Solution of Ordinary Differential Equation with Initial Condition Using New Elzaki Transform

International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056

Volume: 10 Issue: 04 | Apr 2023 www.irjet.net

Solution of Ordinary Differential Equation with Initial Condition Using New Elzaki Transform

1Assistant professor, Department of Mathematics, Rabindra nath Tagore University,

2Proffessor, Department of Mathematics, Govt. Motilal Vigyan Mahavidyalya, Bhopal(M.P.)***

Abstract- In this paper the solution of Ordinary Differential Equation with initial condition by using the new Elzaki transform is introduced. The new Elzaki transform is modified version of Laplace and Sumudu transform, and it also expand the nth order derivatives by mathematical induction method. Also in this paper we have explained the properties of Elzaki transform, with inversion form of the transform. With this application we can generate simple formula for solving First order first degree, and Second order first degree ordinary differential equations, with constant coefficients.

Keywords: Elzaki transform Ordinary differential equations.

AMS Subject Classification: 44AXX, 34A30, 34A12.

1. INTRODUCTION

Ordinary differential equations are used in many areas of engineering and basic science like application Beams, Electrical circuits, Dynamics, etc the Laplace transform are some of the well known ordinary differential equations used in these fields. Many type of Ordinary differential equations also can be solved with aid of integral transforms such as Laplace transform, Fourier transform, Sumudu transform[1,5]. In this paper, we havestudiedtoobtainaformulaforaspecialsolutionof in the most general case nth order Ordinary Differential Equations with constant coefficient. Also found the solution of first order first degree & second order first degree linear differential equation with constant coefficient[3,4].

The Elzaki transform method used in several areas of mathematics is an integral transform. We can solve linear differential equation with use Elzaki transform operator moreover partial differential equations, integral equations & integro differential equations. This method can not be suitable for solution of non-linear differential equations. However non-linear differential equation can be solved by using Elzaki transform aid withdifferentialtransformmethod.

Elzaki transform define for function of exponential order,function definewithasetAbelow,

Hear constant M must be real finite number and maybefiniteorinfinite.

ElzakitransformdenotedbyoperatorE,

2. ELZAKI TRANSFORM OF SOME FUNCTIONS

Elzaki transform of first order derivatives of as following:

Weassumethatthetheoremistrueforn=k,

Elzaki transform of second order derivatives of as following:

Nowwehavetoshowthetheoremistrueforn=k+1,

Elzaki transform of third order derivatives of as following:

Now,

Now, find the expansion of nth order derivative after applyElzakitransformandbythisexpansionIhaveused someLemma&examplesand introducethenewresults for first order first degree (FOFD) & second order first degree (SOFD) ordinary differential equations with constantcoefficients.

3. EXPANSION OF NTH ORDER DERIVATIVE WITH ELZAKI OPERATOR

Theorem: 3.1

Proof:

Requiredresult.

Lemma. 3.2

ElzakisolutionofFirstOrderFirstDegree(FOFD)Linear DifferentialEquations.

Proof: GivenFirstOrderFirstDegree(FOFD)LinearDifferential Equations, TakingElzakitransformbothside,

Theoremistrueforn=1.

ItisthesolutionofFOFDlineardifferentialequation. Lemma. 3.3

ElzakisolutionofSecondOrderFirstDegree(SOFD) LinearDifferentialEquations.

ItisthesolutionofSOFDlineardifferentialequation.

Proof:

GivenSecondOrderFirstDegree(SOFD)Linear DifferentialEquations.

TakingElzakitransform,

4. CONCLUSION

InthispaperwehaveappliedElzakitransformwiththe new formula on ordinary differential equation of first orderfirstdegreeandsecondorderfirstdegreewiththe result. This formula is also applicable for nth order, it is very useful and effective method. This process is also very useful for other type of differential equations, appliedmathematics,engineeringandscience.

5. REFERENCES

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