calculus project

1. SOLUTION OF ORDINARY DIFFERENTIAL EQUATION:
A NUMERICAL APPROACH
Department of Pure and Applied Science
MIDNAPORE CITY COLLEGE
Kuturiya, P.O. Bhadutala,
Paschim Medinipur, Pin-721129
West Bengal, India
2022
Submitted by
Arijit Kundu
UNDER GUIDANCE OF
Dr. Sushil Kr. Ghosh

2. CONTENTS
CHAPTER 1: INTRODUCTON
CHAPTER 2: AIMS AND OBJECTIVE
CHAPTER 3: METHODS
CHAPTER 4: SHOOTING METHOD
CHAPTER 5: WORKING PROCEDURE
CHAPTER 6: RESULT
CHAPTER 7: CONCLUSION
CHAPTER 8: FUTURE SCOPE
REFERRENCES

3. INTRODUCTION
A boundary value problem is a system of ordinary differential equation
with solution and derivatives values specified at more than one point .
Most commonly the solution and derivatives are specified at just two
point boundary value problem.

4. Consider the two point boundary value problem
𝑢′′ = 𝑓 𝑥, 𝑢, 𝑢′ , 𝑥 ∈ 𝑎, 𝑏 ………………. (1)
where a prime denotes differentiation with respect to x, with one of the following three
boundary conditions.
Boundary conditions of the first kind:
𝑢 𝑎 = 𝛾1 , 𝑢 𝑏 = 𝛾2 . ………….…..….. (2)
Boundary conditions of second kind:
𝑢′ 𝑎 = 𝛾1 , 𝑢′ 𝑏 = 𝛾2 ……………...….…… (3)

5. Boundary conditions of third kind (or mixed kind):
𝑎0𝑢 𝑎 − 𝑎1𝑢′ 𝑎 = 𝛾1 ………………………. (1.4i)
𝑏0𝑢 𝑏 − 𝑏1𝑢′ 𝑏 = 𝛾2 ……………………. (1.4ii)
where 𝑎0, 𝑏0, 𝑎1, 𝑏1,𝛾1 and 𝛾2 are constants such that
𝑎0𝑎1 ≥ 0 , 𝑎0 + |𝑎1| ≠ 0
𝑏0𝑏1 ≥ 0 , 𝑏0 + 𝑏1 ≠ 0 and 𝑎0 + |𝑏0| ≠ 0

6. Aims and Objective
We are solving ordinary differential equation initial problems numerically so
that I can solve more complicated differential equations in the future with more
accuracy and less errors.

7. METHODS :
(i) Shooting methods
(ii) Finite Difference methods
(iii) Finite element methods

8. SHOOTING METHOD
Shooting method is a famous method for numerical solution of
second order differential equation when boundary condition is
known.
Shooting method convert the given boundary value problem into
initial value problem and solves the problem by using fourth
order Runge Kutta method.

9. WORKING PROCEDURE :
1. As the user executes the program, it asks for boundary values i.e. initial
value of x (x0), initial value of y (y0), final value of x (xn), final value of
y (yn) and the value of increment (h).
2. The second step of calculation is to convert this boundary value problem
into initial value problem.

10. 3. After the conversion into initial value problem, the user has to
input the initial guess value of z (M1) which is known as shooting.
4. Using this guess value of z, the program calculates intermediate
values of z & y . The final value of y obtained is assigned as B1.
5. Again, the user has to shoot i.e. the shooting method program
asks second initial guess value of z (M2).

11. 6. Using M2, new values of y and z are calculated. The final value of y obtained
in second guess is assigned as B2 in the program.
7. (M1, B1), (M2, B2), and ( y0, z0 ) are assumed to be collinear in this C
program and value of z0 determined using following equation,(B2-B1)/(M2-M1)
= (z0-B2)/(y0-M2)
8. Using this new and exact value of z, intermediate values are calculated using
Runge Kuttta Method.
9. Finally, the program prints the result.

12. RESULT
Solution of Ordinary Differential Equation by Shooting Method:
(i)
𝑑2𝑢
𝑑𝑥2 = 𝑢(𝑢 − 1) with conditions 𝑢 0 = 0, 𝑢 1 = 𝑒2 − 1
Graph line—S1
𝑒 = 2.71828
(ii)
𝑑2𝑢
𝑑𝑥2 = 2𝑢 − 𝑢′ with conditions 𝑢 1 = 2𝑒 + 𝑒−2, 𝑢 1 = 2𝑒2 −
𝑒−4 Graph line—S2

14. Solve IVP Of Second Order Differential Equation by Runge-
Kutta Method :
(i)
𝑑2𝑢
𝑑𝑥2 = 𝑢 − 𝑥2 with conditions 𝑢 0 = 0, 𝑢 0 = 0 Graph line—R1
(ii)
𝑑2𝑣
𝑑𝑥2 = 𝑣2 with conditions 𝑣 0 = 0, 𝑣 0 = 1.0 Graph line—R2
(iii)
𝑑2𝑢
𝑑𝑥2 = 𝑢 − 𝑥 with conditions 𝑢 0 = 0.0, 𝑢 0 = 0.0 Graph line—R3
(iv)
𝑑2𝑣
𝑑𝑥2 = 𝑣 with conditions 𝑣 0 = 0, 𝑣 0 = 1.0 Graph line—R4
(v)
𝑑2𝑢
𝑑𝑥2 = 𝑢 − 𝑥2 with conditions 𝑢 0 = 0.0, 𝑢 0 = 1.0 Graph line—R5
(vi)
𝑑2𝑣
𝑑𝑥2 = 𝑣2 with conditions 𝑣 0 = 1.0, 𝑣 0 = 0.0 Graph line—R6
(vii)
𝑑2𝑢
𝑑𝑥2 = 𝑢𝑒𝑥 − 𝑥2 with conditions 𝑢 0 = 0.0, 𝑢 0 = 1.0 Graph line—R7
(viii)
𝑑2𝑣
𝑑𝑥2 = 𝑣2 + 𝑒𝑥 with conditions 𝑣 0 = 0.0, 𝑣 0 = 1.0 Graph line—R8

16. I have solved boundary value problem of second order ordinary differential equations.
My aim is to solve a linear differential equation with two boundary conditions. However,
the method of solving a boundary value problem may not always gives a feasible solution
and that to overcome the anomalies we would like to solve initial value problem by
taking initial guess. The solution obtained these way we check through the given
boundary condition, if not achieved we may change our guess and try to satisfy boundary
condition. Since, with guess of initial condition we proceed to solve a boundary value
problem and again guess arbitary, it attributed to be a shooting and we call it a shooting
method. Another way, we may solve it as to use of Runge-Kutta method. Here, the
theoretical discussion succeeded to the experimental observation as a computer
programming and the graphical representation of the data. This project report implies
that the solution of differential equation either linear or non-linear can be solved by
numerical method for any elaborative solution.
CONCLUSION

17. FUTURE SCOPE
Here, it is to be noted that we have only solved boundary value problem of first kind;
though three kinds of possibilities may occur. A succeeding example/problem may be
generated so as to solve same problems with remaining two cases and compared. This
comparison also sought error analysis as well as stability analysis. Also, in some physical
problem we may encounter non-linear ordinary differential equation with boundary
conditions. Most of the physical phenomenon arises as known initial conditions and
deserves its future course of action. In that mathematical model the present study may be
recognized as the first step and the mathematical methods may upgraded as fourth/fifth
order Runge-Kutta method. A challenging task may be attributed to the solution of non-
linear ordinary simultaneous equations with the use of Runge-Kutta method and the
investigation of accuracy.

18. Atkinson, K. E., An Introduction to Numerical Analysis, John Wiley and Sons, 1978.
C. F. Gerald and P. O. Wheatly (2014) Applied Numerical Analysis, Addison- Wesley
M. K. Jain, S. R. K. Iyengar and R. K. Jain (2016), Numerical Methods for Scientific and Engineering Computation,
New- Age International Publication
M. Pal (2008) Numerical Analysis for Scientist and Engineers,
Narosa
C.H. Edwards and D.E. Penny, Differential Equations and Boundary Value problems Computing and Modeling,
Pearson Education India, 2005.
Boyce and Diprima, Elementary Differential Equations and Boundary Value Problems, Wiley.
Brian Bradie, A Friendly Introduction to Numerical Analysis,
Pearson Education, India, 2007.
E. V. Krishnamurty and S. K. Sen (1999), Numerical Algorithm,
East-West Press Pvt. Ltd. New Delhi.
REFERENCES

19. THANK YOU…