METEORITE SHOOTING AS A DIFFUSION PROBLEM

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Journal for Research | Volume 03| Issue 01 | March 2017 ISSN: 2395-7549

Meteorite Shooting as a Diffusion Problem Dr. Prof. Rashmi R. Keshvani Professor Department of Mathematics Sarvajanik College of Engineering & Technology, Surat, Gujarat, India

Prof. Maulik S. joshi Assist Professor Department of Mathematics Aditya Silver oak Institute of Technology, Ahmedabad, Gujarat, India

Abstract Diffusion problems have been problems of great interest with various initial and boundary conditions. Among those, infinite domain problems have been more interesting. Many of such problems can be solved by various methods but those which can be used for various initial functions with minor changes in the solution obtained are more attractive and efficient. Fourier transforms method and methods obtaining Gauss- Weierstrass kernel play such role among various such methods. To show this feature here in this paper, first the consequences of a local injection of heat to an infinite domain are being discussed. Solutions to such problems at different time are discussed in terms of Gaussian distributions. The theory is then extended to a meteorite shooting problem. Keywords: Fourier Transform, Inverse Fourier Transform, Dirac Delta Function, Gaussian distribution, Mean and variance of a probability distribution, Meteorites, Refraction index _______________________________________________________________________________________________________ I. It is known and can be verified that đ?‘’ −đ?‘–đ?œ”đ?‘Ľ đ?‘’ −đ?‘˜đ?œ” principle of super position, it can be shown that 2 ∞ đ?‘˘(đ?‘Ľ, đ?‘Ą) = âˆŤâˆ’âˆž đ?‘?(đ?œ”) đ?‘’ −đ?‘–đ?œ”đ?‘Ľ đ?‘’ −đ?‘˜đ?œ” đ?‘Ą đ?‘‘đ?œ” is solution of heat equation,

đ?œ•đ?‘˘ đ?œ•đ?‘Ą

=đ?‘˜

đ?œ•2 đ?‘˘ đ?œ•đ?‘Ľ 2

2đ?‘Ą

INTRODUCTION

satisfy heat equation

đ?œ•đ?‘˘ đ?œ•đ?‘Ą

=đ?‘˜

đ?œ•2 đ?‘˘ đ?œ•đ?‘Ľ 2

for all values of đ?œ” So using generalized (1)

where −∞ < đ?‘Ľ < ∞. ∞

The initial condition đ?‘˘(đ?‘Ľ, 0) = đ?‘“(đ?‘Ľ), will be satisfied, if đ?‘˘(đ?‘Ľ, 0) = đ?‘“(đ?‘Ľ) = âˆŤâˆ’âˆž đ?‘?(đ?œ”) đ?‘’ −đ?‘–đ?œ”đ?‘Ľ đ?‘‘đ?œ”. From definitions of Fourier transform and

inverse Fourier transform [1],

∞ âˆŤâˆ’âˆž đ?‘?(đ?œ”) đ?‘’ −đ?‘–đ?œ”đ?‘Ľ đ?‘‘đ?œ”

∞

đ?‘“(đ?‘Ľ) = âˆŤâˆ’âˆž đ?‘?(đ?œ”) đ?‘’ −đ?‘–đ?œ”đ?‘Ľ đ?‘‘đ?œ”

implies that

1 ∞ âˆŤ đ?‘“(đ?‘Ľ)đ?‘’ đ?‘–đ?œ”đ?‘Ľ đ?‘‘đ?‘Ľ 2đ?œ‹ −∞

is inverse Fourier transform of đ?‘“(đ?‘Ľ) and đ?‘?(đ?œ”) = is the Fourier transform of initial temperature function đ?‘“(đ?‘Ľ). 1 ∞ Substituting đ?‘?(đ?œ”) = âˆŤâˆ’âˆž đ?‘“(đ?‘Ľ)đ?‘’ đ?‘–đ?œ”đ?‘Ľ đ?‘‘đ?‘Ľ in (1), and changing dummy variable đ?‘Ľ to đ?‘ĽĚ… in expression for đ?‘?(đ?œ”), now (1) 2đ?œ‹ becomes ∞ 1 ∞ 2 đ?‘˘(đ?‘Ľ, đ?‘Ą) = âˆŤ ( âˆŤ đ?‘“(đ?‘ĽĚ… )đ?‘’ đ?‘–đ?œ”đ?‘ĽĚ… đ?‘‘đ?‘ĽĚ… ) đ?‘’ −đ?‘–đ?œ”đ?‘Ľ đ?‘’ −đ?‘˜đ?œ” đ?‘Ą đ?‘‘đ?œ” 2đ?œ‹ −∞ −∞ 2 ∞ 1 ∞ â&#x;š đ?‘˘(đ?‘Ľ, đ?‘Ą) = âˆŤâˆ’âˆž đ?‘“(đ?‘ĽĚ… ) ( âˆŤâˆ’âˆž đ?‘’ −đ?‘–đ?œ”(đ?‘Ľâˆ’đ?‘ĽĚ… ) đ?‘’ −đ?‘˜đ?œ” đ?‘Ą đ?‘‘đ?œ”) đ?‘‘đ?‘ĽĚ… (2) 2đ?œ‹ ∞

2

2

Also it is known that đ?‘”(đ?‘Ľ) = âˆŤâˆ’âˆž đ?‘’ −đ?‘–đ?œ”đ?‘Ľ đ?‘’ −đ?‘˜đ?œ” đ?‘Ą đ?‘‘đ?œ” is inverse Fourier transform of đ?‘’ −đ?‘˜đ?œ” đ?‘Ą , ∞

2

(a Gaussian Curve). So, đ?‘”(đ?‘Ľ − đ?‘ĽĚ… ) = âˆŤâˆ’âˆž đ?‘’ −đ?‘–đ?œ”(đ?‘Ľâˆ’đ?‘ĽĚ… ) đ?‘’ −đ?‘˜đ?œ” đ?‘Ą đ?‘‘đ?œ” = Substituting this in (2), one gets đ?‘˘(đ?‘Ľ, đ?‘Ą) = It can be shown [1] that lim √ đ?‘Ąâ†’0+

1 4đ?œ‹đ?‘˜đ?‘Ą

√đ?œ‹ √đ?‘˜đ?‘Ą

đ?‘’

Ě… )2 −(đ?‘Ľâˆ’đ?‘Ľ 4đ?‘˜đ?‘Ą

.

∞ −(đ?‘Ľâˆ’đ?‘ĽĚ… )2 1 ∞ 1 √đ?œ‹ −(đ?‘Ľâˆ’đ?‘ĽĚ… )2 âˆŤ đ?‘“(đ?‘ĽĚ… ) ( đ?‘’ 4đ?‘˜đ?‘Ą ) đ?‘‘đ?‘ĽĚ… = âˆŤ đ?‘“(đ?‘ĽĚ… ) √ đ?‘’ 4đ?‘˜đ?‘Ą đ?‘‘đ?‘ĽĚ… 2đ?œ‹ −∞ 4đ?œ‹đ?‘˜đ?‘Ą √đ?‘˜đ?‘Ą −∞

đ?‘’

Ě… )2 −(đ?‘Ľâˆ’đ?‘Ľ 4đ?‘˜đ?‘Ą

= đ?›ż(đ?‘Ľ − đ?‘ĽĚ… ), where đ?›ż(đ?‘Ľ) is the Dirac delta function (Impulse function).[1] The

Dirac delta function, [2] denoted by đ?›ż(đ?‘Ľ), also known as impulse function, is defined as ∞ 0 if đ?‘Ľ ≠0 đ?›ż(đ?‘Ľ) = { ensuring âˆŤâˆ’âˆž đ?›ż(đ?‘Ľ)đ?‘‘đ?‘Ľ = 1 . ∞ if đ?‘Ľ = 0 ∞ Also âˆŤâˆ’âˆž đ?‘“(đ?‘Ľ)đ?›ż(đ?‘Ľ − đ?‘Ž)đ?‘‘đ?‘Ľ = đ?‘“(đ?‘Ž) where đ?‘“ is any continuous function?

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