Mathematics - iii (Differential Calculus) - Vol 2 | 19. Use of Laplace Transforms in Solving PDEs by Abraham | Learn Smarter
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19. Use of Laplace Transforms in Solving PDEs

Laplace Transforms provide a powerful method for solving linear partial differential equations (PDEs), particularly in scenarios involving time-dependent processes by transforming them into ordinary differential equations (ODEs). The method simplifies the resolution of complex PDEs, allowing for efficient retrieval of solutions through inverse transforms. This technique is instrumental across various applications in physics and engineering, including heat conduction, wave propagation, and fluid dynamics.

Sections

  • 19

    Partial Differential Equations

    This section explores the use of Laplace Transforms as a method to solve Partial Differential Equations (PDEs), specifically discussing their application in linear PDEs with initial and boundary conditions.

    Learning Practice

  • 19.1

    Use Of Laplace Transforms In Solving Pdes

    Laplace Transforms simplify the solution of time-dependent Partial Differential Equations (PDEs) by converting them into Ordinary Differential Equations (ODEs).

    Learning Practice

  • 19.2

    Laplace Transforms – A Brief Review

    Laplace Transforms convert partial differential equations (PDEs) into simpler ordinary differential equations (ODEs), especially for time-dependent problems, facilitating their solution.

    Learning Practice

  • 19.2.1

    Basic Idea – Why Use Laplace Transforms In Pdes?

    Laplace transforms simplify solving linear PDEs by converting time-dependent equations into ordinary differential equations (ODEs).

    Learning Practice

  • 192.2.2

    Standard Pde Solvable Via Laplace Transforms

    Laplace Transforms are powerful tools for solving standard linear Partial Differential Equations, such as the heat and wave equations, by transforming them into simpler Ordinary Differential Equations.

    Learning Practice

  • 19.2.3

    Solving A Pde Using Laplace Transform – Step-By-Step

    This section outlines a systematic approach to solve Partial Differential Equations (PDEs), specifically the heat equation, using the Laplace Transform.

    Learning Practice

  • 19.2.4

    Important Application Examples

    This section showcases significant examples of partial differential equations (PDEs) solved using Laplace Transforms.

    Learning Practice

  • 19.2.5

    Advantages Of Using Laplace Transforms

    Laplace Transforms simplify solving Partial Differential Equations (PDEs) by converting them into Ordinary Differential Equations (ODEs), making them particularly effective for time-dependent problems.

    Learning Practice

  • 19.2.6

    Limitations

    Laplace Transforms are effective for solving linear PDEs, but they have limitations including their applicability to only specific types of PDEs.

    Learning Practice

References

Unit_2_ch19.pdf

Class Notes

Memorization

What we have learnt

  • Laplace Transforms facilita...
  • The transformation of time ...
  • Inverse Laplace Transforms ...

Final Test

Revision Tests