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A New Approach on Transforms: Formable Integral Transform and Its Applications

Authors:
Rania Saadeh at Zarqa University
Rania Saadeh
Bayan fu’ad Ghazal at Zarqa University
Bayan fu’ad Ghazal
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Abstract

In this paper, we introduce a new integral transform called the Formable integral transform, which is a new efficient technique for solving ordinary and partial differential equations. We introduce the definition of the new transform and give the sufficient conditions for its existence. Some essential properties and examples are introduced to show the efficiency and applicability of the new transform, and we prove the duality between the new transform and other transforms such as the Laplace transform, Sumudu transform, Elzaki transform, ARA transform, Natural transform and Shehu transform. Finally, we use the Formable transform to solve some ordinary and partial differential equations by presenting five applications, and we evaluate the Formable transform for some functions and present them in a table. A comparison between the new transform and some well-known transforms is made and illustrated in a table.
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Axioms2021,10,332.https://doi.org/10.3390/axioms10040332www.mdpi.com/journal/axioms
Article
ANewApproachonTransforms:FormableIntegralTransform
andItsApplications
RaniaZohairSaadeh*andBayanfu’adGhazal
DepartmentofMathematics,FacultyofScience,ZarqaUniversity,Zarqa13110,Jordan;20209128@zu.edu.jo
*Correspondence:rsaadeh@zu.edu.jo
Abstract:Inthispaper,weintroduceanewintegraltransformcalledtheFormableintegraltrans‐
form,whichisanewefficienttechniqueforsolvingordinaryandpartialdifferentialequations.We
introducethedefinitionofthenewtransformandgivethesufficientconditionsforitsexistence.
Someessentialpropertiesandexamplesareintroducedtoshowtheefficiencyandapplicabilityof
thenewtransform,andweprovethedualitybetweenthenewtransformandothertransformssuch
astheLaplacetransform,Sumudutransform,Elzakitransform,ARAtransform,Naturaltransform
andShehutransform.Finally,weusetheFormabletransformtosolvesomeordinaryandpartial
differentialequationsbypresentingfiveapplications,andweevaluatetheFormabletransformfor
somefunctionsandpresenttheminatable.Acomparisonbetweenthenewtransformandsome
well‐knowntransformsismadeandillustratedinatable.
Keywords:Laplacetransform;Shehutransform;Naturaltransform;ARAtransform;Fourier
transform;Elzakitransform;Sumudutransform;ordinarydifferentialequation;partialdifferential
equation;integraltransform
1.Introduction
Differentialequationsrepresentafieldofmathematicsthathasgreatapplicationsin
science,sincetheyareusedinmathematicalmodeling[1–9]andhenceaidinfindingso‐
lutionsinphysicalandengineeringproblemsinvolvingfunctionsofoneorseveralvaria‐
bles,suchasthepropagationofheatorsound,fluidflow,elasticity,electrostatics,electro‐
dynamics,etc.
Fordecades,methodsforsolvingdifferentialequationshavebeenimportantsubjects
forresearchers,[10–19]becauseoftheirimportantapplicationsinvariousfieldsofscience.
Thetechniqueofusingintegraltransformshasproveditsefficiencyandapplicabilityin
solvingordinaryandpartialdifferentialequations.
Forthefunction𝑔󰇛𝑡󰇜and𝑡 ∈󰇛 ∞,∞ 󰇜,theintegraltransformisobtainedbycom‐
putingtheimproperintegral
where𝑘󰇛𝑠,𝑡󰇜iscalledthekerneloftheintegraltransformandsisthevariableofthetrans‐
form,whichmightberealorcomplexnumberandisindependentofthevariablet.The
theoryofintegraltransformsgoesbacktotheworkofP.S.Laplacein1780[19,20]and
Fourierin1822.Recently,theideaofusingintegraltransformsinsolvingdifferential
equationsandintegralequationshasbeencommonlyusedbymanyresearchersinthe
literature[21–30].
TheLaplacetransformisdefinedas:
Citation:Saadeh,R.Z.;Ghazal,B.f.G.
ANewApproachonTransforms:
FormableIntegralTransformandIts
Applications.Axioms2021,10,332.
https://doi.org/10.3390/
axioms10040332
AcademicEditor:
PalleE.T.Jorgensen
Received:25September2021
Accepted:5November2021
Published:1December2021
Publisher’sNote:MDPIstaysneu‐
tralwithregardtojurisdictional
claimsinpublishedmapsandinstitu‐
tionalaffiliations.
Copyright:©2021bytheauthors.
LicenseeMDPI,Basel,Switzerland.
Thisarticleisanopenaccessarticle
distributedunderthetermsand
conditionsoftheCreativeCommons
Attribution(CCBY)license
(https://creativecommons.org/license
s/by/4.0/).
£ 󰇟𝑔󰇛𝑡󰇜󰇠󰇛𝑠󰇜 𝑘 󰇛𝑠,𝑡󰇜 𝑔󰇛𝑡󰇜𝑑𝑡,
 (1)
Axioms2021,10,3322of22
£ 󰇟𝑔󰇛𝑡󰇜󰇠𝐺󰇛𝑠󰇜 exp󰇛𝑠𝑡󰇜𝑔󰇛𝑡󰇜𝑑𝑡
,(2)
anditshowshighefficiencyinsolvingaclassofdifferentialequations.Byreplacingthe
variable𝑠by𝑖𝑤andmultiplyingEquation(2)by
√weobtainthewell‐knownFourier
integraltransform,definedas𝐹 󰇟𝑔󰇛𝑡󰇜󰇠𝑔󰇛𝑤󰇜
1
√
2𝜋exp󰇛𝑖𝑤𝑡󰇜𝑔󰇛𝑡󰇜𝑑𝑡 .
 (3)
Thesetransformsarebasicinthestudyofintegraltransforms,butthedifferencebe‐
tweenthemisthattheLaplacetransformisapplicableforbothstableandunstablesys‐
tems,buttheFouriertransformisonlydefinedforstablesystems.
Formanyyears,thetheoryofintegraltransformshasbeenverywidelystudiedin
themathematicalliterature,andmanyresearchershaveinvestigatednewtransformssuch
asthez‐transform[29],theMellinintegraltransform[30],theLaplace–Carsontransform
[31]andtheHankeltransform[32,33].
TheSumuduintegraltransform[34]wasintroducedin1993.Itshowedapplicability
insolvingreal‐lifeproblemsandwasusedforsolvingdifferentialequations.TheSumudu
integraltransformisdefinedas:
𝑆󰇟𝑔󰇛𝑡󰇜󰇠󰇛𝑢󰇜𝐺󰇛𝑢󰇜 1
𝑢exp 𝑡
𝑢𝑔󰇛𝑡󰇜𝑑𝑡 .
(4)
In2008,BelgacemandSilambarasanintroducedtheNaturaltransform,asfollows:
𝑁󰇟𝑔󰇛𝑡󰇜󰇠󰇛𝑠,𝑢󰇜𝑅󰇛𝑠,𝑢󰇜 1
𝑢exp 𝑠𝑡
𝑢𝑔󰇛𝑡󰇜𝑑𝑡,𝑠,𝑢0.
(5)
TheElzakiintegraltransformwasobtainedin2011,withthedefinition
𝐸󰇟𝑔󰇛𝑡󰇜 󰇠󰇛𝑢󰇜𝑇󰇛𝑢󰇜 𝑢exp 𝑡
𝑢𝑔󰇛𝑡󰇜𝑑𝑡 .
(6)
ThisiscloselyrelatedtotheLaplaceandSumuduintegraltransforms.
TheShehuintegraltransformisgivenby
𝕊󰇟𝑔󰇛𝑡󰇜󰇠󰇛𝑠,𝑢󰇜𝑉󰇛𝑠,𝑢󰇜exp 𝑠𝑡
𝑢𝑔󰇛𝑡󰇜𝑑𝑡,𝑠,𝑢0
.(7)
Thisalsoshowstheabilitytosolveaclassofdifferentialequationsand,combined
withothernumericalmethodsofsolvingdifferentialequations,toofferanewapproach
indealingwithfractionaldifferentialequations.
In2020,theARAtransformwasintroducedbySaadehetal.andwasimplemented
tosolveawiderangeoffractionalordinaryandpartialdifferentialequations.
TheARAintegraltransformisgivenby
𝒢󰇟𝑔󰇛𝑡󰇜󰇠󰇛𝑠󰇜𝐺󰇛𝑛,𝑠󰇜 𝑠𝑡exp󰇛𝑠𝑡󰇜𝑔󰇛𝑡󰇜𝑑𝑡,𝑠0 .
(8)
Recently,theabovetransformsandothershavebeencombinedwithotheranalytical
methodsinmathematicstosolveawiderangeoflinearandnonlinearfractionalandor‐
dinarydifferentialequations,andothermethodsareshownin.
Inthispaper,weproposeanewintegraltransformcalledtheFormabletransform.
WeintroducethedefinitionandsomepropertiesofthenewtransforminSection2.The
Axioms2021,10,3323of22
dualitiesbetweentheFormableandothertransformsareillustratedinSection3withsome
examples.InSection4,weapplytheFormabletransforminsomeordinaryandpartial
differentialequationstoshowitsefficiencyandaccuracythroughapplications.Finally,
thevaluesoftheFormabletransformforsomespecialfunctionsarepresentedinatable.
2.DefinitionsandTheorems
InthissectionweintroducethedefinitionofthenewtransformcalledtheFormable
transform,togetherwithsometheoremsandpropertiesofthenewtransform.
Definition1.TheFormableintegraltransformofafunction𝑔󰇛𝑡󰇜ofexponentialorderisdefined
overthesetoffunctions
𝑊󰇝𝑔󰇛𝑡󰇜:∃ 𝑁∈󰇛0, ∞󰇜,𝜏0
𝑓
𝑜𝑟 𝑖1,2 , |𝑔󰇛𝑡󰇜| 𝑁exp 󰇡
󰇢,ift ∈󰇟0, ∞󰇜󰇞,
inthefollowingform:
𝑅󰇟𝑔󰇛𝑡󰇜󰇠𝐵󰇛𝑠,𝑢󰇜 𝑠 exp 󰇛 𝑠𝑡󰇜
𝑔󰇛𝑢𝑡󰇜𝑑𝑡(9)
Thisisequivalentto𝑅󰇟𝑔󰇛𝑡󰇜󰇠
exp 󰇛
󰇜
𝑔󰇛𝑡󰇜𝑑𝑡 (10)
𝑅󰇟𝑔󰇛𝑡󰇜󰇠 𝑠𝑢lim
→exp 𝑠𝑡
𝑢𝑔󰇛𝑡󰇜𝑑𝑡
, s 0, 𝑢0
where𝑠 and 𝑢aretheFormabletransform’svariables,𝑥isarealnumberandtheintegral
istakenalongtheline 𝑡 𝑥.Afunctiong(t)issaidtobeofexponentialorder𝑐ifthere
existconstants𝑀and𝑇suchthat|𝑔󰇛𝑡󰇜|𝑀 𝑒 for all 𝑡𝑇.Here,wementionthatwe
chosethename“Formable”forthisnewtransformbecauseofitsflexibilityinsolvingor‐
dinaryandpartialdifferentialequations.Inaddition,ithasadualitywithotherwell‐
knowntransformsthatwillbeconsideredlater.ToshowtheapplicabilityoftheFormable
transform,wecomputethetransformforseveralfunctionsinSection3.Wecomparethe
resultswithothervaluesfromsomewell‐knowntransformsandillustratetheminatable
intheAppendixA.
TheinverseFormabletransformofafunction𝑔󰇛𝑡󰇜isgivenby
𝑅󰇟𝐵󰇛𝑠,𝑢󰇜󰇠𝑔󰇛𝑡󰇜 1
2𝜋𝑖 1
𝑠

 exp 𝑠𝑡
𝑢 𝐵󰇛𝑠,𝑢󰇜𝑑𝑠.
Thatis,fromthedefinitionoftheFouriertransform,weknow
𝐹 󰇟𝑔󰇛𝑡󰇜󰇠𝐹 󰇛𝑤󰇜 1
√
2𝜋 𝑒
 𝑔󰇛𝑡󰇜
 𝑑𝑡
𝐹󰇟𝐹󰇛𝑤󰇜󰇠𝑔󰇛𝑡󰇜1
√
2𝜋 𝑒
 𝐹󰇛𝑤󰇜
 𝑑𝑤.
Then
𝑔󰇛𝑡󰇜1
√2𝜋 𝑒
 󰇩1
√2𝜋 𝑒
 𝑔󰇛𝑡󰇜
 𝑑𝑡󰇪
 𝑑𝑤
1
2𝜋 𝑒
 󰇩 𝑒
 𝑔󰇛𝑡󰇜
 𝑑𝑡󰇪
 𝑑𝑤,
(11)
where𝑔󰇛𝑡󰇜 isafunctiondefinedonthedomain󰇛 ∞,∞ 󰇜,sothatfor𝑡 ∈󰇛∞,0󰇜weas‐
sumethat𝑔󰇛𝑡󰇜0.Hencefort 0,let𝑔󰇛𝑡󰇜𝑔󰇛𝑡󰇜 𝑢(t) 𝑒,where𝑢󰇛𝑡󰇜istheunitstep
functionand𝑐isanyconstant,sothatEquation(11)becomes
Axioms2021,10,3324of22
𝑔󰇛𝑡󰇜 𝑢󰇛𝑡󰇜𝑒
 𝑒 󰇟 𝑒󰇛󰇜 𝑔󰇛𝑡󰇜
𝑑𝑡󰇠
 𝑑𝑤.(12)
MultiplyingbothsidesofEquation(12)by 𝑒,weobtain
𝑔󰇛𝑡󰇜𝑢󰇛𝑡󰇜
 𝑒󰇛󰇜 󰇟 𝑒󰇛󰇜 𝑔󰇛𝑡󰇜
𝑑𝑡󰇠
 𝑑𝑤 (13)
Substituting
𝑐𝑖𝜔,
𝑖𝑑𝑤 and 𝑑𝑤
𝑑𝑠inEquation(13),weobtain
𝑔󰇛𝑡󰇜 𝑢 󰇛𝑡󰇜
 𝑒
󰇟 𝑒
𝑔󰇛𝑡󰇜
𝑑𝑡󰇠

 𝑑𝑠
1
2𝜋𝑖 1
𝑠𝑒
󰇟𝑠𝑢 𝑒

𝑔󰇛𝑡󰇜
𝑑𝑡󰇠

 𝑑𝑠
1
2𝜋𝑖 1
𝑠𝑒
𝐵󰇛𝑠,𝑢󰇜

 𝑑𝑠.
Defining𝑔󰇛𝑡󰇜on󰇛0, ∞󰇜,weobtain
𝑔󰇛𝑡󰇜1
2𝜋𝑖 1
𝑠𝑒
𝐵󰇛𝑠,𝑢󰇜

 𝑑𝑠.
Hence,
𝑅󰇟𝐵󰇛𝑠,𝑢󰇜󰇠 1
2𝜋𝑖 1
𝑠

 exp 𝑠𝑡
𝑢 𝐵󰇛𝑠,𝑢󰇜𝑑𝑠,(14)
and𝑅󰇟𝑅󰇛𝑔󰇛𝑡󰇜󰇜󰇠𝑔󰇛𝑡󰇜.
Theorem1.SufficientconditionsfortheexistenceoftheFormabletransform.
Ifthefunction𝑔󰇛𝑡󰇜isapiecewisecontinuousfunctionineveryfiniteinterval𝑡 ∈
󰇟0, 𝛼󰇠and isofexponentialorder𝛽for 𝑡𝛽,thentheFormabletransform𝐵󰇛𝑠,𝑢󰇜of
𝑔󰇛𝑡󰇜 exists.
Proof.Let𝛼beanypositivenumber,thenwehave
𝐵󰇛𝑠,𝑢󰇜 𝑠𝑢exp 𝑠𝑡
𝑢
𝑔󰇛𝑡󰇜𝑑𝑡
𝑠𝑢exp 󰇛𝑠𝑡
𝑢󰇜
𝑔󰇛𝑡󰇜 𝑑𝑡𝑠𝑢exp 󰇛𝑠𝑡
𝑢󰇜
𝑔󰇛𝑡󰇜𝑑𝑡
Sincethefunction𝑔󰇛𝑡󰇜isapiecewisecontinuousfunctionineveryfiniteinterval
󰇟0, 𝛼󰇠,theintegral
exp 󰇛
󰇜
𝑔󰇛𝑡󰇜 𝑑𝑡exists,andsince𝑔 󰇛𝑡󰇜isofexponentialorder𝛽we
have󰇻
exp 󰇡
󰇢
𝑔󰇛𝑡󰇜𝑑𝑡󰇻󰇻
󰇻󰇻exp 󰇡
󰇢𝑔󰇛𝑡󰇜󰇻
𝑑𝑡
󰇻𝑠𝑢󰇻 exp 𝑠𝑡
𝑢|𝑔󰇛𝑡󰇜|
𝑑𝑡
󰇻𝑠𝑢󰇻 exp 𝑠𝑡
𝑢𝑁 exp 󰇛
𝛽𝑡󰇜𝑑𝑡
=󰇻
󰇻𝑁exp 󰇡𝑡󰇛
𝛽󰇜󰇢
𝑑𝑡
𝑠𝑢 𝑁 exp 󰇡𝑡󰇛 𝑠𝑢𝛽󰇜󰇢
𝑑𝑡
Axioms2021,10,3325of22
𝑁lim
→ 󰇟󰇡󰇛
󰇜󰇢
 󰇠
𝑠𝑢 𝑁 󰇛 1
𝑠𝑢 𝛽 󰇜
𝑠𝑁/󰇛𝑠𝛽𝑢󰇜.
Theproofiscomplete.Now,weintroducesomebasicpropertiesandresultsconcern‐
ingtheFormabletransformwhichenableustosolvemoreapplicationsviathetransform.
Property1(linearityproperty).Let𝛼𝑔󰇛𝑡󰇜and𝛽𝑔󰇛𝑡󰇜 betwofunctionsinaset 𝑊,then
( 𝛼𝑔󰇛𝑡󰇜 𝛽𝑔󰇛𝑡󰇜󰇜 ∈𝑊,where𝛼 and 𝛽arenonzeroarbitraryconstants,and
𝑅󰇟𝛼𝑔󰇛𝑡󰇜 𝛽𝑔󰇛𝑡󰇜󰇠𝛼𝑅󰇟𝑔󰇛𝑡󰇜󰇠 𝛽𝑅󰇟𝑔󰇛𝑡󰇜󰇠.(15)
ProofofProperty1.UsingthedefinitionoftheFormabletransform,wehave
𝑅󰇟𝛼𝑔󰇛𝑡󰇜 𝛽𝑔󰇛𝑡󰇜󰇠𝑠𝑢exp 𝑠𝑡
𝑢󰇛
𝛼𝑔󰇛𝑡󰇜 𝛽𝑔󰇛𝑡󰇜󰇜 𝑑𝑡
𝑠𝑢 exp 𝑠𝑡
𝑢
𝛼𝑔󰇛𝑡󰇜 𝑑𝑡𝑠𝑢 exp 𝑠𝑡
𝑢
𝛽𝑔󰇛𝑡󰇜 𝑑𝑡
𝛼 𝑠𝑢 exp 𝑠𝑡
𝑢
𝑔󰇛𝑡󰇜 𝑑𝑡𝛽𝑠𝑢 exp 𝑠𝑡
𝑢
𝑔󰇛𝑡󰇜 𝑑𝑡
𝛼𝑅󰇟𝑔󰇛𝑡󰇜󰇠 𝛽𝑅󰇟𝑔󰇛𝑡󰇜󰇠.
Theproofiscomplete.
Property2(changeofscale).Letthefunction𝑔(𝛼𝑡󰇜beintheset 𝑊,where𝛼isanarbitrary
constant,then𝑅 󰇟𝑔󰇛𝛼𝑡󰇜󰇠𝐵 󰇛
,𝑢󰇜𝐵󰇛𝑠,𝛼𝑢󰇜.(16)
ProofofProperty2.
𝑅 󰇟𝑔󰇛𝛼𝑡󰇜󰇠 𝑠𝑢exp 󰇛𝑠𝑡
𝑢󰇜
𝑔󰇛𝛼𝑡󰇜𝑑𝑡(17)
Substituting𝛿 𝛼𝑡inEquation(17)wehave
𝑅 󰇟𝑔󰇛𝛼𝑡󰇜󰇠 𝑠𝑢exp 󰇛𝑠𝛿
𝑢𝛼󰇜
𝑔󰇛𝛿󰇜𝑑𝛿
𝛼
𝑠
𝛼𝑢exp 󰇛𝑠𝛿
𝑢𝛼󰇜
𝑔󰇛𝛿󰇜 𝑑𝛿
𝐵󰇛
,𝑢󰇜
= 𝐵󰇛𝑠,𝛼𝑢󰇜.
Property3(Formabletransformofthederivative).Ifthefunction𝑔󰇛 󰇜󰇛𝑡󰇜isthen‐th
derivativeofthefunction 𝑔 󰇛𝑡󰇜,where𝑔󰇛 󰇜󰇛𝑡󰇜∈𝑊, for 𝑛0,1, 2, …withrespectto 𝑡,then
𝑅 󰇟𝑔󰇛󰇜󰇛𝑡󰇜󰇠
𝐵󰇛𝑠,𝑢󰇜 ∑󰇡
󰇢 𝑔󰇛󰇜󰇛0󰇜

 . (18)
ProofofProperty3.For𝑛 1,wehave
Axioms2021,10,3326of22
𝑅 󰇟𝑔󰆒󰇛𝑡󰇜󰇠 𝑠𝑢 exp 𝑠𝑡
𝑢
𝑔󰆒󰇛𝑡󰇜 𝑑𝑡
𝑠𝑢 󰇩lim 󰇟
→exp 𝑠𝑡
𝑢𝑔󰇛𝑡󰇜 󰇠
𝑠𝑢exp 𝑠𝑡
𝑢
𝑔󰇛𝑡󰇜 𝑑𝑡󰇪
󰇟𝑔󰇛0󰇜 𝐵󰇛𝑠,𝑢󰇜󰇠.
Thus𝑅 󰇟𝑔󰆒󰇛𝑡󰇜󰇠 𝑠𝑢 𝐵󰇛𝑠,𝑢󰇜 𝑠𝑢𝑔󰇛0󰇜 . (19)
AssumingthatEquation(18)istruefor𝑛𝑘,thenweshowthatitistruefor𝑛
𝑘 1,byusingthefactthatinEquation(19)wehave
R[𝑔(k+1)(t)]R[󰇛𝑔󰇛󰇜󰇛t󰇜󰇜󰆒󰇠
𝑅 󰇟𝑔󰇛󰇜󰇛t󰇜󰇠
𝑔󰇛󰇜󰇛0󰇜
󰇟
𝐵󰇛𝑠,𝑢󰇜 ∑󰇛
󰇜 𝑔󰇛󰇜󰇛0󰇜

 󰇠
𝑔󰇛󰇜󰇛0󰇜

𝐵󰇛𝑠,𝑢󰇜 ∑󰇛
󰇜 𝑔󰇛󰇜󰇛0󰇜
 .
ThisimpliesthatEquation(17)holdsfor𝑛 𝑘 1,sotheproofiscomplete.
ThefollowingimportantpropertiesareobtainedusingtheLeibnizruleand
Equation(18):
(i) 𝑅[󰇛,󰇜
 󰇠
exp 󰇛
󰇜󰇛,󰇜

𝑑𝑡
󰇟
exp 󰇡
󰇢𝑔󰇛𝑥,𝑡󰇜 𝑑𝑡󰇠
󰇟𝐵󰇛𝑥,𝑠,𝑢󰇜󰇠.
(ii) 𝑅[󰇛,󰇜
󰇠
exp 󰇛
󰇜󰇛,󰇜

𝑑𝑡
󰇟
exp 󰇡
󰇢𝑔󰇛𝑥,𝑡󰇜 𝑑𝑡󰇠
󰇟𝐵󰇛𝑥,𝑠,𝑢󰇜󰇠.
(iii) 𝑅[󰇛,󰇜
󰇠
exp 󰇛
󰇜󰇛,󰇜

𝑑𝑡
󰇟
exp 󰇡
󰇢𝑔󰇛𝑥,𝑡󰇜 𝑑𝑡󰇠
󰇟 𝐵󰇛𝑥,𝑠,𝑢󰇜󰇠.
(20)
Property4(Formabletransformoftheconvolution).If𝐹󰇛𝑠,𝑢󰇜 and 𝐺󰇛𝑠,𝑢󰇜 aretheFormable
transformsofthefunctions𝑓 󰇛𝑡󰇜and𝑔󰇛𝑡󰇜,respectively,then
𝑅󰇟
𝑓
󰇛𝑡󰇜∗𝑔󰇛𝑡󰇜󰇠
𝐹󰇛𝑠,𝑢󰇜 𝐺󰇛𝑠,𝑢󰇜, (21)
where𝑓󰇛𝑡󰇜∗𝑔󰇛𝑡󰇜istheconvolutionofthefunctions𝑓󰇛𝑡󰇜and𝑔󰇛𝑡󰇜definedby
𝑓
󰇛𝑡󰇜∗𝑔󰇛𝑡󰇜
𝑓
󰇛𝜏󰇜 𝑔󰇛𝑡𝜏󰇜𝑑𝜏
 (22)
ProofofProperty4.UsingthedefinitionoftheFormabletransforminEquation(9),we
obtain
𝑅󰇟
𝑓
󰇛𝑡󰇜∗𝑔󰇛𝑡󰇜󰇠𝑠exp󰇛𝑠𝑡󰇜
󰇛
𝑓
∗𝑔󰇜󰇛𝑢𝑡󰇜 𝑑𝑡
𝑠exp󰇛𝑠𝑡󰇜
𝑓
󰇛𝜏󰇜 𝑔󰇛𝑢𝑡𝜏󰇜 𝑑𝜏

𝑑𝑡.
(23)
Letting𝜏𝑢𝑥 and 𝑑𝜏𝑢𝑑𝑥inEquation(23),weobtain
𝑅󰇟
𝑓
󰇛𝑡󰇜∗𝑔󰇛𝑡󰇜󰇠𝑠exp󰇛𝑠𝑡󰇜
𝑓
󰇛𝑢𝑥󰇜 𝑔󰇛𝑢𝑡𝑢𝑥󰇜𝑑󰇛𝑢𝑥󰇜
𝑑𝑡(24)
Axioms2021,10,3327of22
𝑠exp󰇛𝑠𝑡󰇜
𝑓
󰇛𝑢𝑥󰇜 𝑔󰇛𝑢󰇛𝑡𝑥󰇜󰇜𝑢 𝑑𝑥
𝑑𝑡
Letting𝑦𝑡𝑥 and 𝑑𝑦𝑑𝑡 inEquation(24),weobtain
𝑅󰇟
𝑓
󰇛𝑡󰇜∗𝑔󰇛𝑡󰇜󰇠𝑠∬exp𝑠󰇛𝑥𝑦󰇜
𝑓
󰇛𝑢𝑥󰇜 𝑔󰇛𝑢𝑦󰇜 𝑢 𝑑𝑥 𝑑𝑦
𝑠𝑢exp󰇛𝑠󰇛𝑥𝑦󰇜󰇜
𝑓󰇛𝑢𝑥󰇜 𝑔󰇛𝑢𝑦󰇜 𝑑𝑥 𝑑𝑦
𝑠𝑢exp󰇛𝑠𝑥󰇜𝑓󰇛𝑢𝑥󰇜 𝑑𝑥
exp󰇛𝑠𝑦󰇜 𝑔󰇛𝑢𝑦󰇜𝑑𝑦
𝑢𝑠𝑠exp󰇛𝑠𝑥󰇜𝑓󰇛𝑢𝑥󰇜𝑑𝑥
𝑠exp󰇛𝑠𝑦󰇜𝑔󰇛𝑢𝑦󰇜𝑑𝑦
𝐹󰇛𝑠,𝑢󰇜𝐺󰇛𝑠,𝑢󰇜 .
Corollary1.TheFormabletransformof󰇛 𝑓∗𝑔 󰇜󰆒isgivenby
𝐵󰇟󰇛
𝑓
∗𝑔󰇜󰆒󰇠𝐹󰇛𝑠,𝑢󰇜 𝐺󰇛𝑠,𝑢󰇜 (25)
ProofofCorollary1.Applyingthefactsinproperties(3)and(4),weobtain
𝐵󰇟󰇛
𝑓
∗𝑔󰇜󰆒󰇠 𝑠𝑢𝑅
𝑓
󰇛𝑡󰇜∗𝑔󰇛𝑡󰇜𝑠𝑢󰇛
𝑓
∗𝑔󰇜󰇛0󰇜.
But 󰇛𝑓∗𝑔󰇜󰇛0󰇜0,andhence
𝐵󰇟󰇛
𝑓
∗𝑔󰇜󰆒󰇠𝑠𝑢 𝑢𝑠𝐹󰇛𝑠,𝑢󰇜 𝐺󰇛𝑠,𝑢󰇜
𝐹󰇛𝑠,𝑢󰇜 𝐺󰇛𝑠,𝑢󰇜.
Here,ifweput𝑔󰇛𝑡󰇜 𝑓󰇛𝑡󰇜inEquation(25)wehave
𝐵󰇟󰇛
𝑓
∗
𝑓
󰇜󰆒󰇠𝐹󰇛𝑠,𝑢󰇜. (26)
Property5(shiftingons‐domain).Ifthefunction𝑔󰇛𝑡󰇜inaset𝑊ismultipliedwiththeshift
function𝑡,then
𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠 󰇛𝑢󰇜𝑠𝜕
𝜕𝑠󰇩𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠󰇪 . (27)
ProofProperty5.WeshowthatEquation(27)istruefor 𝑛 1.
Putting𝑛 1inEquation(27),wehave
𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠𝑢𝑠 𝜕
𝜕𝑠󰇩𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠󰇪
𝑢
𝑠󰇩𝑠𝜕𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝜕𝑠 𝑅󰇟𝑔󰇛𝑡󰇜󰇠 󰇪
𝑢𝜕𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝜕𝑠 𝑢𝑠 𝑅󰇟𝑔󰇛𝑡󰇜󰇠
(28)
Equation(28)becomes
Axioms2021,10,3328of22
𝜕𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝜕𝑠 1
𝑢𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠1
𝑠𝑅󰇟𝑔󰇛𝑡󰇜󰇠 (29)
IfweproveEquation(29),wearefinished.Westartwiththeleft‐handsideofEqua‐
tion(29)andusingtheLeibnizruleweobtain
𝜕𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝜕𝑠 𝜕
𝜕𝑠󰇟 𝑠𝑢 exp 𝑠𝑡
𝑢𝑔󰇛𝑡󰇜𝑑𝑡
󰇠
𝑠𝑢𝜕
𝜕𝑠𝑒𝑥𝑝
𝑠𝑡
𝑢𝑔󰇛𝑡󰇜𝑑𝑡1
𝑢𝑒𝑥𝑝
𝑠𝑡
𝑢𝑔󰇛𝑡󰇜 𝑑𝑡
𝑠𝑢𝑡
𝑢𝑒𝑥𝑝
𝑠𝑡
𝑢𝑔󰇛𝑡󰇜𝑑𝑡󰇠1
𝑢𝑒𝑥𝑝
𝑠𝑡
𝑢𝑔󰇛𝑡󰇜 𝑑𝑡
1
𝑢 𝑠𝑢𝑡 𝑒𝑥𝑝
𝑠𝑡
𝑢𝑔󰇛𝑡󰇜𝑑𝑡󰇠1
𝑢𝑒𝑥𝑝
𝑠𝑡
𝑢𝑔󰇛𝑡󰇜 𝑑𝑡

𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠
𝑅󰇟𝑔󰇛𝑡󰇜󰇠.
Theproofiscompletefor 𝑛 1.
AssumethatEquation(27)istruefor𝑛suchthat
𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠 󰇛𝑢󰇜𝑠 𝜕
𝜕𝑠󰇩𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠󰇪.
Weshowthat
𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠󰇛𝑢󰇜𝑠 𝜕
𝜕𝑠󰇩𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠󰇪.(30)
UsingthefactsinEquations(28)and(31),wehave
𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠𝑅󰇟𝑡 𝑡𝑔󰇛𝑡󰇜󰇠
𝑢𝑠 𝜕
𝜕𝑠󰇟𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠
𝑠󰇠
𝑢𝑠 𝜕
𝜕𝑠󰇛𝑢󰇜 𝑠 𝜕
𝜕𝑠𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠
𝑠
𝑢𝑠 ⎣
⎢
⎢
⎢
⎡
𝑠󰇩󰇛𝑢󰇜𝑠.𝜕
𝜕𝑠𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠󰇛𝑢󰇜𝜕
𝜕𝑠𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠󰇪󰇛𝑢󰇜𝜕
𝜕𝑠𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠
𝑠⎦
⎥
⎥
⎥
⎤
󰇛𝑢󰇜 𝑠 𝜕
𝜕𝑠
󰇩
𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠
󰇪
.
Remark1.Ifthefunction𝑔󰇛𝑡󰇜hasthenumericalexpansion
𝑔 󰇛𝑡󰇜 ∑𝑎 𝑡
 ,
thentheFormabletransform(seeTableA1inAppendixA)of𝑡𝑔󰇛𝑡󰇜 isgivenby
𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠∑󰇛󰇜!


𝑢𝑠󰇛𝑛1󰇜!𝑎𝑢
𝑠

𝑢𝑠𝜕
𝜕𝑢 𝑛!𝑎𝑢
𝑠
Axioms2021,10,3329of22
𝑢𝑠 𝜕
𝜕𝑢𝑢 𝑛!𝑎𝑢
𝑠

𝑢𝑠 𝜕
𝜕𝑢󰇟𝑢 𝐵󰇛𝑠,𝑢󰇜󰇠.
Thegeneralizationofthepreviousremarkundertheconditionon𝑔󰇛𝑡󰇜givesusan
equivalentformofproperty(5)asfollows:
𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠 𝑢
𝑠 𝜕
𝜕𝑢󰇛 𝑢 𝐵󰇛𝑠,𝑢󰇜 󰇜 (31)
Remark2.If𝑔󰇛󰇜󰇛𝑡󰇜isthen‐thderivativeofthefunction𝑔󰇛𝑡󰇜thatismultipliedwiththeshift
function𝑡,then
𝑅𝑡𝑔󰇛󰇜󰇛𝑡󰇜𝑢𝜕
𝜕𝑢󰇟𝐵󰇛𝑠,𝑢󰇜󰇠 (32)
ProofofRemark2.Considertheright‐handsideofEquation(32).UsingtheLeibnizrule,
weobtain
𝑢 𝜕
𝜕𝑢󰇟𝐵󰇛𝑠,𝑢󰇜󰇠𝑢 𝜕
𝜕𝑢 𝑠exp󰇛𝑠𝑡󰇜
𝑔󰇛𝑢𝑡󰇜𝑑𝑡
𝑢 𝑠exp󰇛𝑠𝑡󰇜
𝜕
𝜕𝑢𝑔󰇛𝑢𝑡󰇜𝑑𝑡
𝑢 𝑠exp󰇛𝑠𝑡󰇜
𝑡𝑔󰇛󰇜󰇛𝑢𝑡󰇜𝑑𝑡
𝑠exp󰇛𝑠𝑡󰇜
󰇛𝑢𝑡󰇜𝑔󰇛󰇜󰇛𝑢𝑡󰇜𝑑𝑡
𝑅
𝑡𝑔󰇛󰇜󰇛𝑡󰇜
.
Property6.Ifthefunction𝑔󰇛𝑡󰇜inaset𝑊isdividedbythemultipleshiftfunction𝑡,then
𝑅󰇩𝑔󰇛𝑡󰇜
𝑡󰇪 𝑠
𝑢⋯𝐵󰇛𝑠,𝑢󰇜
𝑠
󰇛𝑑𝑠󰇜 (33)
ProofofProperty6.Startingwithright‐handsideofEquation(33),weobtain
𝑠
𝑢⋯𝐵󰇛𝑠,𝑢󰇜
𝑠
󰇛𝑑𝑠󰇜 𝑠
𝑢⋯exp󰇛𝑠𝑡󰇜𝑔󰇛𝑢𝑡󰇜 𝑑𝑡
󰇛𝑑𝑠󰇜
𝑠
𝑢𝑔
󰇛𝑢𝑡󰇜
⋯exp 󰇛𝑠𝑡󰇜󰇛𝑑𝑠󰇜
𝑑𝑡
𝑠
𝑢𝑔󰇛𝑢𝑡󰇜
𝑡exp󰇛𝑠𝑡󰇜𝑑𝑡
𝑠 𝑔󰇛𝑢𝑡󰇜
󰇛𝑢𝑡󰇜exp󰇛𝑠𝑡󰇜𝑑𝑡
𝑅
󰇩
𝑔󰇛𝑡󰇜
𝑡
󰇪
.

Axioms2021,10,33210of22
Property7.Letthefunction𝑔󰇛𝑡󰇜bemultipliedwiththeweightfunction 𝑒𝑥𝑝 󰇛𝛼𝑡),then
𝑅󰇟exp󰇛𝛼𝑡󰇜𝑔󰇛𝑡󰇜󰇠𝑠
𝑠𝛼𝑢𝐵󰇣𝑠,𝑢
𝑠𝛼𝑢󰇤 (34)
ProofofProperty7.
𝑅󰇟exp󰇛𝛼𝑡󰇜𝑔󰇛𝑡󰇜󰇠𝑠exp󰇛𝑠𝑡󰇜
exp󰇛𝛼𝑢𝑡󰇜 𝑔󰇛𝑢𝑡󰇜 𝑑𝑡
𝑠exp 󰇛󰇛𝑠𝛼𝑢󰇜𝑡󰇜
𝑔󰇛𝑢𝑡󰇜 𝑑𝑡
(35)
Letting 󰇛𝑠𝛼𝑢󰇜𝑡𝑠𝑤,and𝑑𝑡
𝑑𝑤 inEquation(36),wehave
𝑠exp󰇛𝑠𝑤󰇜
𝑔󰇡𝑢𝑠𝑤
𝑠𝛼𝑢󰇢 𝑠
𝑠𝛼𝑢 𝑑𝑤
𝑠
𝑠𝛼𝑢𝑠exp 󰇛𝑠𝑤󰇜
𝑔󰇡𝑢𝑠𝑤
𝑠𝛼𝑢󰇢𝑑𝑤
𝑠
𝑠
𝛼𝑢
𝐵󰇣 𝑠,𝑢𝑠
𝑠
𝛼𝑢
󰇤.
3.DualitywithTransformsandSomeExamples
Inthissection,weillustratetherelationbetweenthenewtransformandotherwell‐
knowntransforms.AlsowecomputetheFormabletransformforsomefunctionstoshow
itsapplicabilityandsimplicityduringthecomputations.
3.1.DualitiesbetweenFormableTransformandOtherIntegralTransforms
Formable–Laplaceduality:let𝐵󰇛𝑠,𝑢󰇜betheFormabletransformand𝐹󰇛𝑠)bethe
Laplacetransformofthesamefunction𝑔󰇛𝑡󰇜,thenitisclearthat
𝐵 󰇛 𝑠,1
󰇜𝑠𝐹󰇛𝑠󰇜 . (36)
Formable–Elzakiduality:let𝐸 󰇛𝑢󰇜betheElzakitransformof𝑔󰇛𝑡󰇜,then
𝐵 󰇛1, 𝑢󰇜
𝐸 󰇛 𝑢󰇜 (37)
Formable–Sumududuality:let𝐺󰇛𝑢󰇜betheSumudutransformof𝑔󰇛𝑡󰇜,then
𝐵󰇛1, 𝑢󰇜𝐺󰇛𝑢󰇜 (38)
𝐵󰇛𝑠,𝑢󰇜 𝑠𝑢exp 󰇛𝑠𝑡
𝑢󰇜
𝑔󰇛𝑡󰇜𝑑𝑡 .
𝐵󰇛1, 𝑢󰇜1
𝑢𝑒𝑥𝑡
𝑡
𝑢 𝑔󰇛𝑡󰇜𝑑𝑡
𝐺󰇛𝑢󰇜.
Formable–Naturalduality:let𝑅󰇛𝑠,𝑢󰇜betheNaturaltransformof𝑔󰇛𝑡󰇜,then
𝐵󰇛𝑠,𝑢󰇜𝑠𝑅󰇛𝑠,𝑢󰇜 . (39)
Formable–Shehuduality:letV(s,u)betheShehutransformof𝑔󰇛𝑡󰇜,then,
𝐵󰇛𝑠,𝑢󰇜𝑠𝑢𝑉󰇛𝑠,𝑢󰇜 . (40)
Formable–ARAduality:let𝒢(s)betheARAtransformof𝑔󰇛𝑡󰇜,then
𝐵󰇛𝑠,1
󰇜𝒢󰇛𝑠󰇜 . (41)
Axioms2021,10,33211of22
Furthermore,substituting𝑢 1in𝑅 󰇟 𝑡𝑔󰇛 𝑡 󰇜󰇠,weobtain
𝑅 󰇟 𝑡𝑔󰇛 𝑡 󰇜󰇠𝒢󰇟 𝑔󰇛 𝑡 󰇜󰇠(s).(42)
3.2.ExamplesofFormableTransformforSomeFunctions
Inthefollowingarguments,wecomputetheFormabletransformforsomefunctions
todemonstrateitssimplicityandapplicabilitythroughcomputations.
Example1.Letthefunction𝑔󰇛𝑡󰇜1,
Then𝑅󰇟𝑔󰇛𝑡󰇜󰇠 1, (43)
ProofofExample1.
𝑅[1]
exp 󰇛
󰇜
𝑑𝑡
lim 󰇟
→
exp 󰇛
󰇜󰇠
1.
Example2.Letthefunction𝑔󰇛𝑡󰇜𝑡,then
𝑅󰇟𝑔󰇛𝑡󰇜󰇠
. (44)
ProofofExample2.
𝑅[t]
exp 󰇡
󰇢
𝑑𝑡
lim 󰇟
→
𝑡exp 󰇡
󰇢
exp 󰇛
󰇜󰇠
.
Example3.Letthefunction󰇛𝑡󰇜
,then
𝑅 󰇟𝑔󰇛𝑡󰇜󰇠
.(45)
ProofofExample3.
𝑅[
]
exp 󰇡
󰇢
𝑑𝑡
lim 󰇟
→
𝑡exp 󰇡
󰇢2
t exp 󰇛
󰇜2
exp 󰇛
󰇜󰇠
.
Example4.Letthefunction𝑔󰇛𝑡󰇜
!,then
𝑅 󰇟𝑔󰇛𝑡󰇜󰇠
. (46)

Axioms2021,10,33212of22
ProofofExample4.
𝑅[
!]
exp 󰇡
󰇢
!
𝑑𝑡
𝑠
𝑢 𝑛!exp 𝑠𝑡
𝑢 𝑡
𝑑𝑡
1
󰇛𝑛1󰇜!exp 𝑠𝑡
𝑢 𝑡
𝑑𝑡
⋮

exp
󰇡

󰇢
𝑡
𝑑𝑡
.
Example5.Letthefunction𝑔󰇛𝑡󰇜 exp(𝛼𝑡󰇜,then
𝑅 󰇟𝑔󰇛𝑡󰇜󰇠
. (47)
ProofofExample5.
𝑅󰇟𝑒𝑥𝑝(𝛼𝑡󰇜]
exp 󰇡
󰇢 exp󰇛𝛼𝑡󰇜
𝑑𝑡
𝑠𝑢exp 󰇛𝛼𝑢𝑠󰇜𝑡
𝑢
𝑑𝑡
lim 󰇟
→
exp 󰇡󰇛󰇜
󰇢󰇠
 .
Example6.Letthefunction𝑔󰇛𝑡󰇜texp(𝛼𝑡󰇜,then
𝑅 󰇟𝑔󰇛𝑡󰇜󰇠
󰇛󰇜.(48)
ProofofExample6.
𝑅󰇟𝑡 𝑒𝑥𝑝(𝛼𝑡󰇜]
exp 󰇡
󰇢 texp
󰇛𝛼𝑡󰇜
𝑑𝑡
𝑠𝑢t exp 󰇛𝑠𝛼𝑢󰇜𝑡
𝑢
𝑑𝑡
lim 󰇟
→ 
𝑡exp 󰇡󰇛󰇜
󰇢
󰇛󰇜 exp 󰇡󰇛󰇜
󰇢 󰇠

󰇛󰇜 .
Example7.Letthefunction𝑔󰇛𝑡󰇜
!𝑒𝑥𝑝󰇛𝛼𝑡󰇜,then
𝑅󰇟𝑔󰇛𝑡󰇜󰇠 
󰇛󰇜.(49)
ProofofExample7.
𝑅󰇟𝑡exp(𝛼𝑡󰇜]=
𝑡exp 󰇡󰇛󰇜
󰇢
𝑑𝑡
𝑠𝑛
𝑠𝛼𝑢𝑡
exp 󰇛𝑠𝛼𝑢󰇜𝑡
𝑢
𝑑𝑡
𝑠𝑢𝑛󰇛𝑛1󰇜
󰇛𝑠𝛼𝑢󰇜𝑡
exp 󰇛𝑠𝛼𝑢󰇜𝑡
𝑢
𝑑𝑡
Axioms2021,10,33213of22
⋮
𝑠𝑢𝑛!
󰇛
𝑠𝛼𝑢
󰇜
 .
Example8.Letthefunction𝑔󰇛𝑡󰇜sin(𝛼𝑡󰇜,then
𝑅 󰇟𝑔󰇛𝑡󰇜󰇠 
.(50)
ProofofExample8.𝑅󰇟sin󰇛𝛼𝑡󰇜󰇜]
exp 󰇡
󰇢 sin󰇛𝛼𝑡󰇜
𝑑𝑡
𝑠𝑢lim 󰇟
→ exp 𝑠𝑡
𝑢
𝑠
𝑢 sin󰇛𝛼𝑡󰇜 𝛼cos󰇛𝛼𝑡󰇜
𝑠
𝑢 𝛼 󰇠
𝑠𝑢 𝛼
𝑠
𝑢 𝛼

Example9.Letthefunction𝑔󰇛𝑡󰇜cos(𝛼𝑡󰇜,then
𝑅 󰇟𝑔󰇛𝑡󰇜󰇠
.(51)
ProofofExample9.
𝑅[cos(𝛼𝑡󰇜]
exp 󰇡
󰇢 cos󰇛𝛼𝑡󰇜
𝑑𝑡
𝑠𝑢lim 󰇟
→ exp 𝑠𝑡
𝑢
𝑠
𝑢 cos󰇛𝛼𝑡󰇜 𝛼sin󰇛𝛼𝑡󰇜
𝑠
𝑢 𝛼 󰇠
𝑠𝑢 𝑠𝑢
𝑠
𝑢 𝛼
.
Example10.Letthefunction𝑔󰇛𝑡󰇜󰇛󰇜,
,then
𝑅󰇟𝑔󰇛𝑡󰇜󰇠 
.(52)
ProofofExample10.
𝑅[sinh(𝛼𝑡󰇜]
exp 󰇡
󰇢 sinh󰇛𝛼𝑡󰇜
𝑑𝑡
𝑠𝑢lim 󰇟
→ exp 𝑠𝑡
𝑢
𝑠
𝑢 sinh󰇛𝛼𝑡󰇜 𝛼cosh󰇛𝛼𝑡󰇜
𝑠
𝑢 𝛼 󰇠
Axioms2021,10,33214of22
𝑠𝑢 𝛼
𝑠
𝑢 𝛼

.
Example11.Letthefunction𝑔󰇛𝑡󰇜cosh(𝛼𝑡󰇜,then
𝑅 󰇟𝑔󰇛𝑡󰇜󰇠
 (53)
ProofofExample11.
𝑅󰇟cosh󰇛𝛼𝑡󰇜󰇠
exp 󰇡
󰇢 cosh󰇛𝛼𝑡󰇜
𝑑𝑡
𝑠𝑢lim 󰇟
→ exp 𝑠𝑡
𝑢
𝑠
𝑢 cosh󰇛𝛼𝑡󰇜 𝛼sinh󰇛𝛼𝑡󰇜
𝑠
𝑢 𝛼 󰇠
𝑠𝑢 𝑠𝑢
𝑠
𝑢 𝛼
.
Example12.Letthefunction𝑔󰇛𝑡󰇜󰇛󰇜󰇛󰇜,
,then
𝑅󰇟𝑔󰇛𝑡󰇜󰇠 
󰇛󰇜.(54)
ProofofExample12.
𝑅󰇟exp󰇛𝛽𝑡󰇜sin󰇛𝛼𝑡󰇜󰇜]
exp 󰇡
󰇢 exp󰇛𝛽𝑡󰇜sin󰇛𝛼𝑡󰇜
𝑑𝑡
𝑠𝑢exp 󰇧󰇛𝑠𝛽𝑢󰇜
𝑢 𝑡󰇨 𝑠𝑖𝑛󰇛𝛼𝑡󰇜
𝑑𝑡
󰇟 lim 󰇟
→ 
exp 󰇡󰇛󰇜
𝑡󰇢 sin󰇛𝛼𝑡󰇜 󰇠
+
𝛼𝑢
𝑠𝛽𝑢exp 󰇧󰇛𝑠𝛽𝑢󰇜
𝑢 𝑡󰇨 cos󰇛𝛼𝑡󰇜
𝑑𝑡󰇠
𝛼𝑠
𝑠𝛽𝑢exp 󰇧󰇛𝑠𝛽𝑢󰇜
𝑢 𝑡󰇨 cos󰇛𝛼𝑡󰇜
𝑑𝑡

󰇟 lim 󰇟
→ 
exp 󰇡󰇛󰇜
𝑡󰇢 cos 󰇛𝛼𝑡󰇜 󰇠

exp 󰇡󰇛󰇜
𝑡󰇢sin󰇛𝛼𝑡󰇜
𝑑𝑡]
𝛼𝑠
𝑠𝛽𝑢󰇟𝑢
𝑠𝛽𝑢
𝛼𝑢
𝑠𝛽𝑢exp
󰇧
󰇛𝑠𝛽𝑢󰇜
𝑢 𝑡
󰇨
sin󰇛𝛼𝑡󰇜
𝑑𝑡󰇠.
Simplifyingtherequiredintegral,weobtain:
𝑅󰇟exp󰇛𝛽𝑡󰇜𝑠𝑖𝑛󰇛𝛼𝑡󰇜]
󰇛󰇜.(55)

Axioms2021,10,33215of22
Example13.Letthefunction󰇛𝑡󰇜exp󰇛𝛽𝑡󰇜𝑐𝑜𝑠󰇛𝛼𝑡󰇜,then
𝑅󰇟𝑔󰇛𝑡󰇜󰇠 󰇛󰇜
󰇛󰇜.(56)
ProofofExample13.BysimilarcomputationstoExample12,weobtaintheresult.
4.Applications
Inthissection,weintroducesomeapplicationsusingtheFormabletransforminsolv‐
ingordinaryandpartialdifferentialequationsusingseveralpropertiesofthenewtrans‐
form,suchasthederivativeproperty,theconvolutionpropertyandtheshiftingtheorem
oftheFormabletransform.
Example1.Considerthefirstorderdifferentialequation
𝑦󰆒󰇛𝑡󰇜5𝑦󰇛𝑡󰇜0,(57)
subjecttotheinitialcondition𝑦󰇛0󰇜2 .(58)
Solution.ApplyingtheFormabletransformonbothsidesofEquation(57).
𝑅󰇟 𝑦󰆒󰇛𝑡󰇜󰇠𝑅󰇟5𝑦󰇛𝑡󰇜󰇠𝑅󰇟0󰇠,
weobtain
𝐵󰇛𝑠,𝑢󰇜
𝑦󰇛0󰇜5𝐵󰇛𝑠,𝑢󰇜0.(59)
Substitutingtheinitialconditionof(58)andsimplifyingEquation(59),wehave
󰇣
5󰇤𝐵󰇛𝑠,𝑢󰇜2
.
𝐵󰇛𝑠,𝑢󰇜2𝑠
𝑠5𝑢 .(60)
TakingtheinverseFormabletransformofEquation(60),weobtainthesolution
𝑦󰇛𝑡󰇜2exp
󰇛5𝑡󰇜(61)
Example2.Considerthesecondorderdifferentialequation
𝑦󰆒󰆒󰇛𝑡󰇜2𝑦󰆒󰇛𝑡󰇜5𝑦󰇛𝑡󰇜exp󰇛𝑡󰇜sin󰇛𝑡󰇜,(62)
subjecttotheinitialconditions𝑦󰆒󰇛0󰇜1, 𝑦󰇛0󰇜0 . (63)
Solution.ApplyingtheFormabletransforminEquation(62)andusingproperty(3)and
theresultinEquation(55),weobtain
𝑅󰇟 𝑦󰆒󰆒󰇛𝑡󰇜󰇠𝑅󰇟2𝑦󰆒󰇛𝑡󰇜󰇠𝑅󰇟5𝑦󰇛𝑡󰇜󰇠𝑅󰇟exp󰇛𝑡󰇜sin󰇛𝑡󰇜󰇠,
𝐵󰇛𝑠,𝑢󰇜
𝑦󰇛0󰇜
𝑦󰆒󰇛0󰇜2
𝐵󰇛𝑠,𝑢󰇜2
𝑦󰇛0󰇜5𝐵󰇛𝑠,𝑢󰇜
󰇛󰇜
(64)
Substitutingtheinitialconditionsof(63)andsimplifyingEquation(64),weobtain
Axioms2021,10,33216of22
󰇩𝑠2𝑠𝑢5𝑢
𝑢󰇪𝐵󰇛𝑠,𝑢󰇜𝑠𝑢
󰇛𝑠𝑢󰇜𝑢𝑠𝑢.
Hence,𝐵󰇛𝑠,𝑢󰇜
󰇟󰇛󰇜󰇠󰇟󰇛󰇜󰇠
󰇟󰇛󰇜󰇠 (65)
SimplifyingEquation(65),
𝐵󰇛𝑠,𝑢󰇜

󰇟󰇛󰇜󰇠

󰇟󰇛󰇜󰇠. (66)
TakingtheinverseFormabletransformofEquation(66),weobtain
𝑦󰇛𝑡󰇜 1
3exp󰇛𝑡󰇜sin󰇛𝑡󰇜 2
3exp󰇛𝑡󰇜sin󰇛2𝑡󰇜 (67)
Example3:Considerthesecondorderdifferentialequation
𝑦󰆒󰆒󰇛𝑡󰇜3𝑦󰆒󰇛𝑡󰇜2𝑦󰇛𝑡󰇜exp󰇛3𝑡󰇜, (68)
subjecttotheinitialconditions
𝑦󰆒󰇛0󰇜0, 𝑦󰇛0󰇜1 (69)
Solution.ApplyingtheFormabletransformonbothsidesofEquation(68)andusingtheresultin
Equation(47),wehave
𝑅󰇟𝑦󰆒󰆒󰇛𝑡󰇜󰇠𝑅󰇟3𝑦󰆒󰇛𝑡󰇜󰇠𝑅󰇟2𝑦󰇛𝑡󰇜󰇠𝑅󰇟exp󰇛3𝑡󰇜󰇠.
𝐵󰇛𝑠,𝑢󰇜
𝑦󰇛0󰇜
𝑦󰆒󰇛0󰇜3
𝐵󰇛𝑠,𝑢󰇜3
𝑦󰇛0󰇜2𝐵󰇛𝑠,𝑢󰇜

,
(70)
Substitutingtheinitialconditionsof(69)andsimplifyingEquation(70),weobtain
𝐵󰇛𝑠,𝑢󰇜
󰇛󰇜󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜. (71)
AftersimplifyingEquation(71)andtakingtheinverseFormabletransform,wehave
𝑦󰇛𝑡󰇜 5
2exp󰇛𝑡󰇜 2exp
󰇛2𝑡󰇜 1
2exp󰇛3𝑡󰇜. (72)
Example4.ConsidertheBesseldifferentialequation(withpolynomialcoefficients)
𝑡 𝑦󰆒󰆒󰇛𝑡󰇜 𝑦󰆒󰇛𝑡󰇜𝑡𝑦󰇛𝑡󰇜0, (73)
withtheinitialconditions.𝑦󰇛0󰇜1, 𝑦󰆒󰇛0󰇜1. (74)
Solution.Applying,theFormabletransformonbothsidesofEquation(73),weobtain
𝑅󰇟𝑡 𝑦󰆒󰆒󰇛𝑡󰇜󰇠𝑅󰇟 𝑦󰆒󰇛𝑡󰇜󰇠𝑅󰇟𝑡𝑦󰇛𝑡󰇜󰇠𝑅󰇟0󰇠 (75)
UsingthefactsinEquations(18)and(27) inEquation(75),weobtain
𝑢𝑠𝜕
𝜕𝑠𝑠
𝑢𝐵󰇛𝑠,𝑢󰇜𝑠
𝑢𝑦󰇛0󰇜𝑠𝑢 𝑦󰆒󰇛0󰇜
𝑠𝑠𝑢𝐵󰇛𝑠,𝑢󰇜𝑠𝑢𝑢𝑠𝜕
𝜕𝑠󰇩𝐵󰇛𝑠,𝑢󰇜
𝑠󰇪0
Axioms2021,10,33217of22
𝑢𝑠𝜕
𝜕𝑠𝑠
𝑢𝐵󰇛𝑠,𝑢󰇜𝑠
𝑢𝑦󰇛0󰇜1
𝑢 𝑦󰆒󰇛0󰇜𝑠𝑢𝐵󰇛𝑠,𝑢󰇜𝑠𝑢𝑢𝑠𝜕
𝜕𝑠󰇩𝐵󰇛𝑠,𝑢󰇜
𝑠󰇪0
Substitutingtheinitialconditions,weobtain
𝑢𝑠
󰇣
𝐵󰇛𝑠,𝑢󰇜
󰇤
𝐵󰇛𝑠,𝑢󰇜
𝑢𝑠
󰇣󰇛,󰇜
󰇤0. (76)
Aftersimplecomputations,Equation(76)becomes
󰇛,󰇜
󰇛,󰇜
󰇛󰇜𝑑𝑠. (77)
IntegratingbothsidesofEquation(77),weobtain
ln 𝐵󰇛𝑠,𝑢󰇜 ln 𝑠1
2log󰇛𝑠𝑢󰇜ln󰇛𝑐󰇜,
𝐵󰇛𝑠,𝑢󰇜𝑐 𝑠
√
𝑠𝑢.
(78)
TakingtheinverseFormabletransformofEquation(78)andletting𝑐1,weobtain
𝑦󰇛𝑡󰇜
𝐽
󰇛𝑡󰇜. (79)
Example5.Considerthenonhomogeneouspartialdifferentialequation
𝑢𝑢sin 𝜋𝑥 , (80)
withtheinitialboundaryconditions
𝑢󰇛0, 𝑡󰇜𝑢󰇛1, 𝑡󰇜0
𝑢󰇛𝑥,0
󰇜𝑢󰇛𝑥,0
󰇜0 (81)
Solution.ApplyingtheFormabletransformonbothsidesofEquation(80)andusingthefactsin
Equations(18)and(20),weobtain
𝑠
𝑢𝐵󰇛𝑥,𝑠,𝑢󰇜𝑠
𝑢𝑢󰇛𝑥,0
󰇜𝑠𝑢𝑢󰇛𝑥,0󰇜𝜕
𝜕𝑥𝐵󰇛𝑥,𝑠,𝑢󰇜sin 𝜋𝑥(82)
Substitutingtheinitialconditionsof(81)inEquation(82),wehave
𝐵󰇛𝑥,𝑠,𝑢󰇜
𝐵󰇛𝑥,𝑠,𝑢󰇜sin 𝜋𝑥 . (83)
ThegeneralsolutionofthedifferentialEquation(83)canbewrittenas
𝐵󰇛𝑥,𝑠,𝑢󰇜𝐵󰇛𝑥,𝑠,𝑢󰇜𝐵󰇛𝑥,𝑠,𝑢󰇜, (84)
where𝐵󰇛𝑥,𝑠,𝑢󰇜𝐶exp 󰇡
𝑥󰇢𝐶exp 󰇛
𝑥󰇜isthehomogeneouspartofthegeneralso‐
lutionofEquation(83)and𝐵󰇛𝑥,𝑠,𝑢󰇜𝐴sin 𝜋𝑥𝐵cos 𝜋𝑥isthenonhomogeneouspart
ofthegeneralsolutionofEquation(83).
TofindAandBin𝐵󰇛𝑥,𝑠,𝑢󰇜,wesubstitute𝐵󰇛𝑥,𝑠,𝑢󰇜inEquation(83)togive
𝐵󰇛𝑥,𝑠,𝑢󰇜
 sin 𝜋𝑥,
since
𝐴
𝑢
𝑠𝜋𝑢 ,and 𝐵0.
Hence,Equation(84)becomes
𝐵󰇛𝑥,𝑠,𝑢󰇜𝐶exp 󰇡
𝑥󰇢 𝐶exp 󰇡
𝑥󰇢
 sin 𝜋𝑥. (85)
Axioms2021,10,33218of22
Substitutingtheboundaryconditionsof(81)inEquation(85),weobtain𝐶 𝐶0,
andtherefore𝐵󰇛𝑥,𝑠,𝑢󰇜
 sin 𝜋𝑥.
𝐵󰇛𝑥,𝑠,𝑢󰇜𝑢𝑠 𝑢𝑠 𝑠
𝑠𝜋𝑢 sin 𝜋𝑥
(86)
InEquation(86),weconsider
𝐹󰇛𝑠,𝑢󰇜
→
𝑓
󰇛𝑡󰇜𝑡, and𝐺󰇛𝑠,𝑢󰇜
 →𝑔󰇛𝑡󰇜 cos 𝜋𝑡 .
Hence,takingtheinverseFormabletransformofbothsidesofEquation(86),andus‐
ingtheconvolutionproperty,weobtain
𝑢󰇛𝑥,𝑡󰇜󰇛
𝑓
󰇛𝑡󰇜∗𝑔󰇛𝑡󰇜󰇜 sin 𝜋𝑥.
sin 𝜋𝑥𝜏 cos 𝜋󰇛𝑡𝜏󰇜𝑑𝜏
sin 𝜋𝑥
𝜋
󰇟1cos 𝜋𝑡󰇠.
Hence,thesolutionofEquation(80)withtheconditionsof(81)is
𝑢󰇛𝑥,𝑡󰇜 sin 𝜋𝑥
𝜋󰇟1cos 𝜋𝑡󰇠.(87)
5.Conclusions
Inthisarticle,wepresentedanewintegraltransformcalledtheFormabletransform.
Weintroducedthesufficientconditionsfortheexistenceofthenewtransform.Theduality
withothertransformswasexplained,andsomeessentialpropertieswereproved.Theap‐
plicabilityandaccuracyofthenewtransformwereshownbysolvingexamplesforboth
ordinaryandpartialdifferentialequations.Inaddition,wepresentedtablesintheAppen‐
dixAtocomparetheFormabletransformwithotherwell‐knowntransformsandtoillus‐
tratethesimplicityandabilityofthenewtransformthroughapplications.Inthefuture,
weintendtosolvefractionaldifferentialequationsandintegralequationsusingtheForm‐
abletransform.Furthermore,weplantocombinethetransformwithotheranalytical
methodstosolvenonlinearproblemssuchasDuffingoscillatorandMEMSoscillator
problemsandsomefractionaldifferentialequationsintheconformablesense.
AuthorContributions:Conceptualization,R.Z.S.andB.f.G.;methodologyR.Z.S.andB.f.G.;valida‐
tion,R.Z.S.andB.f.G.;formalanalysis,R.Z.S.;writing—originaldraftpreparation,R.Z.S.andB.f.G.;
writing—reviewandediting,R.Z.S.;supervision,R.Z.S.;projectadministration,R.Z.S.andB.f.G.
Allauthorshavereadandagreedtothepublishedversionofthemanuscript.
Funding:Thisresearchreceivednoexternalfunding.
DataAvailabilityStatement:Thereisnodataneeded.
ConflictsofInterest:Theauthorsdeclarenoconflictofinterest.
AppendixA
TableA1.Formabletransformofsomespecialfunctions.
No.𝒈󰇛𝒕󰇜𝑩󰇛𝒔,𝒖󰇜
111
2𝑡𝑢
𝑠
3𝑡
𝑛
!;
𝑓
𝑜𝑟 𝑛0,1,2, …𝑢
𝑠
Axioms2021,10,33219of22
4exp 󰇛𝛼𝑡󰇜𝑠
𝑠
𝛼𝑢
5𝑡
𝑛!exp 󰇛𝛼𝑡󰇜𝑠𝑢
󰇛
𝑠𝛼𝑢
󰇜

6sin 󰇛𝛼𝑡󰇜
𝛼
𝑠𝑢
𝑠𝛼𝑢
7cos 󰇛𝛼𝑡󰇜𝑠
𝑠𝛼𝑢
8sinh 󰇛𝛼𝑡󰇜
𝛼
𝑠𝑢
𝑠𝛼𝑢
9cosh 󰇛𝛼𝑡󰇜𝑠
𝑠
𝛼
𝑢
10exp 󰇛βt󰇜sin 󰇛𝛼𝑡󰇜
𝛼
𝑠𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢
11exp 󰇛βt󰇜cos 󰇛𝛼𝑡󰇜𝑠󰇛𝑠𝛽𝑢󰇜
󰇛
𝑠𝛽𝑢
󰇜
𝛼𝑢
12exp 󰇛βt󰇜sinh 󰇛𝛼𝑡󰇜
𝛼
𝑠𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢
13exp 󰇛βt󰇜cosh 󰇛𝛼𝑡󰇜𝑠󰇛𝑠𝛽𝑢󰇜
󰇛𝑠𝛽𝑢󰇜𝛼𝑢
14𝑒𝑥𝑝󰇛𝛽𝑡󰇜𝑒𝑥𝑝󰇛𝛼𝑡󰇜
𝛽
𝛼 ; 𝛼𝛽𝑠𝑢
󰇛𝑠𝛽𝑢󰇜󰇛𝑠𝛼𝑢󰇜
15𝛽𝑒𝑥𝑝󰇛𝛽𝑡󰇜𝛼𝑒𝑥𝑝󰇛𝛼𝑡󰇜
𝛽𝛼 ; 𝛼𝛽𝑠
󰇛
𝑠𝛽𝑢
󰇜
󰇛
𝑠𝛼𝑢
󰇜
16𝑡 sin 󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢
󰇛
𝑠𝛼𝑢
󰇜
17𝑡 sin 󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢󰇛3𝑠𝛼𝑢󰇜
󰇛
𝑠𝛼𝑢
󰇜
18𝑡 cos 󰇛𝛼𝑡󰇜𝑠𝑢󰇛𝑠𝛼𝑢󰇜
󰇛
𝑠𝛼𝑢
󰇜
19𝑡cos󰇛𝛼𝑡󰇜
2𝑠𝑢󰇛𝑠3𝛼𝑢󰇜
󰇛
𝑠𝛼𝑢
󰇜
20𝑡 sinh 󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢
󰇛
𝑠𝛼𝑢
󰇜
21𝑡 sinh 󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢󰇛3𝑠𝛼𝑢󰇜
󰇛
𝑠𝛼𝑢
󰇜
22𝑡 cosh 󰇛𝛼𝑡󰇜𝑠𝑢 󰇛𝑠𝛼𝑢󰇜
󰇛
𝑠𝛼𝑢
󰇜
23𝑡cosh󰇛𝛼𝑡󰇜
2𝑠𝑢󰇛𝑠3𝛼𝑢󰇜
󰇛
𝑠𝛼𝑢
󰇜
24sin󰇛𝛼𝑡󰇜𝛼𝑡cos󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢
󰇛
𝑠𝛼𝑢
󰇜
25sin󰇛𝛼𝑡󰇜𝛼𝑡cos󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢
󰇛
𝑠𝛼𝑢
󰇜
26cos󰇛𝛼𝑡󰇜1
2𝛼𝑡sin󰇛𝛼𝑡󰇜𝑠
󰇛
𝑠𝛼𝑢
󰇜
27sinh󰇛𝛼𝑡󰇜𝛼𝑡cosh󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢
󰇛
𝑠𝛼𝑢
󰇜
28𝛼𝑡cosh󰇛𝛼𝑡󰇜sinh󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢
󰇛
𝑠𝛼𝑢
󰇜
29cosh󰇛𝛼𝑡󰇜1
2𝛼𝑡sinh󰇛𝛼𝑡󰇜𝑠
󰇛
𝑠𝛼𝑢
󰇜
30sinh󰇛𝛼𝑡󰇜sin󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢
𝑠𝛼𝑢
Axioms2021,10,33220of22
31sinh󰇛𝛼𝑡󰇜sin󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢
𝑠
𝛼
𝑢
32cosh󰇛𝛼𝑡󰇜cos󰇛𝛼𝑡󰇜
2𝛼𝑠𝑢
𝑠
𝛼
𝑢
33cosh󰇛𝛼𝑡󰇜cos󰇛𝛼𝑡󰇜
2𝑠
𝑠𝛼𝑢
34
𝐽
󰇛𝛼𝑡󰇜𝑠
√
𝑠𝛼𝑢
35𝐼󰇛𝛼𝑡󰇜𝑠
√
𝑠𝛼𝑢
36𝑡
𝐽
󰇛𝛼𝑡󰇜𝑠𝑢
󰇛
𝑠𝛼𝑢
󰇜
37𝑡 𝐼󰇛𝛼𝑡󰇜𝑠𝑢
󰇛
𝑠𝛼𝑢
󰇜
38𝐶𝑖󰇛𝛼𝑡󰇜1
2log 𝑠𝛼𝑢
𝛼
𝑢
39𝑆𝑖󰇛𝛼𝑡󰇜𝑡𝑎𝑛𝛼𝑢
𝑠
40𝛿󰇛𝑡󰇜𝑠
𝑢
41𝛿󰇛𝑡𝛼󰇜𝑠
𝑢
exp 󰇡𝛼𝑠
𝑢
󰇢
42𝑈󰇛𝑡𝛼󰇜exp 󰇡𝛼𝑠
𝑢󰇢
TableA2.GeneralpropertiesofFormabletransform.
No.PropertyDefinition
1Definition𝐵󰇛𝑠,𝑢󰇜 𝑠𝑢exp 󰇛𝑠𝑡
𝑢󰇜
𝑔󰇛𝑡󰇜𝑑𝑡
2Inverse𝑔󰇛𝑡󰇜


 exp
󰇡

󰇢
𝐵󰇛𝑠,𝑢󰇜𝑑𝑠
3Derivative𝑅󰇟𝑔󰇛𝑡󰇜󰇠𝑠
𝑢𝐵󰇛𝑠,𝑢󰇜 󰇛𝑠𝑢󰇜 𝑔󰇛󰇜󰇛0󰇜

4Productshift𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠 𝑢
𝑠𝜕
𝜕𝑢󰇟𝑢𝐵󰇛𝑠,𝑢󰇜󰇠
󰇛𝑢󰇜𝑠𝜕
𝜕𝑠
󰇩
𝑅󰇟𝑔󰇛𝑡󰇜󰇠
𝑠
󰇪
5Productshiftand
derivative𝑅󰇟𝑡𝑔󰇛𝑡󰇜󰇠𝑢𝜕
𝜕𝑢󰇟𝐵󰇛𝑠,𝑢󰇜󰇠
6Divisionshift𝑅
󰇩
𝑔󰇛𝑡󰇜
𝑡
󰇪
𝑠
𝑢⋯𝐵󰇛𝑠,𝑢󰇜
𝑠
󰇛𝑑𝑠󰇜
7Convolution𝐵󰇟
𝑓
∗𝑔󰇠
𝐹󰇛𝑠,𝑢󰇜𝐺󰇛𝑠,𝑢󰇜
TableA3.Importantfunctionsanddefinitions.
No.Function Definition
1Besselfunction
𝐽
󰇛𝑥󰇜
󰇛󰇜󰇥1
󰇛󰇜
.󰇛󰇜󰇛󰇜⋯󰇦
2ModifiedBessel
function𝐼󰇛𝑥󰇜𝑖
𝐽
󰇛𝑖𝑥󰇜
󰇛󰇜󰇥1
󰇛󰇜
.󰇛󰇜󰇛󰇜⋯󰇦
3Sineintegral𝑆𝑖󰇛𝑡󰇜
𝑑𝑢

4Cosineintegral𝐶𝑖󰇛𝑡󰇜
𝑑𝑢

Axioms2021,10,33221of22
TableA4.Someintegraltransforms.
No.IntegralTransformDefinition
1Laplacetransform £ 󰇟𝑔󰇛𝑡󰇜󰇠𝐺󰇛𝑠󰇜 exp󰇛𝑠𝑡󰇜𝑔󰇛𝑡󰇜𝑑𝑡

2Fouriertransform𝐹 󰇟𝑔󰇛𝑡󰇜󰇠𝑔󰇛𝑤󰇜
√exp󰇛𝑖𝑤𝑡󰇜𝑔󰇛𝑡󰇜𝑑𝑡
 
3Mellintransform𝑀󰇟𝑔󰇛𝑠󰇜;𝑠󰇠󰇛𝑠󰇜𝑔∗󰇛𝑠󰇜𝑥 𝑔󰇛𝑥󰇜𝑑𝑥

4Elzakitransform 𝐸󰇟𝑔󰇛𝑡󰇜 󰇠󰇛𝑢󰇜𝑇󰇛𝑢󰇜 𝑢exp
󰇡

󰇢
𝑔󰇛𝑡󰇜𝑑𝑡

5Sumudutransform𝑆󰇟𝑔󰇛𝑡󰇜󰇠󰇛𝑢󰇜𝐺󰇛𝑢󰇜
exp 󰇡
󰇢𝑔󰇛𝑡󰇜𝑑𝑡

6Naturaltransform𝑁󰇟𝑔󰇛𝑡󰇜󰇠󰇛𝑠,𝑢󰇜𝑅󰇛𝑠,𝑢󰇜
exp
󰇡

󰇢
𝑔󰇛𝑡󰇜𝑑𝑡,𝑠,𝑢0

7Shehutransform𝕊󰇟𝑔󰇛𝑡󰇜󰇠󰇛𝑠,𝑢󰇜𝑉󰇛𝑠,𝑢󰇜exp
󰇡

󰇢
𝑔󰇛𝑡󰇜𝑑𝑡,𝑠,𝑢0

8ARAtransform𝒢󰇟𝑔󰇛𝑡󰇜󰇠󰇛𝑠󰇜𝐺󰇛𝑛,𝑠󰇜 𝑠𝑡exp󰇛𝑠𝑡󰇜𝑔󰇛𝑡󰇜𝑑𝑡,𝑠0
TableA5.ComprehensivelistoftheFormabletransformsB(s,u)andtheirrelationshipwiththeNaturaltransforms𝑅󰇛𝑠,𝑢󰇜,
theShehutransforms𝑉󰇛𝑠,𝑢󰇜andtheARAtransforms𝐺󰇛𝑚,𝑠󰇜.
No.𝒈󰇛𝒕󰇜𝑩󰇛𝒔,𝒖󰇜𝑽󰇛𝒔,𝒖󰇜𝑹󰇛𝒔,𝒖󰇜𝑮󰇛𝒎,𝒔󰇜
111𝑢𝑠1
𝑠Γ󰇛𝑚󰇜
𝑠

2𝑡𝑢𝑠𝑢
𝑠
𝑢
𝑠Γ󰇛𝑚1󰇜
𝑠
3𝑡
𝑛!;
𝑓
𝑜𝑟 𝑛
0,1,2, …𝑢
𝑠𝑢
𝑠𝑢
𝑠𝑠 Γ󰇛𝑚𝑛󰇜
𝑛!
4exp 󰇛𝛼𝑡󰇜𝑠
𝑠𝛼𝑢𝑢
𝑠𝛼𝑢1
𝑠𝛼𝑢𝑠 Γ󰇛𝑚󰇜
󰇛
𝑠𝛼
󰇜
5sin 󰇛𝛼𝑡󰇜
𝛼𝑠𝑢
𝑠𝛼𝑢𝑢
𝑠𝛼𝑢𝑢
𝑠𝛼𝑢𝑠
2𝛼𝑖 Γ󰇛𝑚󰇜1
󰇛
𝑠𝑖𝛼
󰇜
1
󰇛
𝑠𝑖𝛼
󰇜
6cos 󰇛𝛼𝑡󰇜𝑠
𝑠𝛼𝑢𝑠𝑢
𝑠𝛼𝑢𝑠
𝑠𝛼𝑢𝑠
2𝑖 Γ󰇛𝑚󰇜1
󰇛
𝑠𝑖𝛼
󰇜
1
󰇛
𝑠𝑖𝛼
󰇜
7sinh 󰇛𝛼𝑡󰇜
𝛼𝑠𝑢
𝑠𝛼𝑢𝑢
𝑠𝛼𝑢𝑢
𝑠𝛼𝑢𝑠
2𝛼 󰇧𝛼
𝑠𝑠󰇨Γ󰇛𝑚󰇜󰇟󰇛1𝛼𝑠󰇜
󰇛𝛼𝑠
𝑠󰇜󰇠
8cosh 󰇛𝛼𝑡󰇜𝑠
𝑠𝛼𝑢𝑠𝑢
𝑠𝛼𝑢𝑠
𝑠𝛼𝑢𝑠2 󰇛𝑠𝛼󰇜Γ󰇛𝑚󰇜󰇟󰇛𝑠|𝛼|󰇜
󰇛𝑠|𝛼|󰇜
󰇠
9exp 󰇛βt󰇜sin 󰇛𝛼𝑡󰇜
𝛼𝑠𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑠𝛼 󰇛𝑠𝛽󰇜Γ󰇛𝑚󰇜󰇛1
𝛼
󰇛
𝛽𝑠
󰇜
󰇜
sin󰇛𝑚tan 𝛼
𝑠𝛽󰇜
10exp 󰇛βt󰇜cos 󰇛𝛼𝑡󰇜𝑠󰇛𝑠𝛽𝑢󰇜
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑢󰇛𝑠𝛽𝑢󰇜
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑠𝛽𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑠 󰇛𝑠𝛽󰇜Γ󰇛𝑚󰇜󰇛1
𝛼
󰇛
𝛽𝑠
󰇜
󰇜
cos󰇛𝑚tan 𝛼
𝛽𝑠󰇜
11exp 󰇛βt󰇜sinh 󰇛𝛼𝑡󰇜
𝛼𝑠𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢
𝑠
2𝛼 󰇛𝑠𝛽󰇜Γ󰇛𝑚󰇜󰇛1𝛼
󰇛𝛽𝑠󰇜󰇜 󰇟󰇛1
𝛼
𝛽𝑠󰇜󰇛1
𝛼
𝑠
𝛽
󰇜󰇠
12exp 󰇛βt󰇜cosh 󰇛𝛼𝑡󰇜𝑠󰇛𝑠𝛽𝑢󰇜
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑢󰇛𝑠𝛽𝑢󰇜
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑠𝛽𝑢
󰇛𝑠𝛽𝑢󰇜𝛼𝑢𝑠2 󰇛𝑠𝛽󰇜Γ󰇛𝑚󰇜󰇟󰇛1
√
𝛼
𝛽𝑠󰇜 󰇛1
√
𝛼
𝑠
𝛽
󰇜󰇠
Axioms2021,10,33222of22
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... Remark If we substitute (r ) = r into (3.1), we acquire the Formable transform as investigated in Saadeh and Bayan (2021). ...
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Solution of partial differential equations by new double integral transform (Laplace - Sumudu transform)
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The primary purpose of this research is to demonstrate an efficient new double transform mentioned since the double Laplace - Sumudu transform (DLST) solve partial differential equations. The theorems handling fashionable properties of the double Laplace - Sumudu transform are proved, the convolution theorem with evidence is mentioned, then, via the usage of these outcomes the solution of partial differential equations is made. The results showed that the double Laplace - Sumudu transform was more efficient and useful to handle such these kinds of equations.
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Solution of Integral Differential Equations by New Double Integral Transform (Laplace-Sumudu Transform)
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The primary purpose of this research is to demonstrate an efficient replacement double transform named the Laplace–Sumudu transform (DLST) to unravel integral differential equations. The theorems handling fashionable properties of the Laplace–Sumudu transform are proved; the convolution theorem with an evidence is mentioned; then, via the usage of these outcomes, the solution of integral differential equations is built. 1. Introduction Double integral transform and their characteristics and theories are nevertheless new and below studies [1–3], in which the preceding research treated some components of them along with definitions, simple theories, and the answer of normal and partial differential equations [4–16]; additionally, some researchers addressed these transforms and combine them with exclusive mathematical method such as differential transform approach, homotopy perturbation technique, Adomian decomposition method, and variational iteration method [7–16] so that we can solve the linear and nonlinear fractional differential equations. In this paper, we are ready to spotlight the way during which the Laplace–Sumudu transform is blend to solve the integral differential equations. A wide range of linear integral differential equations are considered which include the Volterra integral equation (Section 3.1), the Volterra integro-partial differential equation (Section 3.2), and the partial integro-differential equation (Section 3.3). Definition 1. The double Laplace–Sumudu transform of the function of two variables and is denoted by and defined as Clearly, double Laplace–Sumudu transform is a linear integral transformation as shown below: where and are constants. Definition 2. The inverse double Laplace–Sumudu transform is defined by the following form: 2. Double Laplace–Sumudu Transform of Basic Functions (1)Let then (2)Let then If and are positive integral, then (3)Let then Similarly, Consequently, (4) Let Recall that Therefore, (5) Let then where is the modified Bessel function of order zero. (6) Let then 2.1. Existence Condition for the Double Laplace–Sumudu Transform If is an exponential order, then and as , and if a positive constant such that then and we write as Or, equivalently, The function is called an exponential order as , and clearly, it does not grow faster than as . Theorem 3. If a function is a continuous function in every finite interval and of exponential order , then the double Laplace–Sumudu transform of exists for all and provided and Proof. From the Definition 1., we have Then, from Eq. (16) we have or . 2.2. Basic Derivative Properties of the Double Laplace–Sumudu Transform If , then Proof. Using integration by parts, let then Proof. Using integration by parts, let then . Similarly, we can prove Theorem 4. If , then where is the Heaviside unit step function defined by Proof. We have, by Definition 1., that is, by putting 2.3. Convolution Theorem of Double Laplace–Sumudu Transform Definition 5. The convolution of and is denoted by and defined by Theorem 6. (convolution theorem) If and then Proof. From the definition 1., we have which is, using the Heaviside unit step function, that is, by Theorem 4 gives 3. Application of Laplace–Sumudu Transform (DLST) of Integral Differential Equations In this section, we apply the double Laplace–Sumudu transform (DLST) method to linear integral differential equations. 3.1. Volterra Integral Equation Consider the linear Volterra integral equation as form where is the unknown function, is a constant, and and are two known functions. Applying the double Laplace–Sumudu transform (DLST) with linearity to both sides of equation (32) and using Theorem 6 (convolution theorem), we get Consequently, Taking for equation (34), we obtain the solution of equation (32). We illustrate the above method by simple examples. (a)Solve the equationwhere and are constant. Applying the double Laplace–Sumudu transform (DLST) of equation (36), we get Consequently, Taking for equation (38), we obtain the solution of equation (36). (b)Solve the equationwhere is a constant. Applying (DLST) of equation (40), we get Or Taking for equation (42), we obtain the solution of equation (40). (c)Solve the equation Applying (DLST) of equation (44), we get Simplifying and taking for equation (45), we obtain 3.2. Volterra Integro-Partial Differential Equations Consider the linear Volterra integro-partial differential equation as form with the conditions: where is the unknown function, is a constant, and and are two known functions. Applying (DLST) to both sides of (47) and single (LT) and (ST) for equation (48) and simplification, we get Applying to (49), we obtain the solution of (47) in the form We illustrate the above method by a simple example. (d)Solve the equation By substituting in (47), we have got with the conditions: Substituting in (50) and simplifying, we get the solution of (51) 3.3. Partial Integro-Differential Equation Consider the linear partial integro-differential equation as form with the conditions: Applying (DLST) to both sides of (55) and single (LT) and (ST) for equation (56) and simplification, we get Applying to (57), we obtain the solution of (55) in the form We illustrate the above method by a simple example. (e)Solve the equation: By substituting in (55), we have got with the conditions: Substituting in (58) and simplifying, we get a solution of (59) 4. Conclusion In this paper, the Laplace–Sumudu transform approach for solving integral differential equations is studied. We provided the theorems and popular properties for this new double transform and furnished some examples. The examples show that the Laplace–Sumudu transform approach is powerful in solving the equations of taken into consideration type, and a couple of advanced problems in linear and nonlinear partial differential equations and nonlinear integral differential equations could be discussed during a later paper. Data Availability No data were used to support this study. Conflicts of Interest The authors declare that they have no conflicts of interest.
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Fractal N/MEMS: From pull-in instability to pull-in stability
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  • Oct 2020
Pull-in instability, as an inherent nonlinear problem, continues to become an increasingly important and interesting topic in the design of electrostatic Nano/Micro-electromechanical systems (N/MEMS) devices. Generally, the pull-in instability was studied in a continuous space, but when the electronic devices work in a porous medium, they need to be analyzed in a fractal partner. In this paper, we establish a fractal model for N/MEMS, and find a pull-in stability plateau, which can be controlled by the porous structure, and the pull-in instability can be finally converted to a stable condition. As a result, the pull-in instability can be completely eliminated, realizing the transformation of pull-in instability into pull-in stability.
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