Mathematics - iii (Differential Calculus) - Vol 2 | 12. One-Dimensional Heat Equation by Abraham | Learn Smarter
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12. One-Dimensional Heat Equation

The One-Dimensional Heat Equation is a critical model for understanding heat diffusion in materials. It highlights the importance of boundary and initial conditions in deriving solutions through methods such as separation of variables. The equation also finds applications in various fields, from engineering to financial mathematics, underscoring its broad relevance.

Sections

  • 12

    Partial Differential Equations

    This section introduces the One-Dimensional Heat Equation, a key application of PDEs in heat conduction.

    Learning Practice

  • 12.1

    Derivation Of The One-Dimensional Heat Equation

    This section covers the derivation of the One-Dimensional Heat Equation, highlighting its significance in heat conduction.

    Learning Practice

  • 12.2

    Boundary And Initial Conditions

    This section covers the critical concepts of boundary and initial conditions necessary for solving the One-Dimensional Heat Equation.

    Learning Practice

  • 12.3

    Solution Of The Heat Equation By Separation Of Variables

    This section explains the method of separation of variables for solving the One-Dimensional Heat Equation.

    Learning Practice

  • 12.4

    Fourier Series And Initial Condition

    This section introduces the methodology for determining Fourier coefficients from the initial temperature distribution in the one-dimensional heat equation.

    Learning Practice

  • 12.5

    Example Problem

    This section presents a specific example problem related to the One-Dimensional Heat Equation, demonstrating the process of finding the solution.

    Learning Practice

  • 12.6

    Physical Interpretation

    The heat equation models the diffusion of heat over time in a given medium, demonstrating the behavior of temperature changes in a one-dimensional rod.

    Learning Practice

  • 12.7

    Applications

    The primary applications of the One-Dimensional Heat Equation in various fields, including engineering and mathematical modeling, are discussed.

    Learning Practice

References

Unit_2_ch12.pdf

Class Notes

Memorization

What we have learnt

  • The One-Dimensional Heat Eq...
  • Separation of variables tra...
  • Boundary conditions are ess...

Final Test

Revision Tests