Volume 6, Issue 8, August – 2021 International Journal of Innovative Science and Research Technology

ISSN No:-2456-2165

Kharrat-Toma Transform and its Application in

Solving Some Ordinary Differential Equations with
Initial Boundary Conditions
MUSTAPHA R. A.1 , SALAU A. M.2 , BABATUNDE Idowu.3 OGABI C. O.4 , IDOWU G. A.5
1,2,5
Department of Mathematics, Lagos State University, Ojo, Lagos, Nigeria
3,4
Department of Physics, Lagos State University, Ojo, Lagos, Nigeria

Abstract:- In this paper, we adopted a new integral transform in [8], Laguerre transform by Edmond Laguerre in
transform called Kharrat-Toma Transform which can [9], G.K. watugula introduced the Sumudu transform in
be considered to be a basis for a number of potential new [10]. Natural Transform was initiated by Khan and Khan in
integral transforms. Some fundamental properties about [11]. The Aboodh transform was presented by Khalid .S.
this new integral transform were used in this work, Aboodh in [12]. The Elzaki transform was presented by
includes the existence theorem, transportation theorem, Tarig M. Elzaki in [13]. The new integral transform “M-
convolution theorem and inversion equation. The major transform” was suggested by Srivastava in [14]. The ZZ
advantage of this new technique is that it solves ordinary transform was devised by Zafar in [15], A. Kamal and H.
differential equation with variable and constant Sedeeq proposed the Kamal Transform in [16], the Yang
coefficients. Some relevant examples were solved to show transform was introduced by Xiao-jun Yang in [17]. And
the efficiency of this technique. finally R. Saaduh et al introduced ARA transform in [18].

Keywords:- Kharrat-Toma Transform, Ordinary Differential Kharrat et al also interested in integral transform
Equations, Exact Solution. methods, where they applied the Differential transform to
solve boundary value problems represented by differential
I. INTRODUCTION equations from higher orders and also to solve a system of
differential equation [19-21]. In addition, they suggested
Differential equations plays vital role in engineering, hybridization the homotopy perturbation method with
physics, mathematics, Applied mathematics, chemistry and sumundu transform to solve initial value problems for
physiology Applications; In the research work, we suggest nonlinear partial differential equation [22], They also
to develop a new technique for obtaining solution which introduced the hybridization of the Natural transform
approximate the exact solution. method with the homotopy perturbation method to solve
Van Der pol oscillator problem [23].
There are many integral transforms widely used to
solve the differential equation and thus there are several The purpose of this work is to show the efficiency and
works on the differential transform such as laplace applicability of the new integral transform and applied it to
transform was introduced by P.S. Laplace in 1780’s [1], solve Ordinary differential equations with variable and
laplace transform is the oldest integral transform and the constant coefficients as proposed by Kharrat et al in [24].
most widely used. The Stieltjes in [2], was the first to give a The rest of the paper is as follows: we present the basic idea
systematic formulation of the mellin transformation in [3], of Kharrat-Toma transform in Section 2. In Section 3,
The Mohomd Transform was introduced by mohomd Kharrat-Toma transform of some functions is introduced
M.Mahgoub in [4], S. Ahmad et al proposed a new integral and we proof some properties, In section 4, the application
transform to solve higher order linear Laguerre and Hermite for solving ordinary differential equation is shown and
differential equations in [5], D. Hilbert Suggested the conclusion in section 5.
Hilbert transform in [7], J. Radon founded the radon

Kharrat-Toma Transform: Definition 1. The function 𝑓(𝑥) is said to have exponential order on every finite interval in [0, +∞)
If there exist a positive number M that satisfying: |𝑓(𝑥)| ≤ 𝑀𝑒 ∝𝑥 , 𝑀 > 0, ∝> 0, ⩝ 𝑥 ≥ 0
Definition 2: The Kharrat-Toma integral transform and inversion is defined by.
∞
−𝒙
𝟑∫
𝑩[𝒇(𝒙)] = 𝑮(𝒔) = 𝒔 𝒇(𝒙) 𝒆 𝒔𝟐 𝒅𝒙 , 𝒙 ≥ 𝟎
𝟎
∞
−𝒙
𝒇(𝒙) = 𝑩−𝟏 [𝑮(𝑺)] = 𝑩−𝟏 [𝒔𝟑 ∫ 𝒇(𝒙) 𝒆 𝒔𝟐 𝒅𝒙]
𝟎

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Volume 6, Issue 8, August – 2021 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
The B integral transform states that, if 𝑓(𝑥) is piecewise continuous on [0, +∞) and has exponential order. The 𝑩−𝟏 will be
the inverse of the 𝐵 integral transform.

Theorem 1: [Sufficient Condition for Existence of a Kharrat-Toma Transform]: The Kharrat-Toma transform 𝑩[𝒇(𝒙)] exists
𝒃
if it has exponential order and ∫𝟎 |𝒇(𝒙) | 𝒅𝒙 exists for any 𝑏 > 0.
Proof:
∞ 𝒏 ∞
−𝒙 −𝒙 −𝒙
𝟑∫ |𝒇(𝒙) 𝒆 𝒔𝟐 | 𝒅𝒙 𝟑 ∫ |𝒇(𝒙) 𝒆 𝒔𝟐 | 𝒅𝒙 𝟑∫ |𝒇(𝒙) 𝒆 𝒔𝟐 | 𝒅𝒙
𝒔 =𝒔 +𝒔
𝟎 𝟎 𝒏

𝒏 ∞
−𝒙
𝟑 ∫|𝒇(𝒙) | 𝒅𝒙 + 𝒔𝟑 ∫|𝒇(𝒙) | 𝒆 𝒔𝟐 𝒅𝒙
≤𝒔
𝟎 𝟎

𝒏 ∞
−𝒙
𝟑 ∫|𝒇(𝒙) 𝟑∫
≤𝒔 | 𝒅𝒙 + 𝑴𝒔 𝒆∝𝒙 𝒆 𝒔𝟐 𝒅𝒙
𝟎 𝟎

𝒏 ∞
𝟏
𝟑 ∫|𝒇(𝒙) | 𝟑∫ −( 𝟐 −∝)𝒙
=𝒔 𝒅𝒙 + 𝑴𝒔 𝒆 𝒔 𝒅𝒙
𝟎 𝟎

𝒏
𝑩
𝟑 ∫|𝒇(𝒙) |
𝑴𝒔𝟑 𝟏
−( −∝)𝒙 𝟏
=𝒔 𝒅𝒙 + 𝐥𝐢𝐦 [𝒆 𝒔𝟐 ] ; 𝟐 >∝
𝟏 𝑩→∞ 𝒔
𝟎 − ( 𝟐 −∝) 𝟎
𝒔
𝒏
𝟑 ∫|𝒇(𝒙)
𝑴𝒔𝟑
=𝒔 | 𝒅𝒙 +
𝟏
𝟎 −∝
𝒔𝟐
−𝒙
𝟏 ∞
The first integral exists, and the second term is finite for >∝ , so the integral 𝒔𝟑 ∫𝟎 𝒇(𝒙) 𝒆 𝒔𝟐 𝒅𝒙 converges absolutely and the
𝒔𝟐
Kharrat-Toma 𝑩[𝒇(𝒙)] exists.

Kharrat-Toma Transform of Some Functions: In this section we find Kharrat-Toma transform of some functions;
𝑩
𝒇(𝒙) = 𝟏 ←→ 𝑮(𝒔) = 𝒔𝟓 (1)
𝑩−𝟏

𝑩
𝒇(𝒙) = 𝒙𝒏 ←→ 𝑮(𝒔) = 𝒔𝟐𝒏+𝟓 . 𝒏! (2)
𝑩−𝟏

𝑩
𝒌𝒔𝟕
𝒇(𝒙) = 𝒔𝒊𝒏(𝒌𝒙) ←→ 𝑮(𝒔) = (3)
𝟏+𝒌𝟐 𝒔𝟒
𝑩−𝟏

𝑩
𝒔𝟓
𝒇(𝒙) = 𝒄𝒐𝒔(𝒌𝒙) ←→ 𝑮(𝒔) = (4)
𝟏+𝒌𝟐 𝒔𝟒
𝑩−𝟏

𝑩
𝒌𝒔𝟕
𝒇(𝒙) = 𝒔𝒊𝒏𝒉(𝒌𝒙) ←→ 𝑮(𝒔) = 𝟏−𝒌𝟐𝒔𝟒 (5)
𝑩−𝟏

𝑩
𝒔𝟓
𝒇(𝒙) = 𝒄𝒐𝒔𝒉(𝒌𝒙) ←→ 𝑮(𝒔) = (6)
𝟏−𝒌𝟐 𝒔𝟒
𝑩−𝟏

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Proof:
−𝒙 −𝒙 𝑩 −𝒙
∞ ∞
𝑩[1] = 𝒔𝟑 ∫𝟎 𝒆 𝒔𝟐 𝒅𝒙 = −𝒔𝟓 𝐥𝐢𝐦 [𝒆 𝒔𝟐 ] = 𝒔𝟓 𝑩[𝒔𝒊𝒏𝒉(𝒌𝒙)] = 𝒔𝟑 ∫𝟎 𝒔𝒊𝒏𝒉(𝒌𝒙)𝒆 𝒔𝟐 𝒅𝒙
𝑩→∞ 𝟎
𝑢 = 𝒔𝒊𝒏𝒉(𝒌𝒙) ⇒ 𝑑𝑢 = 𝑘𝒄𝒐𝒔𝒉(𝒌𝒙)𝑑𝑥
−𝒙 −𝒙
Where 𝒔𝟐 > 𝟎 𝑑𝑣 = 𝒆 𝒔𝟐 d𝑥 ⇒ 𝑣 = −𝒔𝟐 𝒆 𝒔𝟐
(1)
Then we get,
−𝒙 −𝒙 ∞
𝑛] 𝟑 ∞
𝑩[𝑥 =𝒔 ∫𝟎 𝑥 𝑛
𝒆 𝒅𝒙 𝒔𝟐 𝑩[𝒔𝒊𝒏𝒉(𝒌𝒙)] = 𝒔𝟑 [−𝒔𝟐 𝒔𝒊𝒏𝒉(𝒌𝒙) 𝒆 𝒔𝟐 | +
𝟎
𝑢 = 𝑥 ⇒ 𝑑𝑢 = 𝑛𝑥 𝑛−1 𝑑𝑥
𝑛
𝟐 ∞
−𝒙
−𝒙 −𝒙
𝟐 𝒔𝟐
𝒌𝒔 ∫𝟎 𝒄𝒐𝒔𝒉(𝒌𝒙)𝒆 𝒔𝟐 𝒅𝒙]
𝒔𝟐
𝑑𝑣 = 𝒆 d𝑥 ⇒ 𝑣 = −𝒔 𝒆 −𝒙
∞
(2) = 𝒌𝒔𝟓 ∫𝟎 𝒄𝒐𝒔𝒉(𝒌𝒙)𝒆 𝒔𝟐 𝒅𝒙 ; 𝒔𝟐 > 𝟎
Then we get, (5)
−𝒙 ∞ −𝒙
∞
𝑩[𝑥 𝑛]
=𝒔 𝟑 [−𝒔𝟐 𝑛
𝑥 𝒆 | + 𝒏𝒔𝒔𝟐 𝟐
∫𝟎 𝑥 𝑛−1 𝒔𝟐
𝒆 𝒅𝒙] 𝑢 = 𝒄𝒐𝒔𝒉(𝒌𝒙) ⇒ 𝑑𝑢 = 𝑘𝒔𝒊𝒏𝒉(𝒌𝒙)𝑑𝑥
𝟎 −𝒙 −𝒙

𝟓 ∞
−𝒙
𝑑𝑣 = 𝒆 𝒔𝟐 d𝑥 ⇒ 𝑣 = −𝒔𝟐 𝒆 𝒔𝟐
= 𝒏𝒔 ∫𝟎 𝑥 𝑛−1
𝒆 𝒅𝒙 𝒔𝟐
Yields
𝑛−1
𝑢=𝑥 ⇒ 𝑑𝑢 = (𝑛 − 1)𝑥 𝑛−2 𝑑𝑥 −𝒙 ∞
−𝒙 −𝒙
𝒔𝟐
𝑑𝑣 = 𝒆 d𝑥 ⇒ 𝑣 = −𝒔𝟐 𝒆 𝒔𝟐 𝑩[𝒔𝒊𝒏𝒉(𝒌𝒙)] = 𝒌 𝒔𝟓 [−𝒔𝟐 𝒄𝒐𝒔𝒉(𝒌𝒙) 𝒆 𝒔𝟐 |
𝟎
∞
−𝒙
Yields
+ 𝒌𝒔𝟐 ∫ 𝒔𝒊𝒏𝒉(𝒌𝒙)𝒆 𝒔𝟐 𝒅𝒙]
−𝒙 ∞
𝑩[𝑥 𝑛 ] = 𝒏𝒔𝟓 [−𝒔𝟐 𝑥 𝑛−1 𝒆 𝒔𝟐 | + (𝒏 𝟎
𝟎 𝒌
∞ = 𝒌𝒔𝟓 [𝒔𝟐 + 𝑩[𝒔𝒊𝒏𝒉(𝒌𝒙)]]
−𝒙 𝒔
− 𝟏)𝒔𝟐 ∫ 𝑥 𝑛−2 𝒆 𝒔𝟐 𝒅𝒙]
Then we get,
𝟎 𝒌𝒔𝟕
−𝒙
𝟕 ∞ 𝑛−2 𝑩[𝒔𝒊𝒏𝒉(𝒌𝒙)] = 𝟏−𝒌𝟐𝒔𝟒
= 𝒏(𝒏 − 𝟏)𝒔 ∫𝟎 𝑥 𝒔𝟐
𝒆 𝒅𝒙 = ⋯ =
𝟐𝒏+𝟓 (6) Proof the same way as in (5)
𝒔 . 𝒏! Theorem 2:
Let 𝑩[𝒇𝟏 (𝒙)] = 𝑮𝟏 (𝒔), … , 𝑩[𝒇𝒏(𝒙)] = 𝑮𝒏 (𝒔) and the
Where 𝒔𝟐 > 𝟎
constant 𝒄𝟏 …,𝒄𝒏 , then
(3) 𝒏 𝒏
−𝒙
∞
𝑩[𝒔𝒊𝒏(𝒌𝒙)] = 𝒔𝟑 ∫𝟎
𝒔𝒊𝒏(𝒌𝒙)𝒆 𝒅𝒙 𝒔𝟐 𝑩 [∑ 𝒄𝒊 𝒇𝒊 (𝒙)] = ∑ 𝒄𝒊 𝑩 [𝒇𝒊(𝒙)]
𝑢 = 𝒔𝒊𝒏(𝒌𝒙) ⇒ 𝑑𝑢 = 𝑘𝒄𝒐𝒔(𝒌𝒙)𝑑𝑥 𝒊=𝟏 𝒊=𝟏
−𝒙 −𝒙 Proof:
𝒔𝟐 𝟐 𝒔𝟐 ∞ 𝒏
𝑑𝑣 = 𝒆 d𝑥 ⇒ 𝑣 = −𝒔 𝒆 𝒏
−𝒙
𝟑∫
𝑩 [∑ 𝒄𝒊 𝒇𝒊 (𝒙)] = 𝒔 ∑ 𝒄𝒊 𝒇𝒊 (𝒙) 𝒆 𝒔𝟐 𝑑𝑥
Then we get, 𝒊=𝟏 𝟎 𝒊=𝟏
−𝒙 ∞
𝒏 ∞
𝑩[𝒔𝒊𝒏(𝒌𝒙)] = 𝒔𝟑 [−𝒔𝟐 𝒔𝒊𝒏(𝒌𝒙) 𝒆 𝒔𝟐 | + −𝒙
𝟎
∞
−𝒙 = ∑ 𝒄𝒊 (𝒔𝟑 ∫ 𝒇𝒊 (𝒙) 𝒆 𝒔𝟐 𝑑𝑥)
𝒌𝒔𝟐 ∫𝟎 𝒄𝒐𝒔(𝒌𝒙)𝒆 𝒅𝒙] 𝒔𝟐 𝒊=𝟏 𝟎
∞
−𝒙
= ∑𝒏𝒊=𝟏 𝒄𝒊 𝑩[ 𝒇𝒊 (𝒙)]
= 𝒌𝒔𝟓 ∫𝟎 𝒄𝒐𝒔(𝒌𝒙)𝒆 𝒅𝒙 ; 𝒔𝟐 > 𝟎 𝒔𝟐

𝑢 = 𝒄𝒐𝒔(𝒌𝒙) ⇒ 𝑑𝑢 = −𝑘𝒔𝒊𝒏(𝒌𝒙)𝑑𝑥 Translation Property, First Shifting Property,

−𝒙 −𝒙
𝑑𝑣 = 𝒆 d𝑥 ⇒ 𝑣 =𝒔𝟐 −𝒔𝟐 𝒆 𝒔𝟐 Convolution Theorem, Kharrat-Toma Transform of
Yields 𝑥 𝑛 𝑓(𝑥); 𝑛 ≥ 1, Kharrat-Toma Transform of Derivatives
−𝒙 ∞ was exhaustively proved in [24].
𝑩[𝒔𝒊𝒏(𝒌𝒙)] = 𝒌 𝒔𝟓 [−𝒔𝟐 𝒄𝒐𝒔(𝒌𝒙) 𝒆 𝒔𝟐 |
𝟎 Application: In this section, we introduce the methodology
∞
−𝒙 of application of Kharrat-Toma transform for solving initial
− 𝒌𝒔𝟐 ∫ 𝒔𝒊𝒏(𝒌𝒙)𝒆 𝒔𝟐 𝒅𝒙] value problem. This new integral transform can be used as
𝟎 an effective tool for solving ordinary differential equations
𝒌 with initial conditions, we also compared the result of
= 𝒌𝒔𝟓 [𝒔𝟐 − 𝑩[𝒔𝒊𝒏(𝒌𝒙)]] Kharrat-Toma transform with the traditional way of solving
𝒔
ordinary differential equations(Exact solution). We
Then we get, considered some examples to show the use and efficiency
𝒌𝒔𝟕
𝑩[𝒔𝒊𝒏(𝒌𝒙)] = 𝟏+𝒌𝟐𝒔𝟒 of this integral transform.
(4)
Proof in the same way as in (3) Example 1:
We consider the initial value problem

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′ 𝑑
𝑦 = 2𝑦 + 3 𝑒 −6𝑥 𝑦 = 𝑒 −6𝑥
{ 𝑑𝑥
𝑦(0) = 1 𝑑
(1) ∫ 𝑑𝑥 𝑒 −6𝑥 𝑦 = ∫ 𝑒 −6𝑥 𝑑𝑥
1
𝑒 −6𝑥 𝑦(𝑥) = − 6 𝑒 −6𝑥 + 𝑐
Using analytical approach (Exact solution), 𝑦 = 1 When 𝑥 = 0
By using integrating factor(IF) 𝑒 ∫ 𝑓(𝑥)𝑑𝑥 1
1(1) = − 6 + 𝑐
𝑦 ′ (𝑥) + 𝑓(𝑥)𝑦(𝑥) = 𝑔(𝑥) 7
𝑒 ∫ −2 𝑑𝑥 = 𝑒 −2𝑥 𝑐=6
𝑒 −2𝑥 𝑦 ′ (𝑥) − 2𝑒 −2𝑥 𝑦(𝑥) = 3𝑒 −2𝑥 7
∴ 𝑦(𝑥) = 6 𝑒 2𝑥 − 6
1
𝑑 −2𝑥
𝑑𝑥
𝑒 𝑦 = 3𝑒 −2𝑥
𝑑 Using Kharrat-Toma for example 2,
∫ 𝑑𝑥 𝑒 −2𝑥 𝑦 = ∫ 3𝑒 −2𝑥 𝑑𝑥
3 𝑦 ′ = 6𝑦 + 1
𝑒 −2𝑥 𝑦(𝑥) = − 2 𝑒 −2𝑥 + 𝑐 𝑦 ′ − 6𝑦 = 1
𝑦 = 1 When 𝑥 = 0 𝐵[𝑦 ′ − 6𝑦] = 𝐵[1]
3
1(1) = − + 𝑐 𝐵[𝑦 ′ ] − 6𝐵[𝑦] = 𝐵[1]
2
5
𝑐= Applying the kharrat-Toma transform on (1), we get
2
5 3
∴ 𝑦(𝑥) = 2 𝑒 2𝑥 − 2 1
𝐺 (𝑆) − 𝑠 3 𝑦(0) − 6𝐺 (𝑠) = 𝑠 5
𝑠2
1
Using Kharrat-Toma for example 1, 𝐺(𝑆) − 𝑠 3 − 6𝐺(𝑠) = 𝑠 5
𝑦 ′ = 2𝑦 + 3 𝑠2
1
𝑦 ′ − 2𝑦 = 3 𝐺(𝑆) [ 2 − 6] = 𝑠 5 + 𝑠 3
𝐵[𝑦 ′ − 2𝑦] = 𝐵[3] 𝑠
𝑠7 + 𝑠5
𝐵[𝑦 ′ ] − 2𝐵[𝑦] = 𝐵[3] 𝐺(𝑆) =
1 − 6𝑠 2
Applying the kharrat-Toma transform on (1), we get 𝑠7 𝑠5
= +
1 1 − 6𝑠 2 1 − 6𝑠 2
𝐺(𝑆) − 𝑠 3 𝑦(0) − 2𝐺(𝑠) = 3𝑠 5 𝑠 5 𝑠 5 − 7𝑠 7 𝑠5
𝑠2 = − +
1 1 1 − 6𝑠 2 1 − 6𝑠 2
𝐺(𝑆) − 𝑠 3 − 2𝐺(𝑠) = 3𝑠 5 1 5 7 𝑠5
𝑠2 =− 𝑠 +
1 6 6 (1 − 6𝑠 2 )
𝐺(𝑆) [ 2 − 2] = 3𝑠 5 + 𝑠 3
𝑠 1 7
3𝑠 7 + 𝑠 5 = 𝐵 [− ] + 𝐵 [ 𝑒 6𝑥 ]
𝐺(𝑆) = 6 6
1 − 2𝑠 2
3𝑠 7 𝑠5 Applying the inverse Kharrat-Toma transform then the exact
= + solution for the Initial value problem is
1 − 2𝑠 2 1 − 2𝑠 2
𝑠 5 𝑠 5 − 5𝑠 7 𝑠5 1 7
= − + 𝑦(𝑥) = − + 𝑒 6𝑥
1 1 − 2𝑠 2 1 − 2𝑠 2 6 6
3 5 5 𝑠5
=− 𝑠 + Example 3:
2 2 (1 − 2𝑠 2 )
3 5 We consider the initial value problem
= 𝐵 [− ] + 𝐵 [ 𝑒 2𝑥 ] 𝑦 ′ = −3𝑦 + 5
2 2 {
𝑦(0) = 1
Applying the inverse Kharrat-Toma transform then the exact (2)
solution for the Initial value problem is
3 5 Using analytical approach (Exact solution),
𝑦(𝑥) = − + 𝑒 2𝑥 By using integrating factor(IF) 𝑒 ∫ 𝑓(𝑥)𝑑𝑥
2 2
𝑦 ′ (𝑥) + 𝑓(𝑥)𝑦(𝑥) = 𝑔(𝑥)
Example 2: 𝑒 ∫ 3𝑑𝑥 = 𝑒 3𝑥
We consider the initial value problem 𝑒 3𝑥 𝑦 ′ (𝑥) + 3𝑒 3𝑥 𝑦(𝑥) = 5𝑒 3𝑥
𝑦 ′ = 6𝑦 + 1 𝑑 3𝑥
𝑒 𝑦 = 5𝑒 3𝑥
{ 𝑑𝑥
𝑦(0) = 1 𝑑
(2) ∫ 𝑑𝑥 𝑒 3𝑥 𝑦 = ∫ 5𝑒 3𝑥 𝑑𝑥
5
𝑒 3𝑥 𝑦(𝑥) = 𝑒 3𝑥 + 𝑐
3
Using analytical approach (Exact solution), 𝑦 = 1 When 𝑥 = 0
By using integrating factor(IF) 𝑒 ∫ 𝑓(𝑥)𝑑𝑥 5
1(1) = 3 + 𝑐
𝑦 ′ (𝑥) + 𝑓(𝑥)𝑦(𝑥) = 𝑔(𝑥) 2
𝑒 ∫ −6 𝑑𝑥 = 𝑒 −6𝑥 𝑐 = −3
𝑒 −6𝑥 𝑦 ′ (𝑥) − 6𝑒 −6𝑥 𝑦(𝑥) = 𝑒 −6𝑥

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Volume 6, Issue 8, August – 2021 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
2 5 [2]. Stieltjes, T.S., 1894. Recherches sur les fractions
∴ 𝑦(𝑥) = − 𝑒 −3𝑥 +
3 3
continues, Annales de la Faculte des Sciences de
Using Kharrat-Toma for example 2,
Toulouse, pp: 1-123.
𝑦 ′ = −3𝑦 + 5
[3]. Mellin, H., 1896. ´lUber die fundamentale
𝑦 ′ + 3𝑦 = 5
Wichtgkeit des Satzes von Cauchy f´lur die Theorien
𝐵[𝑦 ′ + 3𝑦] = 𝐵[5]
der Gammaund der hypergeometrischen funktionen,
𝐵[𝑦 ′ ] + 3𝐵[𝑦] = 𝐵[5] Acta Soc. Fennicae, 21: 1-115.
[4]. La Fourier, J., 1822. ThÂt'eorie Analytique de la
Applying the kharrat-Toma transform on (1), we get Chaleur, English Translation by A. Freeman, Dover
1
𝐺(𝑆) − 𝑠 3 𝑦(0) + 3𝐺(𝑠) = 5𝑠 5 Publications.
𝑠2 [5]. Hilbert, D., 1912. Grundz´luge einer allgemeinen
1
𝐺(𝑆) − 𝑠 3 + 3𝐺(𝑠) = 5𝑠 5 Theorie der linearen Integralgeichungen, pp: 75-77.
𝑠2 [6]. Radon, J., 1917. ´lUber die Bestimmung von
1
𝐺(𝑆) [ 2 + 3] = 5𝑠 5 + 𝑠 3 Funktionen durch ihre Integralw ´larte l´langs
𝑠 gewisser manningfaltigkreiten, Ber Verh. Akad. Wiss.
5𝑠 7 + 𝑠 5
𝐺(𝑆) = Math. - Nat., Leipzig, 69: 262-277.
1 + 3𝑠 2 [7]. Debnath, L., 1960. On Laguerre transform, Bull.
5𝑠 7 𝑠5 Calcutta Math. Soc., 55: 69-77.
= +
1 + 3𝑠 2 1 + 3𝑠 2 [8]. Watugula, G.K., 1993. Sumudu transform: a new
𝑠 5 𝑠 5 − 2𝑠 7 𝑠5 integral transform to solve differential equations and
= − +
1 1 + 3𝑠 2 1 + 3𝑠 2 control engineering problems, International Journal of
5 5 2 𝑠5 Mathematical Education in Science and Technology,
= 𝑠 −
3 3 (1 + 3𝑠 2 ) 24(1): 35-43.
5 2 [9]. Khan Z.H. and W.A. Khan, 2008. Natural transform-
= 𝐵 [ ] − 𝐵 [ 𝑒 −3𝑥 ] properties and applications, NUST Journal of
3 3
Engineering Sciences, 1: 127-133.
Applying the inverse Kharrat-Toma transform then the [10]. Elzaki, T.M., 2011. The new integral transform
exact solution for the Initial value problem is "Elzaki transform", Global Journal of Pure and
5 2 Applied Mathematics, 7(1): 57-64.
𝑦(𝑥) = − 𝑒 −3𝑥 [11]. Aboodh, K.S., 2013. The new integral transform
3 3
"Aboodh Transform" Global Journal of Pure and
II. CONCLUSION Applied Mathematics, 9(1): 35-43.
[12]. Srivastava, H.M., 2015. A new integral transform and
The main aim of this paper is to present some its applications, Acta Mathematica Scientia, 35B:
fundamental properties of newly defined integral transform 1386-1400.
“Kharrat-Toma Transform” as proposed by Kharrat et al in [13]. Zafar, Z., 2016. ZZ transform method, International
[24] . Convolution theorem is also proved for Kharrat-Toma Journal of Advanced Engineering and Global
transform. It provides a new mathematical tool to solve Technology, 04(01).
ordinary differential equations of variable and constant [14]. Kamal, A. and H. Sedeeg, 2016. The new integral
coefficients with initial condition. We validated the new transform "Kamal Transform" Advances in Theoretical
integral transform method by giving a sophisticated and Applied Mathematics, 11(4): 451-458.
analytical solution to solve ordinary differential equations. [15]. Xiao-Jun, Y., 2016. A New Integral Transform
Method For Solving Steady Heat-Transfer Problem,
FUTURE WORK Thermal Science, Vol. 20, Suppl., 3: S639-S642.
[16]. Mohand M.A. Mahgoub, 2017. The new integral
We are already improving on the existing method for transform "Mohand Transform" Advances in
its ability to solve non-linear differential equations and the Theoretical and Applied Mathematics, 12(2): 113-120.
existence of it’s orthogonality property which was not [17]. Ahmadi, S.A.P., H., Hosseinzadeh, A.Y.A. Cherati,
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[22]. Kharrat, B.N. and G. Toma, 2018. Modified
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