Mathematics - iii (Differential Calculus) - Vol 1 | 18. Application to Integral Equations by Abraham | Learn Smarter
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18. Application to Integral Equations

Integral equations, particularly Volterra-type equations, can be effectively solved using Laplace Transforms, leveraging the Convolution Theorem. This technique transforms complex integral equations into simpler algebraic forms, facilitating the solution process. The methodology encompasses applying the Laplace Transform, solving algebraically in the s-domain, and then using the inverse transform to yield the final solution in the time domain, proving to be efficacious across various engineering applications.

Sections

  • 18

    Understanding Integral Equations

    Integral equations, particularly Volterra type, can be efficiently solved using Laplace Transforms.

    Learning Practice

  • 18.1

    Laplace Transform Approach

    The Laplace Transform method offers a powerful technique for solving Volterra integral equations, particularly benefiting from the Convolution Theorem.

    Learning Practice

  • 18.1.1

    Convolution Theorem

    The Convolution Theorem transforms integral equations into algebraic equations, simplifying their resolution using Laplace Transforms.

    Learning Practice

  • 18.2

    Step-By-Step Solution Using Laplace Transforms

    This section outlines the step-by-step application of Laplace Transforms to solve Volterra Integral Equations of the second kind.

    Learning Practice

  • 18.2.1

    Step 1: Apply Laplace Transform To Both Sides

    This section discusses how to apply the Laplace Transform to both sides of a Volterra integral equation to simplify and solve it.

    Learning Practice

  • 18.2.2

    Step 2: Solve Algebraically For 𝐹(𝑠)

    This section describes the algebraic process of solving for 𝐹(𝑠) in the context of Volterra Integral Equations using Laplace Transforms.

    Learning Practice

  • 18.2.3

    Step 3: Apply The Inverse Laplace Transform To Find 𝑓(𝑡)

    This section focuses on applying the inverse Laplace Transform to determine the function 𝑓(𝑡) after solving linear integral equations.

    Learning Practice

  • 18.3

    Example Problems

    This section demonstrates the application of Laplace Transforms to solve Volterra-type integral equations through worked examples.

    Learning Practice

  • 18.4

    Applications In Engineering

    This section discusses how Laplace Transforms are utilized to solve linear integral equations, particularly Volterra-type equations found in engineering applications.

    Learning Practice

  • 18.5

    Summary

    Laplace Transforms simplify solving Volterra-type integral equations, converting complex integrals into straightforward algebraic forms.

    Learning Practice

References

Unit 1 ch18.pdf

Class Notes

Memorization

What we have learnt

  • Integral equations can be s...
  • The Convolution Theorem is ...
  • The solution process involv...

Final Test

Revision Tests