Admission Open

Differential Equations Course in Mianwali

Module 1: Introduction to Differential Equations

1.1. Definition and Classification

Introduction to differential equations (ODEs and PDEs)
Order, degree, and linearity of differential equations
Classification based on variables and functions
1.2. Solutions and Initial Value Problems (IVPs)

General solutions vs. particular solutions
Initial value problems and boundary value problems
Existence and uniqueness of solutions
1.3. Applications of Differential Equations

Modeling physical phenomena (growth and decay, motion, circuits)
Engineering applications (heat transfer, fluid dynamics)
Economic and biological models
Module 2: First-Order Differential Equations

2.1. Separable Differential Equations

Definition and solution method for separable ODEs
Applications in population growth and radioactive decay
2.2. Linear First-Order Differential Equations

Integrating factor method for solving linear ODEs
Applications in exponential growth and decay
2.3. Exact Differential Equations

Conditions for exactness and integrating factors
Solving exact ODEs and applications in physics and engineering
Module 3: Second-Order Differential Equations

3.1. Homogeneous Second-Order ODEs

Definitions and solution methods for homogeneous linear ODEs
Characteristic equation, roots, and general solutions
3.2. Non-Homogeneous Second-Order ODEs

Methods of undetermined coefficients and variation of parameters
Applications in forced oscillations and resonance
3.3. Applications of Second-Order ODEs

Mechanical vibrations and harmonic motion
Electrical circuits and LRC circuits
Damped and forced oscillations
Module 4: Systems of Differential Equations

4.1. Introduction to Systems

Definition and classification of systems of ODEs
Matrix notation and vector form of systems
4.2. Solution Techniques

Eigenvalues, eigenvectors, and diagonalization
Phase plane analysis and stability of equilibrium points
Applications in population dynamics and chemical reactions
Module 5: Laplace Transforms

5.1. Definition and Properties

Laplace transform of functions and derivatives
Linearity, shifting, scaling, and differentiation properties
5.2. Inverse Laplace Transforms

Finding inverse transforms using partial fractions and convolution
Application of Laplace transforms in solving ODEs and systems
Module 6: Fourier Series and Partial Differential Equations

6.1. Fourier Series

Representation of periodic functions using Fourier series
Even and odd functions, half-range expansions
Applications in heat conduction and vibration analysis
6.2. Introduction to Partial Differential Equations (PDEs)

Classification of PDEs (elliptic, parabolic, hyperbolic)
Method of separation of variables for solving PDEs
Applications in physics, engineering, and mathematics
Module 7: Numerical Methods for Differential Equations

7.1. Euler’s Method and Runge-Kutta Methods

Introduction to numerical methods for ODEs
Euler’s method, improved Euler’s method, and Runge-Kutta methods
7.2. Finite Difference Methods for PDEs

Finite difference approximations for partial derivatives
Explicit and implicit methods for solving PDEs
Applications in computational fluid dynamics and heat transfer
Module 8: Boundary Value Problems and Green’s Functions

8.1. Boundary Value Problems (BVPs)

Definition and examples of BVPs
Shooting method and finite difference methods for BVPs
8.2. Green’s Functions

Definition and properties of Green’s functions
Applications in solving non-homogeneous differential equations and integral equations
Module 9: Applications and Advanced Topics

9.1. Applications in Engineering and Physics

Vibrations of mechanical systems
Heat conduction and diffusion equations
Electrical circuits and RLC circuits
9.2. Advanced Topics

Stability analysis of nonlinear systems
Chaos theory and dynamical systems
Introduction to stochastic differential equations

Admission Open for this course 
Contact Number: 03307615544

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