Hydraulic Engineering - Vol 3 | 13. Domain of Dependence and Range of Influence by Abraham | Learn Smarter
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13. Domain of Dependence and Range of Influence

The chapter delves into the various domains of influence and dependence pertaining to elliptic, parabolic, and hyperbolic partial differential equations. It further categorizes physical problems into equilibrium, propagation, and Eigen problems, highlighting the significance of boundary conditions in solving these equations. A pivotal technique discussed is the finite difference method, which approximates differential equations via truncated Taylor series, providing insights into analytical versus numerical solutions.

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Sections

  • 1

    Domain Of Dependence And Range Of Influence

    This section discusses the concepts of domain of dependence and range of influence in the context of partial differential equations, particularly focusing on elliptical, parabolic, and hyperbolic types.

    Learning Practice

  • 1.1

    Elliptic Partial Differential Equation

    This section introduces elliptical partial differential equations, focusing on the concepts of domain of dependence and range of influence.

    Learning Practice

  • 1.2

    Parabolic Partial Differential Equation

    This section introduces parabolic partial differential equations (PDEs), distinguishing their domains of dependence and influence, and classifying physical problems represented by these equations.

    Learning Practice

  • 1.3

    Hyperbolic Partial Differential Equation

    This section discusses the concept of hyperbolic partial differential equations (PDEs) and their classification along with the domain of dependence and range of influence.

    Learning Practice

  • 2

    Classification Of Physical Problems

    This section discusses the classification of physical problems into equilibrium, propagation, and eigen problems, highlighting their connection with partial differential equations.

    Learning Practice

  • 2.1

    Equilibrium Problems

    This section explores equilibrium problems, mainly focusing on partial differential equations (PDEs) and classifies physical problems into different categories.

    Learning Practice

  • 2.2

    Propagation Problems

    This section discusses the concepts of the domain of dependence and range of influence in the context of different types of partial differential equations (PDEs), specifically focusing on propagation problems.

    Learning Practice

  • 2.3

    Eigen Problems

    This section covers Eigen problems, highlighting their distinct characteristics in relation to other types of partial differential equations, particularly how solutions depend on certain parameter values known as eigenvalues.

    Learning Practice

  • 3

    Discretization Technique

    This section introduces discretization techniques in the context of partial differential equations, emphasizing the concepts of domain of dependence and range of influence.

    Learning Practice

  • 3.1

    Finite Difference Method

    This section introduces the Finite Difference Method, discussing its applications in solving partial differential equations (PDEs), specifically through the concepts of domain of influence and dependence.

    Learning Practice

  • 3.2

    Taylor Series Formulation

    This section covers the application of Taylor Series in the approximation of derivatives within partial differential equations (PDEs).

    Learning Practice

  • 3.3

    Analytical Vs Numerical Solution

    This section discusses the differences between analytical and numerical solutions in the context of partial differential equations (PDEs), emphasizing their applications across various types of PDEs including elliptic, parabolic, and hyperbolic equations.

    Learning Practice

References

55b.pdf

Class Notes

Memorization

What we have learnt

  • Elliptic partial differenti...
  • Physical problems can be cl...
  • The finite difference metho...

Final Test

Revision Tests