MA 204 Study Notes

MA 204 - Numerical Methods

Course Details

• Instructor: Niraj
• Date: 18 March 2026
• Course Code: MA204N

Recommended Books

1. SS Sastry, Introductory Methods for Numerical Analysis, PHI Learning, 2012
2. SRK Iyengar & RK Jain, Numerical Methods, New Age International, 2009

Numerical Solution of Ordinary Differential Equations (ODE)

Initial Value Problem (IVP)

• Consider the equation:
  $y' = f(x, y)$
• Example: Find the solution for $y' = x - y^2$ given that $y(0) = 1$.
• The Taylor series for $y(x)$ around $x = 0$ is given by:
  $y(x) = y_0 + (x - 0)y'(0) + \frac{(x - 0)^2}{2!}y''(0) + \frac{(x - 0)^3}{3!}y'''(0) + ext{…}$
• To compute $y(0.1)$ to four decimal places involves several derivatives calculated at the point of interest.

Taylor Series Method

Derivation of Taylor Series Expansion

• The Taylor series for a function $y(x)$ is:
  $Y(x) = Y_0 + (x - x_0)Y'(x_0) + \frac{(x - x_0)^2}{2!}Y''(x_0) + ext{…}$
• Higher derivatives can be calculated recursively using the property of derivatives, e.g., $y'' = f' = \frac{ ext{d}}{ ext{dx}}(f)$
    - Total derivative in multivariable functions:
  $\frac{ ext{d}y}{ ext{d}x} = f_x + f_y \frac{ ext{d}y}{ ext{d}x}$

Example 1: Finding $y(0.1)$ for $y' = x - y^2$

• Given:
    - $y'(0) = 0$
    - Substitute into the Taylor series:
    - Step-By-Step Calculation:
      - Calculate successive derivatives at the given point:
        1. $y'(x) = x - y^2$
        2. $y''(x) = 1 - 2yy'$
        3. $y'''(x) = -2y'y' - 2yy''$
• The Taylor series becomes:
  $Y(x) = 1 + 0.1 - \frac{1}{6}0.1^2 + …$
• This expands to a quadratic or cubic approximation which allows the calculation of $y(0.1)$.

Picard’s Method of Successive Approximations

• General formulation:
• Start with an initial guess $Y_0$; use the integral form of the function defined by the ODE to obtain the next approximation.
• The sequence for approximations can be expressed recursively:
     y^{(n)} = Y_0 + rac{1}{h} extstyle egin{pmatrix} ext{integral vertex}_1 ext{d}x… + ext{integral vertex}_2 ext{d}x ext{… trusted values}... ext{and further combinations} ext{…} ext{to the } n-1 ext{th term…} ext{until convergence}\
         ext{Repeating this will yield a successive approximation to the IVP.}
  

Euler’s Method

• Start with initial condition $Y(0) = Y_0$

1. For each step from $x_n$ to $x_{n+1}$, calculate:
   $Y_{n+1} = Y_n + h f(x_n, Y_n)$
2. Repeat this step for a predefined number of iterations or until a desired accuracy is achieved.

Example: Solve $y' = -4$ with $y(0) = 1$

• Using Euler's method, we'll apply to obtain successive points and how they converge to an approximation of the solution.
• Important steps include calculations for incremental $h$:
  $y(0.1) ext{, } y(0.2), y(0.3) ext{ etc… }$

Modified Euler’s Method

• Improved calculation using a midpoint approach:
• At each step calculate a midpoint (or use a trapezoidal rule):
  $Y_{n+1} = Y_n + \frac{h}{2} \big[f(x_n, Y_n) + f(x_n + h, Y_n + hf(x_n, Y_n))\big]$

Runge-Kutta Method

• A highly reliable method; generally implemented of second to fourth order conventional improvements to improve error:

Example: Fourth Order Runge-Kutta Method

1. Compute:
     - $k_1 = h f(x_n, Y_n)$
     - $k_2 = h f(x_n + \frac{h}{2}, Y_n + \frac{k_1}{2})$
     - $k_3 = h f(x_n + \frac{h}{2}, Y_n + \frac{k_2}{2})$
     - $k_4 = h f(x_n + h, Y_n + k_3)$
2. Calculate the next term:
   $Y_{n+1} = Y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)$

Predictor-Corrector Methods

• Have Predictor which estimates; subsequently refined via Corrector which refines the estimate for better accuracy.

Final Exercise Examples

• There are many theoretical and practical exercises to consider regenerative function values at multiple points as discussed above.
• Each exercise aims to reinforce and integrate the knowledge acquired in various numerical methods towards solving IVPs successfully.

Conclusion

• The essence of numerical methods is their iterative approximation leading to higher precision in solving ordinary differential equations through well-framed mathematical theories and methods.