Differential equations and integal transforms

In summary, different transforms can be used for different types of problems. Laplace transforms can be used for problems with initial conditions, but Hankel and Mellin transforms are more specialized.
  • #1
Yegor
147
1
Differential equations and integral transforms

Hi!

I have some general questions on using integral transforms for solving differential equations.

Also, I know that Fourier and Laplace transforms are useful means for solving linear ODE's and PDE's.
1. Are there cases, when one of them is more powerful as another one?
2. What about Hankel and Mellin transforms? Are there also very powerful?
3. I think, that Laplace transforms can be applied only for Cauchy problems (initial conditions must be given). Is it correct?

Thank you in advance
 
Last edited:
Physics news on Phys.org
  • #2
Should i ask such questions in Homework Forum?! :confused:
It's not a homework at all... just hoping that someone have more experience with such things...
 
  • #3
It is a pretty broad question. To answer this would be to place one's self out on a limb and speculate. But, here is an attempt.

The last two types you mentioned are sort of specialized. Hankel transforms are related to Fourier transforms in that they are FT of radially symmetric functions. Mellin are even more specialized and have some questions of convergence. (Some well-known functions can be viewed as a Mellin transform.)

Very roughly though (and I am sure there are cases where some may disagree) FT and LT are most common: The one you choose would depend on what sort of kernel (exponential) you want. The kernel for the FT has modulus 1 and the kernel for the LT has exponential decay. The exponential decay of the LT is nice since you can transform some functions that do not have a FT and you can do the so-called Heaviside calculus (procedure which puts the initial conditions into the problem and transforms the differential equation to an algebraic equation). Try looking all these up on Mathworld
http://mathworld.wolfram.com/

So, there is an answer in a nutshell, but the question is bigger than a watermelon.
 
  • #4
Thank you very much for interesting information!
 

Related to Differential equations and integal transforms

1. What is the difference between a differential equation and an integral transform?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. In contrast, an integral transform is a mathematical operation that converts a function into another representation, such as a series or an integral. Differential equations are used to model physical systems, while integral transforms are used to solve these equations.

2. How are differential equations and integral transforms used in science?

Differential equations and integral transforms are used in a wide range of scientific fields, including physics, chemistry, biology, and engineering. They are used to model and analyze complex systems and phenomena, such as the motion of objects, chemical reactions, and electrical circuits.

3. What are the most common types of differential equations?

The most common types of differential equations are ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve a single independent variable, while PDEs involve multiple independent variables. Other types of differential equations include linear and nonlinear equations, as well as systems of equations.

4. What are some applications of integral transforms?

Integral transforms have a wide range of applications in science and engineering. They are commonly used to solve differential equations, as well as to analyze and manipulate signals, images, and other types of data. Some examples of integral transforms include the Fourier transform, Laplace transform, and Z-transform.

5. Can differential equations and integral transforms be solved analytically?

While some differential equations and integral transforms can be solved analytically, many cannot. In these cases, numerical methods must be used to approximate the solutions. However, for simpler equations and transforms, analytical solutions can be found using techniques such as separation of variables, variation of parameters, and Laplace transforms.

Similar threads

A Applying the Laplace transform to solve Differential equations
    • Differential Equations
Replies
5
Views
2K
I Solving a differential equation using Laplace transform
    • Differential Equations
Replies
5
Views
2K
I Solution of delayed forcing function
    • Differential Equations
Replies
5
Views
1K
General solution vs particular solution
    • Differential Equations
2
Replies
52
Views
1K
A Generalization of hypergeometric type differential equation
    • Differential Equations
Replies
1
Views
2K
I Can all differential equations be turned into algebraic equations via the FT?
    • Differential Equations
Replies
1
Views
1K
A Integrability of Systems of Differential Equations
    • Differential Equations
Replies
5
Views
1K
I Solving Differential Equation Using Reduction of Order
    • Differential Equations
Replies
2
Views
1K
I On Laplace transform of derivative
    • Differential Equations
Replies
17
Views
948
A Partial differential equation containing the Inverse Laplacian Operator
    • Differential Equations
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top