On the convergence of triple Elzaki transform

In this work, the convergence properties of triple Elzaki transform was examine and the results presented in the form of theorems on convergence, absolute convergence and uniform convergence of triple Elzaki transform. The triple Elzaki transform of triple integral examined for integral evaluation. Finally, Volterra integro-partial differential equation solved by using triple Elzaki transform.

In this paper, we have discussed the various convergence properties of triple Elzaki transformation. The triple Elzaki transform is a very useful technique to solve some differential equations [17][18][19][20], partial differential equations, and integral equations. This transform used as a very efficient tool in simplifying the calculations in many disciplines of engineering and mathematics. Triple Elzaki transform: Let f (x, y, t) be a function that can be expressed as convergent infinite series, and let (x, y, t) ∈ R + 3 , then, the triple Elzaki transform is denoted by: where x, y, t > 0 and ,s,δ are transform variables for x, y and t respectively, whenever the improper integral is convergent.

Convergence theorems of triple Elzaki integral
In this section, we prove the convergence theorem of triple Elzaki integral.

Theorem 2.1
If �(x, y, t) is a continuous function on the positive of the x, y, t-plane. If the integral converges at ρ = ρ • , s = s • , δ = δ • then, the integral: Converges for at ρ < ρ • , s < s • , δ < δ • For the proof, we will use the following lemmas.

Lemma 2.2 If the integral,
1 approach a limit and, and then: let, R 1 → ∞ . If, s < s • , the first term on the right approaches zero, 1 and R 1 such that 0 < 1 < R 1 , then: the theorem proved if the integral on the right converges.
Directly by using the limit test for convergence (see [17]) we see, therefore, integral on the right of (4) converges for, s < s • , hence the integral, h(x, s, t)dx converges at ρ = ρ • , by using fundamental theorem of calculus Eq. (7), becomes, choose ∈ 2 and R 2 and so that 0 < ∈ 2 < R 2 now let ϵ 2 → 0. Both terms on the right which depend on ϵ 2 approach a limit and, now let R 2 → ∞, If s < s • , the first term on the right approaches zero.

converges.
Directly by means of the limit test for convergence (see [17]) we see, therefore, integral on the right of (6) converges for, ρ < ρ • .
Hence, the given integral ρ ∫
the given theorem proved if the integral on the right converges.
Now by using the limit test for convergence (see [17]) we consider, therefore, integral on the right of (12) converges for, δ < δ • , hence, the given integral δ ∫ The proof of the Theorem 2.1 is as follows: therefore, form given hypothesis, ρsδ Therefore the integral (1) converges absolutely for,

Uniform convergence
In this section, we prove the uniform convergence of triple Elzaki Transform.
For the proof, we will use the following lemmas: Hence triple Elzaki transform of, f converges uniformly We now prove the differentiability of triple Elzaki transform.
for the proof, we will use the following lemmas.
x m y n f (x, y, t)dxdy converges uniformly on ρ, ∞), [s, ∞) if The validation is a similar to Lemmas 3.6 and 3.7.

Application of triple Elzaki transform in Volterra integro-partial differential equation
In this section, we use the triple Elzaki transform to solve the Volterra integro-partial differential equation, which already solved in [19] by means of differential transform method.  (29), simplifying, and we obtain, So ū(ρ, s, δ) = 4ρ 3 s 3 δ 3 , applying triple inverse Elzaki transform, to ascertain the exact solution in the shape, Which is the same result that obtained by using differential transform method, see [19].
paper is to study the convergence, absolute convergence and uniform convergence of triple Elzaki transform.

Conclusion
We have demonstrated the convergence, absolute convergence and uniform convergence of triple Elzaki transform they have been shown. Other than these, we got triple Elzaki transform of triple integral and use to illuminate Volterra integro-partial differential equation.

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.
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