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Submission declined on 19 May 2026 by EatingCarBatteries (talk). This draft provides insufficient context for those unfamiliar with the subject. Please see the guide to writing better articles for how to improve your writing. If you would like to continue working on the submission, click on the "Edit" tab at the top of the window. If you have not resolved the issues listed above, your draft will be declined again and potentially deleted. If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors. Please do not remove reviewer comments or this notice until the submission is accepted. Where to get help If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews. If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed. How to improve a draft Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia. Help:Wikitext – how to use the markup Help:Referencing for beginners – how to include references Wikipedia:Article development – how to develop your article Wikipedia:Writing better articles – how to improve your article Wikipedia:Verifiability – make sure your article includes reliable third-party sources You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Add tags to your draft Editor resources Easy tools: Citation bot (help) | Advanced: Fix bare URLs Declined by EatingCarBatteries 20 days ago. Last edited by ~2026-27919-40 4 days ago. Reviewer: Inform author. This draft has been resubmitted and is currently awaiting re-review.

Submission declined on 18 May 2026 by Devonian Wombat (talk). This draft provides insufficient context for those unfamiliar with the subject. Please see the guide to writing better articles for how to improve your writing. This draft reads like an essay or opinion piece. Wikipedia is not a place for original research or personal opinions. The draft should: summarize secondary sources: do not offer your own analysis or arguments; be written from a neutral point of view: represent the subject without bias, avoiding praise, criticism, or persuasive or promotional language; not contain original research: do not include new theories, unpublished ideas, or personal experiences. Instead, only summarize in your own words a range of independent, reliable, published sources that discuss the subject. Declined by Devonian Wombat 20 days ago.

• Comment: Please remove the username tags from within the article's text, those are not supposed to be there. Devonian Wombat (talk) 12:22, 18 May 2026 (UTC)

In mathematics, the ${\displaystyle {\mathbb {J} }}$-transform.^[1] is an effective integral transform developed as a modification of the well-known Sumudu transform^[2] and the N-transform^[3] for solving differential equations arising in applied physical sciences and engineering. The ${\displaystyle {\mathbb {J} }}$-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 ${\displaystyle {\mathbb {J} }}$-transform has been widely employed in obtaining solutions to both ordinary differential equations (ODEs) and partial differential equations (PDEs) with practical real-world applications. ^{[4] [5] [6] [7][8][9][10][11] [12][13]}

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

by

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

Here, ${\displaystyle \mu >0}$ and ${\displaystyle \nu >0}$, provided the limit of the integral exists, and ${\displaystyle \mu \,\,}$and ${\displaystyle \,\,\nu }$ are the ${\displaystyle {\mathbb {J} }}$-transform variables.^[14]. The ${\displaystyle {\mathbb {J} }}$-transform converges to Laplace transform when the variable ${\displaystyle \nu }$=1.

Let ${\displaystyle F(\mu ,\nu )}$ be the ${\displaystyle {\mathbb {J} }}$-transform of the function ${\displaystyle f(t)}$, then the inverse ${\displaystyle {\mathbb {J} }}$-transform is defined as^[15]

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

Equivalently, the complex inverse ${\displaystyle {\mathbb {J} }}$-transform is defined as^[16] ${\displaystyle {\mathbb {J} ^{-1}}\left[F(\mu ,\nu )\right]=f(t)=\lim _{\beta \rightarrow \infty }{\frac {1}{2\pi i}}\int _{\alpha -i\beta }^{\alpha +i\beta }{\frac {1}{\nu ^{2}}}\exp \left({\frac {\mu t}{\nu }}\right)F(\mu ,\nu )d\mu }$

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

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

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

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

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are two constant parameters^[17]

First translation or shifting property:

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

	${\displaystyle {\mathbb {J} }\left[\exp \left(\alpha t\right)f(t)\right]={\frac {\mu -\alpha \nu }{\mu }}F\left(\mu ,{\frac {\mu \nu }{\mu -\alpha \nu }}\right).}$[18]

Moreover, the shifting property provides results based on certain variable transformations^[19]

$${\displaystyle {\mathbb {J} }\left[\exp \left(\alpha t\right)f(t)\right]={\begin{cases}{\frac {\mu -\alpha }{\mu }}F\left(\mu ,{\frac {\mu }{\mu -\alpha }}\right),\,\,{\text{when}}\,\,\nu =1,\,\,{\text{Laplace transform}}&\\(1-\alpha \nu )F\left(1,{\frac {\nu }{1-\alpha \nu }}\right),\,\,{\text{when}}\,\,\mu =1,\,\,{\text{Elzaki transform}}.\end{cases}}}$$ It is evident that for ${\displaystyle \nu =1,}$ we have the Laplace transform^[20] and for ${\displaystyle \mu =1}$ we have the Elzaki transform ^[21] correspondingly.

Scaling property:

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

${\displaystyle {\mathbb {J} }\left[f(\alpha t)\right]={\frac {1}{\alpha }}F\left({\frac {\mu }{\alpha }},\nu \right).}$^[22]

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

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

${\displaystyle {\mathbb {J} }\left[f^{(n)}(t)\right]={\frac {\mu ^{n}}{\nu ^{n}}}F(\mu ,\nu )-\sum _{k=0}^{n-1}{\frac {\mu ^{n-(k+1)}}{\nu ^{n-(k+2)}}}f^{(k)}(0).}$^[23]

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

Let the functions ${\displaystyle f(t)}$ and ${\displaystyle g(t)}$ be in set A. If ${\displaystyle F(\mu ,\nu )}$ and ${\displaystyle W(\mu ,\nu )}$ are the respective ${\displaystyle {\mathbb {J} }}$-transforms of the functions ${\displaystyle f(t)}$ and ${\displaystyle g(t)}$. Then the convolution theorem of ${\displaystyle {\mathbb {J} }}$-transform is defined as^[24]

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

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

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