OntheSolutionofWave-SchrodingerEquation

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© 2019. Wanchak Satsanit. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Global Journal of Science Frontier Research: F Mathematics and Decision Sciences

Volume 19 Issue 3 Version 1.0 Year 2019

Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals

Online ISSN: 2249-4626 & Print ISSN: 0975-5896

On the Solution of Wave-Schrodinger Equation

By Wanchak Satsanit

Maejo University Abstract- In this paper, we are finding solution of fraction Wave-Schrodinger equation by Laplace transform in sense of Caputo fractional derivative. It was found that the fundamental solution of the equation related to Wright function.

Keywords: dirac delta distribution, laplacian operator, wright function.

GJSFR-F Classification: MSC 2010: 35L05

OntheSolutionofWave-SchrodingerEquation

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On the Solution of Wave-Schrodinger

Equation

Wanchak Satsanit

Author:Department of Mathematics,Faculty of Science, Maejo University,Chiang Mai, 50290 Thailand.e-mail: [email protected]

Abstract-In this paper, we are finding solution of fraction Wave-Schrodinger equation by Laplacetransform in sense of

Caputo fractional derivative. It was found that the fundamentalsolution of the equation related to Wright function.

dirac delta distribution, laplacian operator, wright function.

The Laplacian operator 4k _{iterated k− times is defined by}

4k₌ µ ∂2 ∂x2 1 + ∂ 2 ∂x2 2 + . . . + ∂ 2 ∂x2 n ¶k , (1.1)

where n is the dimension of space Rn_{, k is a nonnegative integer. A. Kananthai[1]}

has proved that the generalized function (−1)k_S

2k(x) is an elementary solution of the

operator 4k_{, that is}

4k₍₋₁₎k_S

2k(x) = δ,

where δ is the Dirac-delta distribution and S2k(x) is defined by

S2k(x) = π−n₂₂−2k_Γ¡n−2k 2 ¢ (x2 1+ x22+ ... + x2n) 2k−n 2 Γ (k) , (1.2)

In 2002, A.Kananthai, S. Suantai, V. Longani[2] have first introduced the operator 4k

i and is defined by 4k i = Ã _p X i=1 ∂2 ∂x2 i + i p+q X j=p+1 ∂2 ∂x2 j ! , i =√−1 (1.3)

They have proved the function (−1)k_(−i)q₂_S

2k(x) is an elementary solution of the

operator 4k

i and S2k(x) is defined by (1.2).It is well known the linear Schrodinger

equation can be written as the following form

Keywords: 1. K an an th ai , On t h e S olu ti on o f t he n -D im en si on al D iam on d O p er at or , A p p lie d M at h em at ic s an d C om p u tat ion al , E lsev ier S ci en ce I n c. Ne w Yor k , 19 97, p p . 27 -37.

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On the Solution of Wave-Schrodinger Equation

∂ ∂tu(x, t) = i ∂2 ∂x2u(x, t), i = √ −1 (1.4)

with the initial condition

u(x, 0) = f (x).

The Schrodinger equation has been widely in application in science and engineering, there are several integral transform such as Laplace transform, Fourier transform, Wavelet transform etc. for solving the equation.

The purpose of this work is to introduce a new function where related the Wright function [3] and studied Laplace transform of a new function.After that,we are solving the fundamental solution of the wave-schrodinger equation as follows:

∂α ∂tαφ(x, t) + i ∂2 ∂x2φ(x, t) = 0 , i = √ −1 , 1 < α ≤ 2 (1.5) with the initial condition

φ(x, 0) = 0 , φt(x, 0) = δ(x),

where δ is the Dirac-delta distribution and ∂α

∂tα is the Caputo derivative. Before going

that point, the following definitions and some important concepts are needed.

Let f (t) be a function an exponential order and piecewise continuous. The Laplace transform of the function f is given by

L[f (t)] =

Z _∞

0

e−stf (t)dt (2.1)

Let f (t) be a function of the Schwart space the Fourier transform of f (t) is given by b f (w) = Z R f (t)eiwtdt (2.2)

For m to be the smallest integer that exceeds α , the Caputo fractional derivatives of order is defined by

Dα_{u(x, t) =} ∂αu(x, t) ∂tα = ( 1 Γ(m−α) R_t 0(t − τ )m−n−1 ∂ m ∂tmu(x, t) ∂m ∂tmu(x, t) , n = m (2.3)

The Laplace transform of the Caputo fractional derivative is defined by L[Dαf (t)] = sαF (s) − n−1 X k=0 sα−k−1f(k)(0) , n − 1 < α < n (2.4) II. Preliminaries De¯nition 2.1 De¯nition 2.2 De¯nition 2.3 De¯nition 2.4 3. I. P o dl ub ny . F rac tio an l D i® er en tial Eq u at ion , A n i n tr oduc tio n t o fr a ct io na l d er iva tiv es. A ca d em ic P ress 1 9 9 9 .

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The Wright function Wα,β is defined by Wα,β = ∞ X n=0 Zn n!Γ(nα + β) , α > −1 , β ∈ C (2.5) where Γ(x) is the Euler Gamma function is given by the integral

Γ(x) =

Z _∞

0

tx−1e−6dt (2.6)

The function γ(a, t) is defined by the following expressions

γ(a, t) = t−α+1W−α , 2−α(at−α) (2.7)

and Laplace transform of γ(a, t) is given by

L[γ(a, t)] = sα−2_easα . By (2.1) , we have L[γ(a, t)] = Z _∞ 0 e−st_t−α+1_W −α , 2−α(at−α)dt = Z _∞ 0 e−st_t−α+1 ∞ X k=0 (at−α₎k k!Γ(−αk + 2 − α)dt = ∞ X k=0 ak k!Γ(−αk + 2 − α) Z _∞ 0 e−st_t−α−αk+1_dt = ∞ X k=0 ak k!Γ(−αk + 2 − α)L[t −α−αk+1_] = ∞ X k=0 ak k!Γ(−αk + 2 − α) Γ(−α − αk + 2) s−α−αk+2 = ∞ X k=0 ak k!s−α−αk+2 = sα−2 ∞ X k=0 (asα₎k k! = sα−2easα (2.8)

That completes the proof.

Consider the Fractional Wave-Schrodinger equation ∂α ∂tαφ(x, t) + i ∂2 ∂x2φ(x, t) = 0 , i = √ −1 , 1 < α ≤ 2 (3.1) with the initial condition

De¯nition 2.5

Lemma 2.1

III. Main Results

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φ(x, 0) = 0 , φt(x, 0) = δ(x)

where δ(x) is the dirac delta distribution.By the Laplace and Fourier transform,we obtain the fundamental solution of the equation (3.1) is given by

φ(x, t) = 1 2 √ it−α 2+1W₋α 2 , 2−α2(− √ i|x|t−α 2) (3.2)

where Wα,β is the Wright function is defined by (2.5). If we put α = 2 in (3.1) the

fractional Wave-Schrodinger equation reduced to ∂2

∂t2φ(x, t) + i

∂2

∂x2φ(x, t) = 0 (3.3)

and the solution of (3.3) is given by φ(x, t) = 1 2 √ iW−1,1(− √ i|x|t−1) (3.4) By (3.1) , we have ∂α ∂tαφ(x, t) + i ∂2 ∂x2φ(x, t) = 0 (3.5)

Taking Laplace transform both sides of (3.5) and we get by definition 2.1 Lh ∂ α ∂tαφ(x, t) i + iLh ∂ 2 ∂x2φ(x, t) i = 0 sαφ(x, s) − sα−2δ(x) = −i ∂ 2 ∂x2φ(x, s). (3.6)

Applying Fourier transform respect to variable x both sides of (3.6),we obtained sα_{Fφ(x, s) − s}α−2_{F[δ(x)] = −iF} ∂2 ∂x2φ(x, s) sα_{φ(ω, s) − s}α−2 _{= iω}2_{φ(ω, s)} φ(ω, s) = s α−2 sα_{+ (−i)ω}2 = isα−2 isα_{+ ω}2. (3.7)

Applying inverse Fourier transform both sides of (3.7) , we obtain φ(x, s) = F−1h is α−2 isα_{+ ω}2 i = √ isα−2_e−|x|√isα2 2sα2 = 1 2 √ isα2−2e−|x| √ isα2 .

On the Solution of Wave-Schrodinger Equation

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By Lemma 2.1,we obtain the solution of (3.1) as follows φ(x, t) = 1 2 √ ir(−√i|x|, t) = 1 2 √ it−α2+1W₋α 2,2− α 2(− √ i|x|t−α2) (3.8)

If we put α = 2 in (3.1) and (3.8) respectively, the equation reduced to the Wave-Schrodinger equation

∂2

∂t2φ(x, t) + i

∂2

∂x2φ(x, t) = 0, (3.9)

and the solution of (3.9) is given by φ(x, t) = 1 2 √ iW−1,1(− √ i|x|t−1). (3.10)

That completes the proof.

Acknowledgement

The authors would like to thank the Thailand Research Fund , Graduate School, Maejo University, Chiang Mai, Thailand for ¯nancial support and Prof. Luis Guillermo Romero for the helpful of discussion.

References Références Referencias

1. Kananthai, On the Solution of the n-Dimensional Diamond Operator, Applied

Mathematics and Computational, Elsevier Science Inc. New York, 1997, pp. 27-37. 2. A. Kananthai, S. Suantai, V, Longani. On the operator ©k related to the wave

equation and Laplacian, Applied Mathematics and Computational, Elsevier Science Inc. New York, 2002, pp 219-229.

3. I. Podlubny. Fractioanl Di®erential Equation, An introduction to fractional derivatives. Academic Press 1999.