Draft:J-transform

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In mathematics, the ${\mathbb {J} }$-transform is an integral transform introduced as a modification of the well-known Sumudu transform and the Natural transform for solving differential equations arising in applied physical sciences and engineering. The ${\mathbb {J} }$-transform possesses several useful properties that distinguish it from the Sumudu transform and the natural transform, and it can be applied to problems that may be difficult to handle using either transform independently. The ${\mathbb {J} }$-transform was introduced by Shehu Maitama and Weidong Zhao in 2020. Since its introduction, it has been applied to the solution of various ordinary differential equations (ODEs) and partial differential equations (PDEs), including problems involving fractional-order models and real-life applications in the physical science and engineering.

Formal definition

The ${\mathbb {J} }$-transform of the function $f(t)$ 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 that $f{(t)}$$\in$$A$ possesses a ${\mathbb {J} }$-transform, and let $f^{(n)}(t)$ denote its nth derivative. Then, the $\mathbb {J}$-transform of its nth derivative of $f{(t)}$ is given by:

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

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

Let the functions $f(t)$ and $g(t)$ be in set $A$. Suppose that $F(\mu ,\nu )$ and $W(\mu ,\nu )$ denote the respective ${\mathbb {J} }$-transforms of function $f(t)$ and $g(t)$. Then, the convolution theorem for the ${\mathbb {J} }$-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

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