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
1. Introduction 2. Types of Equations 3. Errors in Computing 4. Bisection Method 5. Method of False Position (Regula Falsi) 6. Newton-Raphson Method 7. Fixed Point Iteration 8. Solution of System of Nonlinear Equations Key Takeaways – Unit 1
1. Introduction to Transaction A transaction is a logical unit of work that performs one or more database operations (like INSERT, UPDATE, DELETE) and ensures data integrity. Example:When transferring money between two accounts: Both operations must succeed together — otherwise, neither should happen. 2. Properties of a Transaction (ACID) Transactions must satisfy ACID properties to…
1. Query Processing Overview Steps in Query Processing: 2. Measuring Query Cost Purpose: Helps the optimizer choose the best query plan. 3. Selection Operation SQL Example: SELECT * FROM Students WHERE Age > 18; 4. Sorting Example: SELECT * FROM Students ORDER BY Name ASC; 5. Joining Tables 6. Query Optimization 7. Database Security Overview…
1. Introduction As databases grow larger and more complex, performing the same SQL operations repeatedly or ensuring data integrity manually becomes inefficient.To handle this, SQL provides Stored Procedures, Triggers, and Indexes — advanced features to improve automation, consistency, and performance. A. Stored Procedures 1. Definition A Stored Procedure is a precompiled set of SQL statements…
1. Introduction Once database and tables are created, we need to manipulate (insert/update/delete) and query (retrieve) data stored in them.For this, we use DML (Data Manipulation Language) and DQL (Data Query Language) commands. 2. Major SQL DML Commands Command Purpose Example INSERT Adds new rows of data INSERT INTO Students VALUES (…); SELECT Retrieves data…