Mathematics (Civil Engineering -1) | 16. Partial Differential Equations – Basic Concepts by Abraham | Learn Smarter
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16. Partial Differential Equations – Basic Concepts

Partial Differential Equations (PDEs) are essential for modeling various physical phenomena in engineering, particularly in Civil Engineering. This chapter provides an overview of PDE definitions, classifications, the formation of PDEs from relations, and standard forms of PDEs. It emphasizes the application of PDEs in stress analysis, fluid flow, and heat transfer, highlighting the significance of both analytical and numerical methods for solving real-world problems.

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Sections

  • 16

    Partial Differential Equations – Basic Concepts

    This section introduces Partial Differential Equations (PDEs), their definitions, classifications, methods of formation, and applications, particularly in civil engineering.

    Learning Practice

  • 16.1

    Definition And Notation

    Partial Differential Equations (PDEs) involve partial derivatives of a function with respect to multiple independent variables, forming a fundamental tool in mathematical modeling.

    Learning Practice

  • 16.2

    Order And Degree Of A Pde

    This section defines the order and degree of partial differential equations (PDEs), offering examples to clarify these concepts.

    Learning Practice

  • 16.3

    Formation Of Partial Differential Equations

    This section discusses the formation of partial differential equations (PDEs) by eliminating arbitrary constants and functions from given relations.

    Learning Practice

  • 16.3.A

    By Eliminating Arbitrary Constants

    This section discusses the formation of partial differential equations (PDEs) by eliminating arbitrary constants from functions that relate multiple variables.

    Learning Practice

  • 16.3.B

    By Eliminating Arbitrary Functions

    This section discusses the formation of partial differential equations (PDEs) by eliminating arbitrary functions from relationships involving multiple variables.

    Learning Practice

  • 16.4

    Classification Of Second-Order Pdes

    This section covers the classification of second-order partial differential equations (PDEs) based on their discriminant.

    Learning Practice

  • 16.5

    Linear And Nonlinear Pdes

    This section distinguishes between linear and nonlinear partial differential equations (PDEs), providing definitions and examples for each.

    Learning Practice

  • 16.6

    Standard Forms Of First-Order Pdes

    Standard forms of first-order partial differential equations (PDEs) provide essential tools for modeling various physical phenomena.

    Learning Practice

  • 16.7

    Solution Of First-Order Linear Pde – Lagrange’s Method

    This section explores Lagrange's method for solving first-order linear partial differential equations (PDEs), emphasizing the integration of auxiliary equations.

    Learning Practice

  • 16.8

    Types Of Solutions Of Pdes

    This section outlines the different types of solutions to partial differential equations (PDEs), specifically focusing on complete integrals, general solutions, and particular solutions.

    Learning Practice

  • 16.9

    Applications In Civil Engineering

    Partial Differential Equations (PDEs) are crucial in various applications in civil engineering, such as stress analysis, fluid flow, heat distribution, and vibrations.

    Learning Practice

  • 16.10

    Canonical (Standard) Forms Of Second-Order Pdes

    This section discusses how to convert second-order partial differential equations (PDEs) into canonical forms, facilitating easier analytical solutions.

    Learning Practice

  • 16.11

    Method Of Separation Of Variables

    The method of separation of variables is a technique for solving linear partial differential equations by assuming that the solution can be written as a product of functions, each dependent on a single variable.

    Learning Practice

  • 16.12

    Worked Example – Wave Equation

    This section presents a worked example of solving the one-dimensional wave equation with specified boundary and initial conditions.

    Learning Practice

  • 16.13

    Common Pdes In Civil Engineering Practice

    This section outlines the primary partial differential equations (PDEs) commonly applied in civil engineering, highlighting their mathematical forms and applications in real-world scenarios.

    Learning Practice

  • 16.14

    Numerical Methods (Overview)

    This section provides an overview of the numerical methods employed to solve partial differential equations (PDEs) in practical engineering applications.

    Learning Practice

References

Chapter_16_Parti.pdf

Class Notes

Memorization

What we have learnt

  • PDEs involve partial deriva...
  • Classification of PDEs into...
  • Numerical methods such as F...

Final Test

Revision Tests