1. Introduction to PDEs A Partial Differential Equation (PDE) involves partial derivatives of a function of two or more independent variables. General form: [F(x, y, z, p, q, r, s, t) = 0]where( z = f(x, y) ),( p = \frac{\partial z}{\partial x} ),( q = \frac{\partial z}{\partial y} ),( r = \frac{\partial^2 z}{\partial x^2}…

1. Introduction An Ordinary Differential Equation (ODE) is an equation involving one dependent variable and its derivatives with respect to one independent variable. Example:[\frac{dy}{dx} = f(x, y)] Goal → To find approximate (y) for (x = x_1, x_2, …) 2. Methods for Solving ODEs There are two types: 3. Euler’s Method The simplest numerical method….

1. Introduction 2. Existence and Types of Solutions A system ( A\mathbf{x} = \mathbf{b} ) may have: 3. Consistency of Linear System A system is consistent if it has at least one solution, i.e.,[\text{Rank}(A) = \text{Rank}([A|B])]If (\text{Rank}(A) < \text{Rank}([A|B])), then the system is inconsistent. 4. Direct Methods These methods give exact (or nearly exact) solutions…

1. Introduction 2. Numerical Differentiation 3. Newton’s Differentiation Formulas 4. Numerical Integration 5. Romberg Integration 6. Numerical Double Integration Key Takeaways – Unit 3

1. Introduction 2. Errors in Polynomial Interpolation 3. Lagrange’s Interpolation 4. Newton’s Interpolation 5. Cubic Spline Interpolation 6. Least Squares Method Key Takeaways – Unit 2