Mathematics - iii (Differential Calculus) - Vol 2 | 15. Fourier Series Solutions to PDEs by Abraham | Learn Smarter
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15. Fourier Series Solutions to PDEs

The Fourier Series technique is essential in solving Partial Differential Equations (PDEs), particularly for scenarios involving heat flow, vibrations, and potential theory. This method transforms PDEs into solvable ordinary differential equations (ODEs) using an infinite series of sine and cosine functions. By leveraging orthogonality and convergence under Dirichlet conditions, the Fourier Series allows for practical applications in various engineering and physics problems.

Sections

  • 15

    Partial Differential Equations

    Partial Differential Equations (PDEs) are solved effectively using Fourier series, particularly for boundary value problems.

    Learning Practice

  • 15.1

    Basics Of Fourier Series

    This section introduces the Fourier series, a powerful tool used to represent periodic functions through sums of sine and cosine functions.

    Learning Practice

  • 15.2

    Fourier Series In Solving Pdes

    This section covers the application of Fourier series in solving partial differential equations (PDEs), specifically focusing on the Heat Equation, Wave Equation, and Laplace's Equation.

    Learning Practice

  • 15.2.A

    Heat Equation

    This section focuses on the application of Fourier series in solving the heat equation, a foundational concept in partial differential equations.

    Learning Practice

  • 15.2.B

    Wave Equation

    The Wave Equation describes the relationship between wave propagation, and Fourier series provide a robust technique to solve this equation under various boundary and initial conditions.

    Learning Practice

  • 15.2.C

    Laplace Equation (Steady State Heat)

    The Laplace equation is used in steady-state heat conduction problems, where it simplifies complex heat distribution analysis.

    Learning Practice

  • 15.3

    Half-Range Expansions

    This section introduces half-range expansions, which adapt the Fourier series method for functions defined on limited intervals, facilitating solutions with non-periodic boundary data.

    Learning Practice

  • 15.4

    Key Observations

    This section outlines the importance and function of Fourier series in solving Partial Differential Equations (PDEs).

    Learning Practice

References

Unit_2_ch15.pdf

Class Notes

Memorization

What we have learnt

  • The Fourier series can expr...
  • This method allows the tran...
  • Critical conditions must be...

Final Test

Revision Tests