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Integral Transform: Overview & Definition

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What is the Integral Transform?

The integral transform is a way to solve challenging problems by transforming coordinates into another space, via integration, where the problem is easier to solve. The solution is then transformed back into the original coordinates with an inverse transform.
integral transform flow 2

Integral transforms are applied to Initial Boundary Value Problems (IBVP) as follows [1]:

  1. Apply a transform to one independent variable of the partial differential equation. This eliminates any partial derivatives associated with that variable.
  2. Solve the transformed PDE (if this isn’t possible, the equation can be transformed again).
  3. Invert the transforms to convert the solution to the solution of the original IBVP.

Types of Integral Transform

There are an infinite number of transforms. The most commonly used transforms in calculus are the Laplace transform and the Fourier transform. Fourier transforms are widely used in engineering and physics, while the Laplace transform is an efficient way of solving some ordinary and partial differential equations. However, there is really only one difference between the two methods: Laplace transforms can be defined for unstable systems with the Fourier transform cannot [2]. They are named after the mathematicians who originally worked on them: Laplace and Fourier.


Other types of commonly used transformations include:

  • Fourier-cosine transform,
  • Fourier-sine transform,
  • Hankel transform,
  • Mellin transform.

Which one you use is usually determined by the boundary condition. For example, the ideal transformation a problem defined on a semi-infinite domain (0, ∞) in space with a Dirichlet boundary condition is the Fourier sine transform.


Formal Definition of Integral Transform

More formally, the integral transform with integral kernel K is a mapping that takes function f(t) to a function f(x) with the rule [3]:
integral transform

Where a and b are real numbers (including infinity and negative infinity).

References

[1] Lambers, J. (2013). Integral Transforms (Sine and Cosine Transforms). Retrieved May 10, 2021 from: https://www.math.usm.edu/lambers/mat417/lecture8.pdf
[2] M. Shehu & Weidong, Z. New Integral Transform: Shehu Transform a Generalization of Sumudu and Laplace Transform for Solving Differential Equations. Retrieved May 10, 2021 from: http://arxiv-export-lb.library.cornell.edu/pdf/1904.11370
[3] Green, L. (2020). The Laplace Transform. Retrieved May 9, 2021 from: https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/6%3A_Power_Series_and_Laplace_Transforms/6.6%3A_The_Laplace_Transform

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Stephanie Glen. "Integral Transform: Overview & Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/integral-transform-overview-definition/
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