Mathematics - iii (Differential Calculus) - Vol 2 | 3. Linear and Non-linear PDEs by Abraham | Learn Smarter
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3. Linear and Non-linear PDEs

Partial Differential Equations (PDEs) are pivotal in modeling various physical phenomena involving multiple variables and partial derivatives. The chapter distinguishes between linear and non-linear PDEs, classifies them into parabolic, hyperbolic, and elliptic types, and discusses their characteristics and implications for solving real-world problems. Understanding these classifications is essential for applying appropriate mathematical methods in various scientific fields.

Sections

  • 3

    Partial Differential Equations (Pdes)

    This section introduces Partial Differential Equations (PDEs), covering their definitions, classifications, and types, including linear and non-linear equations.

    Learning Practice

  • 3.1

    Linear And Non-Linear Partial Differential Equations

    This section introduces Linear and Non-linear Partial Differential Equations (PDEs), detailing their definitions, characteristics, and significance in modeling physical phenomena.

    Learning Practice

  • 3.1.1

    Definition Of Pde

    A Partial Differential Equation (PDE) is an equation that includes partial derivatives of a function of several variables.

    Learning Practice

  • 3.1.2

    Linear Pdes

    Linear PDEs are equations where the dependent variable and its derivatives appear to the first power, playing a crucial role in mathematical modeling.

    Learning Practice

  • 3.1.3

    Non-Linear Pdes

    This section introduces non-linear partial differential equations (PDEs), highlighting their characteristics and difficulties in solving them.

    Learning Practice

  • 3.2

    Classification Of Second-Order Linear Pdes

    This section explains how to classify second-order linear PDEs into parabolic, hyperbolic, and elliptic types based on the discriminant of the second-order terms.

    Learning Practice

  • 3.3

    Types Of Pdes: Parabolic, Hyperbolic, And Elliptic

    This section categorizes second-order partial differential equations into parabolic, hyperbolic, and elliptic types, relating to their physical applications and solution behaviors.

    Learning Practice

  • 3.3.1

    Parabolic Pdes

    Parabolic PDEs characterize diffusion-like processes, with the heat equation exemplifying their behavior.

    Learning Practice

  • 3.3.2

    Hyperbolic Pdes

    Hyperbolic PDEs describe wave-like phenomena and are characterized by a positive discriminant in their general form.

    Learning Practice

  • 3.3.3

    Elliptic Pdes

    Elliptic PDEs are characterized by their discriminant being less than zero, modeling steady-state systems such as potential fields.

    Learning Practice

  • 3.4

    Summary Table

    The summary table categorizes second-order partial differential equations by their discriminant and provides typical equations along with their physical interpretations.

    Learning Practice

References

Unit_2_ch3.pdf

Class Notes

Memorization

What we have learnt

  • Partial Differential Equati...
  • PDEs can be classified into...
  • The classification into par...

Final Test

Revision Tests