu transform of Eq. 8, which it assumed it exists and satisfies conditions of existence of the double Laplace transform, the proceed as follows.

Let the right-hand side of Eq. 8 be written as:

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Then, taking the inverse of basic analogue of double Sumudu transform of (8) using Corollary (1.2) and Lemma (1.1), we have:

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Image for - Basic Analogue of Double Sumudu Transform and its Applicability in Population Dynamics
(9)

It obtained an approximate solution but it is important to note here that the survival function e-q λ a does not disappear as in Tchuenche21. Thus, in order to obtain for instance e-q λ a, it assumed without loss of reality that u = 1 in the expansion, which gives us an approximation, hence the inequality in (9).

CONCLUSION

The results proved in this study give some contributions to the theory of integral transforms especially q-Sumudu transform and are applicable to the theory of population dynamics. The results proved are believed to be new to the theory of q-calculus and are likely to find certain applications to the solution of the q-integral equations involving various special functions.

REFERENCES

1:  Watugala, G.K., 2002. The Sumudu transform for functions of two variables. Math. Eng. Ind., 8: 293-302.

2:  Tchuenche, J.M. and N.S. Mbare, 2007. An application of the double Sumudu transform. Applied Math. Sci., 1: 31-39.
Direct Link  |  

3:  Purohit, S.D. and S.L. Kalla, 2010. On fractional partial differential equations related to quantum mechanics. J. Phys. A: Math. Theor., Vol. 44, No. 4.
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4:  Saxena, R.K., R. Saxena and S.L. Kalla, 2010. Solution of space-time fractional Schrodinger equation occurring in quantum mechanics. Fract. Calculus Applied Anal., 13: 177-190.
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5:  Watugala, G.K., 1993. Sumudu transform: A new integral transform to solve differential equations and control engineering problems. Integr. Educ., 24: 35-43.
CrossRef  |  Direct Link  |  

6:  Weerakoon, S., 1994. Application of Sumudu transform to partial differential equations. Int. J. Math. Educ. Sci. Technol., 25: 277-283.
CrossRef  |  Direct Link  |  

7:  McLachlan, N.W., 1948. Modern Operational Calculus: With Applications in Technical Mathematics. MacMillan, London

8:  Belgacem, F.B.M., A.A. Karaballi and S.L. Kalla, 2003. Analytical investigations of the Sumudu transform and applications to integral production equations. Math. Prob. Eng., 2003: 103-118.
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9:  Belgacem, F.B.M. and A.A. Karaballi, 2006. Sumudu transform fundamental properties investigations and applications. Int. J. Stochastic Anal., Vol. 2006.
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10:  Abu Risha, M.H., M.H. Annaby, M.E. Ismail and Z.S. Mansour, 2007. Linear q-difference equations. Zeitschrift Anal. Ihre Anwendungen, 26: 481-494.
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11:  Koelink, E., 1996. Quantum groups and q-special functions. Report No. 96-10, Universiteit van Amsterdam.

12:  Purohit, S.D. and R.K. Raina, 2010. Generalized q-Taylor’s series and applications. Gen. Math., 18: 19-28.
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13:  Purohit, S.D. and R.K. Raina, 2011. Certain subclasses of analytic functions associated with fractional q-calculus operators. Math. Scand., 109: 55-70.
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14:  Albayrak, D., S.D. Purohit and F. Ucar, 2013. On q-analogues of Sumudu transform. Anal. Univ. Ovidius Constanta-Seria Matematica, 21: 239-259.
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15:  Ahmad, F., R. Jain and D.K. Jain, 2012. Some transformation formulae of basic analogue of I-function. Asian J. Math. Stat., 5: 158-162.
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16:  Ahmad, A., R. Jain, D.K. Jain and F. Ahmad, 2016. The classical Sumudu transform and its q-image involving mittag-leffler function. Rev. Investig. Operacional, Issue 38, (Forthcoming Accepted).
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17:  Ahmad, A., R. Jain, D.K. Jain and F. Ahmad, 2017. The classical sumudu transform and its q-image of the most generalized hypergeometric and wright-type hypergeometric functions. Int. J. Recent Innov. Trends Comput. Commun., 5: 405-413.
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18:  Ahmad, A., F. Ahmad and S.K. Tripathi, 2016. Application of Sumudu transform in two-parameter fractional telegraph equation. Applied Math. Inform. Sci. Lett., 4: 137-140.
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19:  Jackson, F.H., 1910. On q-definite integrals. Quart. J. Pure Appl. Math., 41: 193-203.

20:  Albayrak, D., S.D. Purohit and F. Ucar, 2014. Certain inversion and representation formulas for q-Sumudu transforms. Hacet. J. Math. Stat., 43: 699-713.
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21:  Tchuenche, J.M., 2003. Solution of an age-physiology dependent population dynamic problem using double laplace transform. Proceedings of the African Mathematical Union International Conference of Mathematical Scientists (AMU-ICMS), November 16-22, 2003, University of Agriculture, Abeokuta, Nigeria, pp: 105-113

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