Partial Differential Equation |KTU Engineering Maths | MAT201 | S3| Module 1|BTech Mathematics Part1

Introduction

This tutorial covers the fundamentals of Partial Differential Equations (PDEs) as presented in the KTU Engineering Maths video by RVS Maths Academy. It will guide you through the formation of partial differential equations by eliminating arbitrary functions, providing a solid foundation for further studies in engineering mathematics.

Step 1: Understanding Partial Differential Equations

• Definition: A Partial Differential Equation is an equation that involves partial derivatives of a function with respect to multiple variables.
• Importance: PDEs are essential in various fields such as physics, engineering, and finance, as they describe a wide range of phenomena including heat conduction, wave propagation, and fluid dynamics.
• Examples of PDEs:
  • Heat equation: ( u_t = k u_{xx} )
  • Wave equation: ( u_{tt} = c^2 u_{xx} )

Step 2: Formulating a Partial Differential Equation

• Identify the Function: Start with a function ( z = f(x, y) ) that depends on two variables ( x ) and ( y ).
• Determine Derivatives: Calculate the necessary partial derivatives:
  • First-order derivatives: ( f_x ) and ( f_y )
  • Second-order derivatives: ( f_{xx}, f_{yy}, ) and ( f_{xy} )
• Eliminate Arbitrary Functions: Use methods such as:
  • Substituting values or expressions into the function.
  • Applying conditions that relate the derivatives to eliminate arbitrary constants.

Step 3: Example of Formation

• Consider a Function: Let’s take ( z = x^2 + y^2 + C ), where ( C ) is an arbitrary constant.
• Take Partial Derivatives:
  • ( \frac{\partial z}{\partial x} = 2x )
  • ( \frac{\partial z}{\partial y} = 2y )
• Formulate the PDE: By eliminating the arbitrary constant ( C ), you can establish a relationship between the derivatives:
  • ( \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = 0 )

Step 4: Common Pitfalls

• Neglecting Boundary Conditions: Always take boundary conditions into account when formulating PDEs.
• Miscalculating Derivatives: Ensure that partial derivatives are computed correctly, keeping variables constant as needed.

Step 5: Real-World Applications

• Engineering: PDEs are used to model systems such as heat exchangers and structural analysis.
• Physics: They describe physical phenomena like electromagnetic fields and fluid dynamics.
• Finance: Options pricing models, such as the Black-Scholes equation, are derived from PDEs.

Conclusion

This guide provides a foundational understanding of Partial Differential Equations and their formulation by eliminating arbitrary functions. By following these steps, you will be well-equipped to tackle problems in engineering mathematics. For further learning, consider exploring specific applications of PDEs in your field of interest or solving more complex examples.