International Journal of Mathematical Archive-5(8), 2014, Available online through ISSN

Transcription

1 International Journal of Mathematical Archive-5(8), 014, Available online through ISSN SIMULATION OF LAPLACE TRANSFORMS WITH PYTHON Gajanan Dhanorkar* and Deepak Sonawane VPCOE, Vidyanagari MIDC, Baramati, Pune, (M.S.) , India. (Received On: ; Revised & Accepted On: ) ABSTRACT In this paper we have given applications of Laplace Transform to analyses signals in time domain to frequency domain using python, solving differential equations with initial conditions and computing the results in graphical format. This technique found useful and create the interest among the students at large. 1. INTRODUCTION Python is very fleible and open source nature, it is widely accepted language in scientific computing community from several years. Different software tools are available for computing mathematical equations such as MATLAB [1], Mathematica, Mapal, C & C++, etc. The major drawback of MATLAB is that it is not free and not open source which make difficult for researcher to share the coding. In the teaching learning paradigm of mathematics it becomes absolutely essential to display the graphical format of the equations and to understand its physical interpretation. In this contet we have attempted to use the power and fleibility of open source, general purpose programming language, Python[]. Python is used with its etension modules NumPy [3], SciPy [4] and Matplotlib [5], and now Sage [6], which adds more power to this scientific computing language to display the graphical results.. LAPLACE TRANSFORM Laplace transform is an integral transform and is a powerful technique to solve differential equations. Particularly useful in solving linear ordinary differential equations (ODE). It finds very wide applications in various areas of physics, electrical engineering, control engineering, optics, mathematics and signal processing. The Laplace transform can be interpreted as a transformation from the time domain where inputs and outputs are functions of time to the frequency domain where inputs and outputs are functions of comple angular frequency. The sufficient condition for any function of time f(t) to be Laplace transformable, it must satisfy the following Dirichlet conditions [7]: 1. f(t) must be piecewise continuous which means that it must be single valued but can have a finite number of finite isolated discontinuities for t > 0. ct. f(t) must be eponential order which means that f(t) must remain less than Me as t approaches to, where S is a positive constant and c is a real positive number. Definition.1: A Laplace Transform of f(t) is F(s) defined by and its inverse is We are using python for finding Lapalace Transform. 3. PRELIMINARIES st L{ f()} t = F() s = e f() t d International Journal of Mathematical Archive- 5(8), August L 1 { Fs ( )} = ft ( ) 3.1. Why Python? i) Very rich scientific computing libraries (a bit less than Matlab, though). ii) Well thought out language, allowing to write very readable and well structured code: we code what we think. iii) Many libraries for other tasks than scientific computing (web server management, serial port access, etc.) iv) Free and open-source software, widely spread, with a vibrant community. Corresponding Author: Gajanan Dhanorkar* 0

2 Gajanan Dhanorkar* and Deepak Sonawane/ Simulation of Laplace Transforms with Python / IJMA- 5(8), August Algebra: using SymPy First we have to import Sympy as "from sympy import *" and define symbol as " = Symbol ('')" To find partial fraction decomposition of 1 ( + + 1)( 1) >>>from sympy import * >>> = Symbol('') >>> apart(1/( (**+*+1)*(**-1) ), ) (perform partial fraction decomposition.) -1/(8*( + 1)) - 1/(4*( + 1)**) - 1/(*( + 1)**3) + 1/(8*( - 1)) 3... To combine fraction of ( + 1) 4( + 1) ( + 1) 8( 1) >>> together(-1/(8*( + 1)) - 1/(4*( + 1)**) - 1/(*( + 1)**3) + 1/(8*( - 1))) (To combine epressions) (-4* - ( - 1)*( + 1)** - *( - 1)*( + 1) + ( + 1)**3 + 4)/(8*( - 1)*( + 1)**3) Epand and separate Find epansion of 3 ( a b) >>> a, b = symbols('a b') >>> ((a-b)**3).epand() a**3-3*a***b + 3*a*b** - b** Find epansion of ( a+ b)( c+ d) >>> separate((a+b)*(c+d)).epand() ac + ad + bc + bd 3.3. Calculus: using SymPy To find limit of the function: sin lim 0 >>> limit(sin()/,, 0) To find limit of lim >>> limit((1/)**(1/),, oo) Derivative of Functions: To find derivative of cos >>> diff(cos(**3), ) -3****sin(**3) To find derivative of >>> diff(sin(**3), ) 3****cos(**3) 3 sin 1 is is 014, IJMA. All Rights Reserved 31

3 Gajanan Dhanorkar* and Deepak Sonawane/ Simulation of Laplace Transforms with Python / IJMA- 5(8), August Series representation of the function: To find series epansion of cos upto 14 >>> cos().series(, 0, 14) ( series representation up to 14.) 1 - **/ + **4/4 - **6/70 + **8/ **10/ **1/ O(**14) To find series epansion of sin upto 14 >>> sin().series(, 0, 14) ( series representation up to 14.) 1 - **/ + **4/4 - **6/70 + **8/ **10/ **1/ O(**14) 3.6. Integration: To find, we can write in python as ( + + 4) d >>> integrate(** + * + 4, ) **3/3 + ** + 4* To find, we can write in python as 5 e sin d >>> integrate(5*** * ep() * sin(), ) 5****ep()*sin()/ - 5****ep()*cos()/ + 5**ep()*cos() - 5*ep()*sin()/ -5*ep()*cos()/ To find π 0 cos e d, we can write in python as >>> integrate (cos()***ep(), (, 0, pi)) -3/5 + 3*ep (pi)/ To find 0 cos e d >>> integrate(cos()***ep(-), (, 0, oo)) 3/ Differential Equation: To solve Differential equation, we can write in python as d f ( ) + 9 f( ) = 0 d First we import, >>> from sympy import Function, dsolve, Eq, Derivative, sin, cos >>> from sympy.abc import >>> f = Function( ' f ' ) >>> dsolve(derivative(f(),,)+9*f(), f()) f() == C1*cos(3*) + C*sin(3*) Solve '' ' t y() t y() t + yt () = e >>> dsolve (Derivative(y(t),t,t)-*Derivative(y(t),t)+ y(t)-ep(*t), y(t)) y(t) = (c1 + ct)e**t + e**t Note: Also we can write above equation in following way diffeq = Eq(y(t).diff(t,t)-*y(t).diff(t)+ y(t), ep(*t)) 014, IJMA. All Rights Reserved 3

4 Gajanan Dhanorkar* and Deepak Sonawane/ Simulation of Laplace Transforms with Python / IJMA- 5(8), August LAPLACE TRANSFORM OF STANDARD FUNCTIONS 4.1. To find L.T. of f(t) = 1 >>> from sympy.integrals import laplace_ transform >>> from sympy.abc import t, s, a >>> laplace_ transform(1, t, s) Outpu: 1, 0, True s 4.. To find L.T. of f(t) = t >>> laplace_ transform(t, t, s) 1, 0, True s 4.3. To find L.T. of c >>> laplace _transform(t**a, t, s) (s**(-a)*gamma(a+1) / s, 0, -re (a) <1) 4.4. To find L.T. of sint >>> laplace _transform(sin(t), t, s) (1/s+1; 0; True) f( t) = t + tsint 4.5. To find L.T. of >>> laplace _transform(t** + t*sin(t), t, s) s s +, 0, True ( s + 1) s 3 5. INVERSE LAPLACE TRANSFORM 5.1. Inverse L.T. of Fs () = e as >>> from sympy.integrals.transforms import inverse_laplace_transform >>> from sympy import ep, Symbol >>> from sympy.abc import s, t >>> a = Symbol('a', positive=true) >>> inverse_laplace_transform(ep(-a*s)/s, s, t) Heaviside(-a + t) 5.. Inverse L.T. of 1 Fs () = s 1 >>> inverse_laplace_transform(1/(s-1), s, t) t e 5.3. Inverse L.T. of s Fs () = s + 1 >>>inverse_laplace_transform(s/(s**+1), s, t) cos(t) 014, IJMA. All Rights Reserved 33

5 Gajanan Dhanorkar* and Deepak Sonawane/ Simulation of Laplace Transforms with Python / IJMA- 5(8), August Inverse L.T. of 1 Fs () = ss ( + 1) >>> inverse_laplace_transform(1/(s*(s**+1)), s, t) (cos( t) 1) 6. APPLICATIONS TO DIFFERENTIAL EQUATIONS y() t y() t + yt () = e, yt () = y() t = 0 (1) Slove Differential equation '' ' t ' >>> y(s)=laplace_transform(derivative(y(t),t,t)-*derivative(y(t),t)+y(t)-ep(*t), t, s) d d 1 s Lt[ yt ()]( s) Lt yt () ( s) Lt yt () () s,, 1 dt + dt s >>> f = 1/((s-)*(s-1)**) >>> inverse_laplace_transform(f, s, t) ( e e te ) t t t 7. GRAPH Graph of solution of above differential equation (1) using python coding is from pylab import * t=arange(0,5,0.01) q=ep(*t)-ep(t)-t*ep(t) plot(t,q,'r') label('time') ylabel(' Charge in ') title ('DE Solution') show () REFERENCES [1] The Math Works MATLAB Release R010b. The Math-Works, Natick, Massachusetts. [] Guido van Rossum and Fred L. Drake Python Language Reference Manual. Network Theory Limited, Bristol, UK. [3] Travis Oliphant Guide To NumPy.Trelgol Publishing, USA. [4] Eric Jones, Travis Oliphant, Pearu Peterson, et al SciPy: Open source scientific tools for Python , IJMA. All Rights Reserved 34

6 Gajanan Dhanorkar* and Deepak Sonawane/ Simulation of Laplace Transforms with Python / IJMA- 5(8), August-014. [5] John D. Hunter Matplotlib: A D graphics environment. Computing In Science and Engineering, 9(3): [6] Sage for Power Users by William Stein. [7] A. D. Poularikas, The Transforms and Applications Hand- book (McGraw Hill, 000), nd ed. [8] Higher Engineering Mathematics, B.V.Ramana. Source of support: Nil, Conflict of interest: None Declared [Copy right 014. This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.] 014, IJMA. All Rights Reserved 35