A class of integral transforms which satisfies certain assumptions is considered. Several examples, including the Fourier, Laplace and Mellin transforms, are in this class as well as transforms which are yet to be named. The assumptions allow for certain properties to exist for these integral transforms, such as the shifting and convolution properties. Further conditions are presented which allow for the integral transform operator to be injective on a specific class of functions. Applications of the results are provided and analytical solutions to differential equations are obtained. Numerical solutions are then compared to the analytical solutions to demonstrate the usefulness of this class of transforms for solving a particular family of differential equations.

考虑满足某些假设的一类积分变换。包括傅立叶、拉普拉斯和梅林变换在内的几个示例以及尚未命名的变换都属于此类。这些假设允许这些积分变换存在某些属性，例如移位和卷积属性。提出了进一步的条件，这些条件允许积分变换算子对特定的函数类进行单射。提供了结果的应用并获得了微分方程的解析解。然后将数值解与解析解进行比较，以证明此类变换对于求解特定微分方程族的有用性。