Mathematics - iii (Differential Calculus) - Vol 2 | 2. Classification of PDEs (Elliptic, Parabolic, Hyperbolic) by Abraham | Learn Smarter
K12 Students

Academics

AI-Powered learning for Grades 8–12, aligned with major Indian and international curricula.

Academics

Grades

Grade 8 Grade 9 Grade 10 Grade 11 Grade 12

Curriculum

CBSE ICSE IB
Professionals

Professional Courses

Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.

Professional Courses
Games

Interactive Games

Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.

games
Typing
Memory
Math
English Adventures
Knowledge
2. Classification of PDEs (Elliptic, Parabolic, Hyperbolic)

Partial Differential Equations (PDEs) are crucial in modeling physical phenomena and are categorized into elliptic, parabolic, and hyperbolic types based on their coefficients and discriminant. The classification relies on the discriminant formula Δ = B² - 4AC, leading to different behaviors and solution methods. Understanding PDE types aids in determining appropriate numerical approaches and initial or boundary conditions necessary for solving complex problems.

Sections

  • 2

    General Form Of Second-Order Pdes

    This section outlines the general form of second-order partial differential equations (PDEs) and the process of classifying them.

    Learning Practice

  • 2.1

    Classification Based On Discriminant

    This section explains how to classify second-order partial differential equations (PDEs) into elliptic, parabolic, and hyperbolic types using the discriminant method.

    Learning Practice

  • 2.2

    Types Of Pdes

    This section classifies second-order partial differential equations into three categories: elliptic, parabolic, and hyperbolic, based on the discriminant of their coefficients.

    Learning Practice

  • 2.2.1

    Elliptic Pdes

    Elliptic PDEs are a class of partial differential equations that describe steady-state processes, characterized by a negative discriminant.

    Learning Practice

  • 2.2.2

    Parabolic Pdes

    Parabolic PDEs are defined by the condition Δ=0, mainly modeling diffusive processes like heat conduction.

    Learning Practice

  • 2.2.3

    Hyperbolic Pdes

    Hyperbolic PDEs are characterized by their discriminant being positive, reflecting phenomena such as wave propagation.

    Learning Practice

  • 2.3

    Characteristic Curves

    This section explores the classification of second-order partial differential equations (PDEs) into elliptic, parabolic, and hyperbolic, focusing on their characteristic curves.

    Learning Practice

  • 2.4

    Canonical Forms

    This section covers the classification of second-order partial differential equations (PDEs) into elliptic, parabolic, and hyperbolic types based on their discriminants and canonical forms.

    Learning Practice

  • 2.5

    Examples For Practice

    This section provides practice problems for classifying second-order partial differential equations based on their discriminants.

    Learning Practice

  • 2.6

    Summary

    This section provides an overview of the classification of partial differential equations (PDEs) into elliptic, parabolic, and hyperbolic types based on their discriminant.

    Learning Practice

References

Unit_2_ch2.pdf

Class Notes

Memorization

What we have learnt

  • The classification of secon...
  • Elliptic PDEs (Δ < 0) repre...
  • Parabolic PDEs (Δ = 0) refe...

Final Test

Revision Tests