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. Watugala^{1} 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 physiologydependent population dynamic problem^{2}.
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 transforms^{3,4}. 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 Watugala^{5} as follow:
Let:
over the set of functions:
It is applied to the solution of ordinary differential equations in control engineering problems. Subsequently, Weerakoon^{6} 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 smultiplied Laplace transform, providing the relation between them^{7,8}. The Sumudu transform is itself linear and preserves linear properties^{9}. In recent past the theory of qanalysis, have been applied in the many areas of mathematics and physics like ordinary fractional calculus, optimal control problems, qtransform analysis, geometric functional theory in finding solutions of the qdifference and qintegral equations ^{1013}. Albayrak et al.^{14} introduced the qanalogues of the Sumudu transform and established several theorems related to qSumudu transform of some functions. The convolution theorem for qSumudu transform has been introduced by Albayrak et al.^{14}. The reader is expected to be familiar with notations of qcalculus. It start with basic definitions and facts from the qcalculus which is necessary for understanding of this study. In this sequel, It assumed that q satisfies the condition 0<q<1. qexponentials have the properties:
The subject deals with the investigations of qintegrals and qderivatives of arbitrary order and has gained importance due to its various applications in the areas like ordinary calculus, solutions of the qdifferential and qintegral equations, qtransform analysis^{1518}. The qintegrals are defined as Jackson^{19}:
The qanalogues of Sumudu transform are defined as follows^{20}:
over the set of functions:
and:
over the set:
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 Mbare^{2}:
Definition 1: The qanalogue of the double Sumudu transform is defined as:
over the set:
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:
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 viceversa. 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 ηaxis (the lower part being represented by Q1 and the upper part Q2, while that separating both the second and fourth quadrants represents the ζaxis (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:
changing variables and applying Fubini’s theorem let:
then we have:
Similarly:
hence (3) becomes:
and for odd functions:
from Eq. 1 and 3, it is obvious that if f is even function. Then:
Lemma 1.1: Let f and g be two real valued functions satisfying^{3}, then:
• 

• 

where, a and b are positive constants:
Corollary 1.2:
• 

• 

The proof is simple by rewriting the left hand side of the equations as:
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:
the inner integral gives:
also:
alternatively:
where, F_{q}(u, 0) = _{0}F_{q}(u) and F_{q}(0,v) = _{0}F_{q}(v). It is obvious from Eq. 5 and 6 that:
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:
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 KermackMackendrick 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:
f_{t}+f_{a}+λ(a)f
where:
taking the qdouble Sumudu transform of Eq. 7 with u, v as the transform variables for t, a, respectively after some little arrangements, we get:
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 righthand side of Eq. 8 be written as:
Then, taking the inverse of basic analogue of double Sumudu transform of (8) using Corollary (1.2) and Lemma (1.1), we have:
It obtained an approximate solution but it is important to note here that the survival function eq λ a does not disappear as in Tchuenche^{21}. Thus, in order to obtain for instance eq 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 qSumudu transform and are applicable to the theory of population dynamics. The results proved are believed to be new to the theory of qcalculus and are likely to find certain applications to the solution of the qintegral equations involving various special functions.