Introduction

Integral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily.

During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms.

Objectives

In this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\_transforms, Pourreza, Aboodh and etc.

Methods

A new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation.

Results

It is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems.

Conclusion

It has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p(s) and q(s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform.

中文翻译：

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求解积分方程的一种新的一般积分变换

介绍

积分变换对于解决实际问题很重要。Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily.

在过去的二十年中，引入了拉普拉斯变换类中的许多积分变换，例如 Sumudu、Elzaki、Natural、Aboodh、Pourreza、Mohand、G_transform、Sawi 和 Kamal 变换。

目标

在本文中，我们介绍了拉普拉斯变换类中的一般积分变换。我们研究这种变换的性质。然后我们将其与拉普拉斯族中少数现有的积分变换如拉普拉斯、苏木杜、埃尔扎基和 G\_transforms、Pourreza、Aboodh 等进行比较。

方法

引入了新的积分变换。然后讨论了这种积分变换的一些性质。这个积分变换用于求解这个新的变换用于求解高阶初值问题、积分方程和分数阶积分方程。

结果

事实证明，过去几十年引入的拉普拉斯变换类中的那些新变换是这种一般变换的特例。结果表明，除非特殊问题，否则这些变换之间没有优势。

结论

它已经表明，这种新的积分变换涵盖了那些现有的变换，例如针对不同p ( s ) 和q ( s )值的 Laplace、Elzaki 和 Sumudu 变换。我们使用这个新的变换来求解 ODE、积分方程和分数积分方程。此外，我们可以通过使用这个新的通用积分变换来引入新的积分变换。