We would like to establish the intrinsic structure and properties of Laplace-typed integral transforms. The methodology of this article is done by a consideration with respect to the common structure of kernels of Laplace-typed integral transform, and
The key motivation for pursuing theories for integral transforms is that it gives a simple tool which is represented by an algebraic problem in the process of solving differential equations [
To begin with, let us see the intrinsic structure of Laplace-typed integral transforms. Of course, the structure is dependent on the kernel, and the form of the kernel in Laplace-typed integral transforms is as follows. Laplace transform is defined by
Normally, integral transforms have a base of exponential function and it gets along with how to integrate from
On the other hand, Kreyszig [
This intrinsic structure in Laplace-typed integral transforms has a meaning which can be directly applied to any situation by choosing an appropriate integer
As mentioned before, let us rewrite the definition of Laplace-typed integral transforms, and we would like to call it
If
Let us first check the shifting theorems.
(1) (
(2) (
(2)
By the similar way, we have
Using
If
Table of Laplace-typed integral transform
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Let a function for Let holds for
By the integration by parts,
Continuing this process by substitution as the above and using induction, (3) follows. Let us minutely establish the validity of the statement of (3) by the mathematical induction. For
Finally,
Show that
Let us check an example in [
Solve
It is a well-known fact that
Let there is an integrable function
We note that the above lemma gives validity to the following equality:
Let us put
It is a well-known fact that convolution helps us to solve integral equations of certain type, mainly Volterra integral equation. Hence, we would like to check the theorem by means of some examples using
Solve the Volterra integral equation of the second kind
Solve the Volterra integral equation of the second kind
Find the solution of
Let us check this by the direct calculation. Expanding the given equation, we have
Similarly, we can easily obtain the solution of integral equations by using
Let us put
(1) Since
(2) Since
(3)
(4) In the definition of
In the above theorem, we note that
where
We consider the function
For given differential equations
Let us check the solution of semi-infinite string by
Find the displacement The string is initially at rest on For Furthermore,
Of course there is no infinite string, but our model describes a long string or rope (of negligible weight) with its right end fixed far out on
This paper has constructed the generalized form of Laplace-typed integral transforms and has established the properties of the generalized Laplace-typed integral transform,
The author declares that there are no conflicts of interest regarding the publication of this paper.
This research was supported by Kyungdong University Research Grant.