Numerical Techniques | 5. Numerical Solutions of Partial Differential Equations by Pavan | Learn Smarter
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5. Numerical Solutions of Partial Differential Equations

Numerical methods such as Finite Difference Methods (FDM) and Finite Element Methods (FEM) are essential for solving Partial Differential Equations (PDEs) that model various physical phenomena. FDM is straightforward and suitable for uniform grids, whereas FEM offers flexibility for complex geometries and varying material properties. Each method has distinctive advantages and limitations regarding implementation, accuracy, and computational cost.

Sections

  • 5

    Numerical Solutions Of Partial Differential Equations

    This section discusses numerical methods for solving Partial Differential Equations (PDEs), focusing on Finite Difference Methods (FDM) and Finite Element Methods (FEM).

    Learning Practice

  • 5.1

    Introduction To Partial Differential Equations (Pdes)

    Partial Differential Equations (PDEs) are equations involving functions of multiple variables and their partial derivatives, utilized for modeling physical phenomena.

    Learning Practice

  • 5.2

    Finite Difference Methods (Fdm)

    Finite Difference Methods (FDM) are numerical techniques used to approximate solutions to partial differential equations by discretizing the problem into a finite grid.

    Learning Practice

  • 5.2.1

    Basic Concept Of Finite Difference Methods

    Finite Difference Methods (FDM) involve discretizing a domain and approximating derivatives, transforming partial differential equations (PDEs) into algebraic equations.

    Learning Practice

  • 5.2.2

    Advantages And Disadvantages Of Fdm

    This section presents the advantages and disadvantages of Finite Difference Methods (FDM) used in solving Partial Differential Equations (PDEs).

    Learning Practice

  • 5.3

    Finite Element Methods (Fem)

    Finite Element Methods (FEM) provide a flexible and advanced technique for solving partial differential equations (PDEs), particularly dealing with complex geometries and material properties.

    Learning Practice

  • 5.3.1

    Basic Concept Of Finite Element Methods

    Finite Element Methods (FEM) provide a sophisticated approach to numerically solving Partial Differential Equations (PDEs) by discretizing complex domains into simpler elements.

    Learning Practice

  • 5.3.2

    Example: Solving A 1d Poisson Equation Using Fem

    This section presents a detailed example of employing the Finite Element Method (FEM) to solve the 1D Poisson equation.

    Learning Practice

  • 5.3.3

    Advantages And Disadvantages Of Fem

    The section discusses the advantages and disadvantages of the Finite Element Method (FEM) in solving partial differential equations (PDEs).

    Learning Practice

  • 5.4

    Comparison Of Finite Difference Methods And Finite Element Methods

    This section compares Finite Difference Methods (FDM) and Finite Element Methods (FEM) highlighting their flexibility, accuracy, implementation complexity, and computational costs.

    Learning Practice

  • 5.5

    Multiphysics And Coupled Problems

    This section discusses the significance of the Finite Element Method (FEM) in solving multiphysics and coupled problems that involve interactions between multiple physical phenomena.

    Learning Practice

  • 5.6

    Summary Of Key Concepts

    This section summarizes the key concepts of Finite Difference Methods (FDM) and Finite Element Methods (FEM) as numerical solutions for Partial Differential Equations (PDEs).

    Learning Practice

References

ee4-nt-5.pdf

Class Notes

Memorization

What we have learnt

  • Partial Differential Equati...
  • Finite Difference Methods (...
  • Finite Element Methods (FEM...

Final Test

Revision Tests