Solving Diff. Eqs. via Laplace Transform

International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 4 Issue 2, February 2020
@ IJTSRD | Unique Paper ID – IJTSRD30197
Solving Differential Equations Including
Leguerre Polynomial
Dr
1Associate Professor, Yogananda College
2Assistant Professor, Jagdish Saran Hindu P. G. College, Amroha
ABSTRACT
The Laplace transformation is a mathematical
differential equations. Laplace transformation makes it easier to solve the
problem in engineering application and make differential equations simple
to solve. In this paper, we will solve differential equations including
Leguerre Polynomial via Laplace Transform Method.
KEYWORDS: Laplace Transform, Differential Equation
SUBAREA: Laplace transformation
BROAD AREA: Mathematics
INRODUCTION
The Laplace transformation is applied in different areas of
science, engineering and technology [1-
transformation is applicable in so many fields and
effectively solving linear differential equations. Ordinary
linear differential equation with constant coefficient and
variable coefficient can be easily solved by the Laplace
transform method without finding their general solutions
[6, 7, 8, 9,]. This paper presents the application of Laplace
transform in solving the differential equations including
Leguerre Polynomial.
DEFINITION
Let F (t) is a well defined function of t for all t
Laplace transformation [10, 11] of F (t), denoted by f (
or L {F (t)}, is defined L {F (t)} =‫׬‬ ݁ି௣௧∞
଴
provided that the integral exists, i.e. convergent. If the
integral is convergent for some value of
Laplace transformation of F (t) exists otherwise not.
Where ‫ ݌‬the parameter which may be real or complex
number and L is the Laplace transformation operator.
Laplace Transformation of Elementary Functions
13]
1. ‫ ܮ‬ሼ1ሽ ൌ
ଵ
௣
, ‫݌‬ ൐ 0
2. ‫ ܮ‬ሼ‫ݐ‬௡ሽ ൌ
௡!
௣೙శభ , ‫݊ ݁ݎ݄݁ݓ‬ ൌ 0,1,2,3 … … …
February 2020 Available Online: www.ijtsrd.com e
30197 | Volume – 4 | Issue – 2 | January-February 2020
ifferential Equations Including
Leguerre Polynomial via Laplace Transform
Dr. Dinesh Verma1, Amit Pal Singh2
Yogananda College of Engineering & Technology, Jammu, Jammu
Jagdish Saran Hindu P. G. College, Amroha, Uttar Pradesh
The Laplace transformation is a mathematical tool used in solving the
differential equations. Laplace transformation makes it easier to solve the
problem in engineering application and make differential equations simple
to solve. In this paper, we will solve differential equations including
Polynomial via Laplace Transform Method.
Laplace Transform, Differential Equation
How to cite this paper
Verma | Amit Pal Singh "Solving
Differential
Equations Including
Leguerre
Polynomial via
Laplace Tr
Published in
International
Journal of Trend in
Scientific Research
and Development (ijtsrd), ISSN: 2456
6470, Volume
2020, pp.1016
www.ijtsrd.com/papers/ijtsrd30197.pdf
Copyright © 2019 by author(s) and
International Journal of Trend in
Scientific Research
and Development
Journal. This is an
Open Access article distributed under
the terms of the Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/licenses/
by/4.0)
The Laplace transformation is applied in different areas of
-5]. The Laplace
transformation is applicable in so many fields and
ctively solving linear differential equations. Ordinary
linear differential equation with constant coefficient and
variable coefficient can be easily solved by the Laplace
transform method without finding their general solutions
resents the application of Laplace
transform in solving the differential equations including
Let F (t) is a well defined function of t for all t ≥ 0. The
Laplace transformation [10, 11] of F (t), denoted by f (‫)݌‬
௣௧
‫ܨ‬ሺ‫ݐ‬ሻ݀‫ݐ‬ ൌ ݂ሺ‫݌‬ሻ,
provided that the integral exists, i.e. convergent. If the
ral is convergent for some value of ‫, ݌‬ then the
Laplace transformation of F (t) exists otherwise not.
the parameter which may be real or complex
number and L is the Laplace transformation operator.
Laplace Transformation of Elementary Functions [12,
….
3. ‫ ܮ‬ሼ݁௔௧ሽ ൌ
ଵ
௣ି௔
, , ‫݌‬ ൐ ܽ
4. ‫ ܮ‬ሼ‫ݐܽ݊݅ݏ‬ሽ ൌ
௔
௣మା௔మ , ‫݌‬ ൐ ܽ
5. ‫ ܮ‬ሼ‫ݐ݄ܽ݊݅ݏ‬ሽ ൌ
௔
௣మି௔మ , ‫݌‬ ൐
6. ‫ ܮ‬ሼܿ‫ݐܽݏ݋‬ሽ ൌ
௣
௣మା௔మ , ‫݌‬ ൐ 0
7. ‫ ܮ‬ሼܿ‫ݐ݄ܽݏ݋‬ሽ ൌ
௣
௣మି௔మ , ‫݌‬ ൐
Proof: By the definition of Laplace transformation, we
know that L {F (t)} =‫׬‬ ݁ି௣௧∞
଴
‫ܨ‬
L { ݁௔௧
} =‫׬‬
∞
଴
= -
ଵ
௣ି௔
ሺ ݁ି∞
െ ݁
ൌ
1
‫݌‬ െ ܽ
ൌ
Laplace Transformation of derivatives
Let F is an exponential order, and that F is a continuous
and f is piecewise continuous on any interval [14, 15, 16,],
then
‫ ܮ‬൛‫ܨ‬′ሺ‫ݐ‬ሻൟ ൌ න
଴
= ሾ0 െ ‫ܨ‬ሺ0ሻሿ െ ‫׬‬
e-ISSN: 2456 – 6470
February 2020 Page 1016
ifferential Equations Including
ia Laplace Transform
Jammu and Kashmir, India
ar Pradesh, India
How to cite this paper: Dr. Dinesh
Verma | Amit Pal Singh "Solving
Differential
Equations Including
Polynomial via
Laplace Transform"
Published in
International
Journal of Trend in
Scientific Research
and Development (ijtsrd), ISSN: 2456-
6470, Volume-4 | Issue-2, February
2020, pp.1016-1019, URL:
www.ijtsrd.com/papers/ijtsrd30197.pdf
Copyright © 2019 by author(s) and
International Journal of Trend in
Scientific Research
and Development
Journal. This is an
Open Access article distributed under
the terms of the Creative Commons
ribution License (CC BY 4.0)
http://creativecommons.org/licenses/
ܽ
|ܽ|
0
|ܽ|
By the definition of Laplace transformation, we
‫ܨ‬ሺ‫ݐ‬ሻ݀‫ݐ‬ then
݁ି௣௧∞
଴
݁௔௧
݀‫ݐ‬
݁ି଴
) =
ଵ
௣ି௔
ሺ0 െ 1ሻ
݂ሺ‫݌‬ሻ, ‫݌‬ ൐ ܽ
Laplace Transformation of derivatives
Let F is an exponential order, and that F is a continuous
and f is piecewise continuous on any interval [14, 15, 16,],
න ݁ି௣௧
∞
଴
‫ܨ‬′ሺ‫ݐ‬ሻ݀‫ݐ‬
ሿ ‫׬‬ െ‫݁݌‬ି௣௧∞
଴
‫ܨ‬ሺ‫ݐ‬ሻ݀‫ݐ‬
IJTSRD30197

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD30197 | Volume – 4 | Issue – 2 | January-February 2020 Page 1017
= െ‫ܨ‬ሺ0ሻ + ‫݌‬ ‫׬‬ ݁ି௣௧∞
଴
= ‫ܮ݌‬ሼ‫ܨ‬ሺ‫ݐ‬ሻሽ − ‫ܨ‬ሺ0ሻ
= ‫݂݌‬ሺ‫݌‬ሻ − ‫ܨ‬ሺ0ሻ
Now, since ‫ ܮ‬൛‫ܨ‬′ሺ‫ݐ‬ሻൟ = ‫ܮ݌‬ሼ‫ܨ‬ሺ‫ݐ‬ሻሽ − ‫ܨ‬ሺ0ሻ
Therefore, ‫ ܮ‬൛‫ܨ‬′′ሺ‫ݐ‬ሻൟ = ‫ܮ݌‬൛‫ܨ‬′ሺ‫ݐ‬ሻൟ − ‫ܨ‬′ሺ0ሻ
‫ ܮ‬൛‫ܨ‬′′ሺ‫ݐ‬ሻൟ = ‫ ݌‬ሼ‫ܮ݌‬ሼ‫ܨ‬ሺ‫ݐ‬ሻሽ − ‫ܨ‬ሺ0ሻሽ − ‫ܨ‬′ሺ0ሻ
‫ ܮ‬൛‫ܨ‬′′ሺ‫ݐ‬ሻൟ = ‫݌‬ଶ
‫ܮ‬ሼ‫ܨ‬ሺ‫ݐ‬ሻሽ − ‫ܨ‬ሺ0ሻ − ‫ܨ‬′ሺ0ሻ
‫ ܮ‬൛‫ܨ‬′′ሺ‫ݐ‬ሻൟ = ‫݌‬ଶ
݂ሺ‫݌‬ሻ − ‫ܨ‬ሺ0ሻ − ‫ܨ‬′ሺ0ሻ
Similarly ‫ ܮ‬൛‫ܨ‬′′′ሺ‫ݐ‬ሻൟ = ‫݌‬ଷ
݂ሺ‫݌‬ሻ − ‫݌‬ଶ
‫ܨ‬ሺ0ሻ − ‫ܨ݌‬′ሺ0ሻ − ‫ܨ‬′
′ሺ0ሻ
And so on.
FORMULATION
Laguerre Polynomial.
The Laguerre polynomial [1-3] is defined as
‫ܮ‬௡ሺ‫ݑ‬ሻ =
௘ೠ
௡!
ௗ೙
ௗ௨೙ (݁ି௨
‫ݑ‬௡
)
We know that by the definition of Laplace Transform
L {F (t)} =‫׬‬ ݁ି௣௧∞
଴
Therefore,
‫ ܮ‬ሼ‫ܮ‬௡ሺ‫ݐ‬ሻሽ =‫׬‬ ݁ି௣௧∞
଴
ቄ
௘೟
௡!
ௗ೙
ௗ௧೙ ሺ݁ି௧
‫ݐ‬௡ሻቅ ݀‫ݐ‬
=
1
݊!
න ݁ିሺ௣ିଵሻ௧
∞
଴
൜
݀௡

݀‫ݐ‬௡
ሺ݁ି௧
‫ݐ‬௡ሻൠ ݀‫ݐ‬
=
ଵ
௡!
[ሺ‫݌‬ − 1ሻ ‫׬‬ ݁ିሺ௣ିଵሻ௧ ௗ೙షభ
ௗ௧೙షభ
∞
଴
ሺ݁ି௧
‫ݐ‬௡ሻ݀‫]ݐ‬
Integrating again,
=
ሺ‫݌‬ − 1ሻଶ
݊!
݀௡ିଶ
݀‫ݐ‬௡ିଶ
∞
଴
ሺ݁ି௧
‫ݐ‬௡ሻ݀‫ݐ‬
Integrating again,
=
ሺ‫݌‬ − 1ሻ௡
݊!
݀௡ି௡
݀‫ݐ‬௡ି௡
∞
଴
ሺ݁ି௧
‫ݐ‬௡ሻ݀‫ ݐ‬
=
ሺ‫݌‬ − 1ሻ௡
݊!
∞
଴
ሺ݁ି௧
‫ݐ‬௡ሻ݀‫ݐ‬
=
ሺ‫݌‬ − 1ሻ௡
݊!
න ݁ି௣௧
∞
଴
‫ݐ‬௡
݀‫ݐ‬
But by the definition of Laplace Transformation
L {F (t)} =‫׬‬ ݁ି௣௧∞
଴
Hence,
ሺ‫݌‬ − 1ሻ௡
݊!
‫ܮ‬ሺ‫ݐ‬௡
ሻ =
ሺ‫݌‬ − 1ሻ௡
݊!
.
݊!
‫݌‬௡ାଵ
Hence,
‫ ܮ‬ሼ‫ܮ‬௡ሺ‫ݐ‬ሻሽ =
ሺ‫݌‬ − 1ሻ௡
‫݌‬௡ାଵ
Solve the differential equations
ሺࡰ૛
− ૜ࡰ + ૛ሻ࢟ = ࡸ૛ሺ࢚ሻ
࢝࢏࢚ࢎ ࢏࢔࢏࢚࢏ࢇ࢒ ࢉ࢕࢔ࢊ࢏࢚࢏࢕࢔࢙
࢟ሺ૙ሻ = ૚ , ࢟′ሺ૙ሻ = ૙
Solution:
Given equation can be written as
‫ݕ‬′′
− 3‫ݕ‬′
+ 2‫ݕ‬ = ‫ܮ‬ଶሺ‫ݐ‬ሻ
Taking Laplace Transform on sides
‫ܮ‬ሼ‫ݕ‬′′
ሽ − 3‫ܮ‬ሼ‫ݕ‬′
ሽ + 2‫ܮ‬ሼ‫ݕ‬ሽ = ‫ܮ‬ሼ‫ܮ‬ଶሺ‫ݐ‬ሻሽ
Because Leguerre polynomial of order 2 is
‫ܮ‬ଶሼ‫ݐ‬ሽ =
1
2
ሼ2 − 4‫ݐ‬ + ‫ݐ‬ଶሽ
[‫݌‬ଶ
‫ݕ‬തሺ‫݌‬ሻ − ‫ݕ݌‬തሺ‫݌‬ሻ − ‫ݕ‬′
ሺ0ሻ] − 3[‫ݕ݌‬തሺ‫݌‬ሻ − ‫ݕ‬ሺ0ሻ] + 2‫ݕ‬തሺ‫݌‬ሻ
=
ሺ‫݌‬ − 1ሻଶ
‫݌‬ଷ
Applying initial conditions, we get
[‫݌‬ଶ
− 3‫݌‬ + 2]‫ݕ‬തሺ‫݌‬ሻ −
ሺ‫݌‬ − 1ሻଶ
‫݌‬ଷ
+ ‫݌‬ − 3
‫ݕ‬തሺ‫݌‬ሻ =
‫݌‬ − 1
‫݌‬ଷሺ‫݌‬ − 2ሻ
+
‫݌‬ − 3
ሺ‫݌‬ − 1ሻሺ‫݌‬ − 2ሻ
Applying inverse Laplace Transform
‫ݕ‬ = ‫ܮ‬ିଵ
൤
‫݌‬ − 1
‫݌‬ଷሺ‫݌‬ − 2ሻ
൨ + ‫ܮ‬ିଵ
൤
‫݌‬ − 3
ሺ‫݌‬ − 1ሻሺ‫݌‬ − 2ሻ
൨ … . ሺ1ሻ
‫ݕ‬ = ܷ + ܸ … … … … . . ሺ2ሻ
ܷ = ‫ܮ‬ିଵ
൤
‫݌‬ − 1
‫݌‬ଷሺ‫݌‬ − 2ሻ
൨
Solving by partial fraction, we get
ܷ = −
1
8
‫ܮ‬ିଵ
൤
1
‫݌‬
൨ −
1
4
‫ܮ‬ିଵ
൤
1
‫݌‬ଶ
൨ +
1
2
‫ܮ‬ିଵ
൤
1
‫݌‬ଷ
൨ +
1
8
‫ܮ‬ିଵ
൤
1
‫݌‬ − 2
൨
ܷ = −
1
8
−
1
4
‫ݐ‬ +
1
4
‫ݐ‬ଶ
+
1
8
݁ଶ௧
And,
ܸ = ‫ܮ‬ିଵ
൤
‫݌‬ − 3
ሺ‫݌‬ − 1ሻሺ‫݌‬ − 2ሻ
൨
Solving by Heaviside’s expansion
Let
‫ܨ‬ሺ‫݌‬ሻ = ‫݌‬ − 3
‫ܩ‬ሺ‫݌‬ሻ = ‫݌‬ଶ
− 3‫݌‬ + 2
Therefore, ‫ܩ‬′ሺ‫݌‬ሻ = 2‫݌‬ − 3
Putting ‫ܩ‬ሺ‫݌‬ሻ = 0, then ‫݌‬ = 1,2
Here, ‫ܩ‬ሺ‫݌‬ሻ have two distinct roots.
Also the degree of ‫ܨ‬ሺ‫݌‬ሻ is less than the degree of‫ܩ‬ሺ‫݌‬ሻ.
Therefore by Heaviside’s expansions
ܸ = ‫ܮ‬ିଵ
ቊ
‫ܨ‬ሺ‫݌‬ሻ
‫ܩ‬ሺ‫݌‬ሻ
ቋ
=
‫ܨ‬ሺ1ሻ
‫ܩ‬′ሺ1ሻ
݁௧
+
‫ܨ‬ሺ2ሻ
‫ܩ‬′ሺ2ሻ
݁ଶ௧
ܸ = 2݁௧
−݁ଶ௧
From (2),
‫ݕ‬ = ܷ + ܸ
࢟ = −
૚
ૡ
−
૚
૝
࢚ +
૚
૝
࢚૛
+
૚
ૡ
ࢋ૛࢚
+ ૛ࢋ࢚
−ࢋ૛࢚

Solve the differential equations
ሺࡰ૛
െ ࡰ െ ૛ሻ࢟ ൌ ࡸ૚ሺ࢚ሻ
࢝࢏࢚ࢎ ࢏࢔࢏࢚࢏ࢇ࢒ ࢉ࢕࢔ࢊ࢏࢚࢏࢕࢔࢙
࢟ሺ૙ሻ ൌ െ૚ , ࢟′ሺ૙ሻ = ૛
Solution:
Given equation can be written as
‫ݕ‬′′
− ‫ݕ‬′
− ‫ݕ‬ = ‫ܮ‬ଵሺ‫ݐ‬ሻ
Taking Laplace Transform on sides
‫ܮ‬ሼ‫ݕ‬′′
ሽ − ‫ܮ‬൛‫ݕ‬′
ൟ − 2‫ܮ‬ሼ‫ݕ‬ሽ = ‫ܮ‬ሼ‫ܮ‬ଵሺ‫ݐ‬ሻሽ
Because Leguerre polynomial of order 1 is
‫ܮ‬ଶሼ‫ݐ‬ሽ = ሼ1 − ‫ݐ‬ሽ
[‫݌‬ଶ
‫ݕ‬തሺ‫݌‬ሻ − ‫ݕ݌‬തሺ‫݌‬ሻ − ‫ݕ‬ᇱ
ሺ0ሻ] − [‫ݕ݌‬തሺ‫݌‬ሻ − ‫ݕ‬ሺ0ሻ] − 2‫ݕ‬തሺ‫݌‬ሻ
=
‫݌‬ − 1
‫݌‬ଶ
Applying initial conditions, we get
[‫݌‬ଶ
− ‫݌‬ − 2]‫ݕ‬തሺ‫݌‬ሻ −
‫݌‬ − 1
‫݌‬ଶ
− ‫݌‬ + 3
‫ݕ‬തሺ‫݌‬ሻ =
‫݌‬ − 1
‫݌‬ଶሺ‫݌‬ − 2ሻሺ‫݌‬ + 1ሻ
−
‫݌‬ − 3
ሺ‫݌‬ + 1ሻሺ‫݌‬ − 2ሻ
Applying inverse Laplace Transform
‫ݕ‬ = ‫ܮ‬ିଵ
൤
‫݌‬ − 1
‫݌‬ଶሺ‫݌‬ − 2ሻሺ‫݌‬ + 1ሻ
൨ − ‫ܮ‬ିଵ
൤
‫݌‬ − 3
ሺ‫݌‬ + 1ሻሺ‫݌‬ − 2ሻ
൨ … . ሺ1ሻ
‫ݕ‬ = ܷ + ܸ … … . . ሺ2ሻ
ܷ = ‫ܮ‬ିଵ
൤
‫݌‬ − 1
‫݌‬ଶሺ‫݌‬ − 2ሻሺ‫݌‬ + 1ሻ
൨
Solving by partial fraction, we get
ܷ = −
3
4
‫ܮ‬ିଵ
൤
1
‫݌‬
൨ +
1
2
‫ܮ‬ିଵ
൤
1
‫݌‬ଶ
൨
+
2
3
‫ܮ‬ିଵ
൤
1
‫݌‬ + 1
൨ +
1
12
‫ܮ‬ିଵ
൤
1
‫݌‬ − 2
൨
ܷ = −
3
4
+
1
2
‫ݐ‬ +
2
3
݁ି௧
+
1
12
݁ଶ௧
And,
ܸ = ‫ܮ‬ିଵ
൤
‫݌‬ − 3
ሺ‫݌‬ + 1ሻሺ‫݌‬ − 2ሻ
൨
Solving by Heaviside’s expansion
Let
‫ܨ‬ሺ‫݌‬ሻ = ‫݌‬ − 3
‫ܩ‬ሺ‫݌‬ሻ = ‫݌‬ଶ
− ‫݌‬ − 2
Therefore, ‫ܩ‬ᇱሺ‫݌‬ሻ = 2‫݌‬ − 1
Putting ‫ܩ‬ሺ‫݌‬ሻ = 0, then ‫݌‬ = −1,2
Here, ‫ܩ‬ሺ‫݌‬ሻ have two distinct roots.
Also the degree of ‫ܨ‬ሺ‫݌‬ሻ is less than the degree of‫ܩ‬ሺ‫݌‬ሻ.
Therefore by Heaviside’s expansions
ܸ = ‫ܮ‬ିଵ
ቊ
‫ܨ‬ሺ‫݌‬ሻ
‫ܩ‬ሺ‫݌‬ሻ
ቋ
=
‫ܨ‬ሺ−1ሻ
‫ܩ‬ᇱሺ−1ሻ
݁௧
+
‫ܨ‬ሺ2ሻ
‫ܩ‬ᇱሺ2ሻ
݁ଶ௧
ܸ = −
1
3
݁ଶ௧
−
4
3
݁ି௧
From (2),
‫ݕ‬ = ܷ + ܸ
࢟ = −
૜
૝
+
૚
૛
࢚ +
૛
૜
ࢋି࢚
+
૚
૚૛
ࢋ૛࢚
−
૚
૜
ࢋ૛࢚
−
૝
૜
ࢋି࢚
CONCLUSION
The solutions of differential equations including Leguerre
Polynomial via Laplace Transform Method are obtained
successfully. It is revealed that the Laplace transform is a
very useful mathematical for obtaining the solutions of
differential equations including Leguerre Polynomial.
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Journal for Innovative Research in Science &
Technology, Volume 4, Issue 11, April 2018.
[8] Dr. Dinesh Verma, Applications of Laplace
Transformation for solving Various Differential
Equations with Variable Coefficients, International
Journal for Innovative Research in Science &
Technology, Volume 4, Issue 11, April 2018.
[9] Dinesh Verma, Rohit Gupta, Amit Pal Singh, Analysis
of integral Equations of convolution type via Residue
Theorem Approach, The International Journal of
analytical and experimental modal analysis, Volume
XII, Issue I, January 2020.Researcher, 10(7), 2018.
[10] Dr. Dinesh Verma, A Laplace Transformation
approach to Simultaneous Linear Differential
Equations, New York Science Journal, 12 (7), 2019.
[11] Rohit Gupta, Amit Pal Singh, Dinesh Verma, Flow of
Heat Through A Plane Wall, And Through A Finite Fin
Insulated At The Tip, International Journal Of
Scientific & Technology Research, Volume 8, Issue
10, October 2019.
[12] Rohit Gupta, Tarun Singhal, Dinesh Verma, Quantum
Mechanical Reflection and Transmission Coefficients

for a Particle through a One-Dimensional Vertical
Step Potential, International Journal of Innovative
Technology and Exploring Engineering, Volume-8
Issue-11, September 2019.
[13] Murray R. Spiegel, Theory and Problems of Laplace
Transforms, Schaum's outline series, McGraw - Hill.
[14] Rahul Gupta and Rohit Gupta, Laplace Transform
method for obtaining the temperature distribution
and the heat flow along a uniform conducting rod
connected between two thermal reservoirs
maintained at different temperatures‖, Pramana
Research Journal, Volume 8, Issue 9, 2018, pp. 47-54.
[15] Dr. Dinesh Verma, Solving Fourier integral problem
by using Laplace transformation, International
journal of innovative research in technology, volume
4, issue 11, April 2018.
[16] Dinesh Verma, Applications of Inverse Laplace
Transformation, Compliance Engineering Journal,
Volume 10, Issue 12, 2019

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