ON CONVOLUTION PROPERTY OF HY TRANSFORM AND ITS APPLICATIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we have given a new application of HY transform. The convolution property for HY transform is obtained. We used this new result to solve integral equations and fractional integral equation. Few examples have been presented to illustrate the efficiency of the property.

In 2019, Ahmadi [9] defined a new integral transform which is called HY transform. However, some properties of this integral transform are not given such as convolution property. Then, the aim of this paper is to prove convolution property of HY transform which is an important property used to solve integral equations. The basic definition of HY transform is given in Section 2. The convolution property is discussed in Section 3, several test examples to show the effectiveness of the proposed property are given in Section 4, and finally the conclusion is summarized in Section 5.

MATHEMATICAL PRELIMINARIES
In this section, we present some basic idea about HY transform [8].
where HY [ f (t)] is called the HY transform of time function. Variables v is the HY transform variable. It converges if the limit of the integral exists, and diverges if not. The HY −1 will be the inverse of the HY transform.
The following useful formulas follow directly from equation (1):

CONVOLUTION PROPERTY FOR HY TRANSFORM
Proof. The convolution of two function f (t) and g(t) is Using HY transform of equation (1), we get Multiplying both sides of equation (4) by v, we obtain Thus This proves the theorem of convolution.
Proof. From equation (2), we have That is,

By Theorem 3.1, we obtain
Therefore

Theorem 3.3. Let HY [ f (t)] = F(v) and HY [g(t)[= G(v). Then HY transform of
Proof. Assuming that equation (6) is true for n = k. From equation (6) and by mathematical induction, we have that

EXAMPLES
Example 4.1 Consider the following Volterra integral equation of first kind (8) f Taking HY transform on both sides of equation (8), we have By Theorem 3.1, we obtain Thus, Example 4.2 Consider the following Volterra integral equation of second kind Taking HY transform on both sides of equation (10), we have By Theorem 3.1, we obtain Then, x 0 e 2(x−t) f (t)dt = x.
Taking HY transform both sides of (12) and by Theorem 3.1, we obtain Then, Example 4.4 Consider the following Volterra integral equation Taking HY transform on both sides of equation (14) and by Theorem 3.1, we obtain Then, Example 4.5 Consider the following integro-differential equation Taking HY transform on both sides of equation (16) and by Theorem 3.1, we obtain Then, Example 4.6 Consider the following fractional integral equation where I α is the well known Riemann-Liouville fractional integral operator. It is defined by t 0 (t − τ) α−1 y(τ)dτ. By substituting 1 Γ(α) t 0 (t − τ) α−1 y(τ)dτ instead of I α in equation (18) and applying the convolution Theorem 3.1, we have Then,

CONCLUSION
In this paper, convolution property of HY transform of is obtained. We have successfully applied HY transform for the solution of integral equations and fractional integral equations.
For further study, HY transform can be applied for solving other singular integral equations and their systems.