Draft:J-transform

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J-transform

In mathematics, the ${\mathbb {J} }$-transform. is an effective integral transform developed as a modification of the well-known Sumudu transform and the N-transform for solving differential equations arising in applied physical sciences and engineering. The ${\mathbb {J} }$-transform offers several advantages over both the Sumudu transform and the natural transform. Notably, it can be successfully employed to solve complex problems that are often beyond the capability of either transform individually. Introduced by Shehu Maitama and Weidong Zhao in 2020, the ${\mathbb {J} }$-transform has been widely employed in obtaining solutions to both ordinary differential equations (ODEs) and partial differential equations (PDEs) with practical real-world applications.

Formal definition

The ${\mathbb {J} }$-transform of the function $f:[0,\infty )\times \mathbb {R}$ of exponential order is defined over the set of functions, $A=\left\{f(t):\exists N,\xi _{1},\xi _{2}>0,\;{\text{such that}}\;\left|f(t)\right|<N\exp \left({\frac {\left|t\right|}{\xi _{j}}}\right)\;{\text{for}}\;t\in (-1)^{j}\times \left[0,\infty \right)\right\}$

by

${\mathbb {J} }\left[f(t)\right](\mu ,\nu )=F(\mu ,\nu )=\nu \int _{0}^{\infty }\exp \left({\frac {-\mu t}{\nu }}\right)f(t)dt=\nu \lim _{\alpha \rightarrow \infty }\int _{0}^{\alpha }\exp \left({\frac {-\mu t}{\nu }}\right)f(t)dt.$

Here, $\mu >0$ and $\nu >0$, provided the limit of the integral exists, and $\mu \,\,$and $\,\,\nu$ are the ${\mathbb {J} }$-transform variables.. The ${\mathbb {J} }$-transform converges to Laplace transform when the variable $\nu$=1.

Inverse ${\mathbb {J} }$-transform

Let $F(\mu ,\nu )$ be the ${\mathbb {J} }$-transform of the function $f(t)$, then the inverse ${\mathbb {J} }$-transform is defined as

${\mathbb {J} ^{-1}}\left[F(\mu ,\nu )\right]=f(t),\,\,{\text{for}}\,\,t\geq 0.$

Equivalently, the complex inverse ${\mathbb {J} }$-transform is defined as ${\mathbb {J} ^{-1}}\left[F(\mu ,\nu )\right]=f(t)=\lim _{\beta \rightarrow \infty }{\frac {1}{2\pi i}}\int _{\alpha -i\beta }^{\alpha +i\beta }{\frac {1}{\nu ^{2}}}\exp \left({\frac {\mu t}{\nu }}\right)F(\mu ,\nu )d\mu$

$=\sum {\text{residues of}}{\frac {1}{\nu ^{2}}}\exp \left({\frac {\mu t}{\nu }}\right)F(\mu ,\nu )\,\,{\text{at the poles of}}\,\,F(\mu ,\nu ).$

Here, $\mu$ is a complex number and $\alpha$ is a real number.

Properties of ${\mathbb {J} }$-transform

Linearity property: Let the functions $f(t)$ and $w(t)$ be in set $A$. Then, the following linearity property holds

	${\mathbb {J} }\left[\alpha f(t)+\beta w(t)\right]=\alpha {\mathbb {J} }\left[f(t)\right]+\beta {\mathbb {J} }\left[w(t)\right],$

where $\alpha$ and $\beta$ are two constant parameters

First translation or shifting property:

Let the function $\exp \left(\alpha t\right)f(t)$ be in set $A$, where $\alpha$ is constant parameter. Then, the first translation or shifting property is defined as

	${\mathbb {J} }\left[\exp \left(\alpha t\right)f(t)\right]={\frac {\mu -\alpha \nu }{\mu }}F\left(\mu ,{\frac {\mu \nu }{\mu -\alpha \nu }}\right).$

Moreover, the shifting property provides results based on certain variable transformations

$${\mathbb {J} }\left[\exp \left(\alpha t\right)f(t)\right]={\begin{cases}{\frac {\mu -\alpha }{\mu }}F\left(\mu ,{\frac {\mu }{\mu -\alpha }}\right),\,\,{\text{when}}\,\,\nu =1,\,\,{\text{Laplace transform}}&\\(1-\alpha \nu )F\left(1,{\frac {\nu }{1-\alpha \nu }}\right),\,\,{\text{when}}\,\,\mu =1,\,\,{\text{Elzaki transform}}.\end{cases}}$$ It is evident that for $\nu =1,$ we have the Laplace transform and for $\mu =1$ we have the Elzaki transform correspondingly.

Scaling property:

Let the function $F(\mu ,\nu )$ be the $\mathbb {J}$-transform of the function $f(t)$, and $\alpha >0$ ($\alpha$ is a nonnegative number). Then, the scaling property is defined as

${\mathbb {J} }\left[f(\alpha t)\right]={\frac {1}{\alpha }}F\left({\frac {\mu }{\alpha }},\nu \right).$

Theorems of ${\mathbb {J} }$-transform

nth derivatives of the $\mathbb {J}$-transform:

Suppose the function $f{(t)}$ in set $A$ has a ${\mathbb {J} }$-transform, and let $f^{(n)}(t)$ denote its nth derivative. Then, the $\mathbb {J}$-transform of its nth derivative is defined as

${\mathbb {J} }\left[f^{(n)}(t)\right]={\frac {\mu ^{n}}{\nu ^{n}}}F(\mu ,\nu )-\sum _{k=0}^{n-1}{\frac {\mu ^{n-(k+1)}}{\nu ^{n-(k+2)}}}f^{(k)}(0).$

Convolution theorem of ${\mathbb {J} }$-transform:

Let the functions $f(t)$ and $g(t)$ be in set $A$. If $F(\mu ,\nu )$ and $W(\mu ,\nu )$ are the respective ${\mathbb {J} }$-transforms of the functions $f(t)$ and $g(t)$. Then the convolution theorem of ${\mathbb {J} }$-transform is defined as

${\mathbb {J} }\left[(f*g)(t)\right]={\frac {1}{\nu }}F(\mu ,\nu )W(\mu ,\nu ),$

where $f*g$ is the convolution of two functions $f(t)$ and $g(t)$ which is defined by

$\int _{0}^{t}f(t)g(t-\alpha )d\alpha =\int _{0}^{t}f(t-\alpha )g(\alpha )d\alpha .$