Simplifying (▲^2 - x▲)(x^3) Using Finite Differences

In summary, finite difference is a numerical method for solving differential equations by dividing the domain into discrete intervals and approximating derivatives. It is used to solve complex equations and provides more accurate solutions. The steps involved in solving it include discretization, approximating derivatives, formulating and solving linear equations, and interpolating the solution. Some limitations include the need for fine discretization and difficulty handling complex boundary conditions. It differs from finite element method in terms of domain division and use of basis functions.
  • #1
irony of truth
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I am finding (▲^2 - x▲)(x^3).

I hope I am correct here...

(▲^2 - x▲)(x^3) = ▲^2(x^3) - x▲(x^3)
= ▲▲(x^3) - x( (x+1)^3 - x^3 )
= ▲( (x+1)^3 - x^3 ) - x( (x+1)^3 - x^3 )
= (x+2)^3 - (x + 1)^3 - (x+1)^3 + x^3 )
- x( (x+1)^3 - x^3 )

Is my solution correct?
 
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  • #2
No. Δ2x3 IS Δ(Δ x3)= Δ((x+1)3- x3)= Δ(3x2+ 3x+ 1)=
3(x+1)2+ 3(x+1)+ 1- (3x2+ 3x+ 1)= 6x+ 6.
 
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  • #3
The work looks right, but you had really ought to simplify the expression. (Oh, and you need to stick an extra parenthesis in front)
 

What is finite difference?

Finite difference is a numerical method used to approximate the solutions to differential equations by dividing the domain into discrete intervals and approximating the derivatives using difference equations.

Why is finite difference used?

Finite difference is used because it allows for the solution of complex differential equations that cannot be solved analytically. It also provides a more accurate solution compared to other numerical methods.

What are the steps involved in solving finite difference?

The steps include:

  1. Discretizing the domain
  2. Approximating the derivatives using difference equations
  3. Formulating a system of linear equations
  4. Solving the system of equations
  5. Interpolating the solution back to the original domain

What are the limitations of finite difference?

Finite difference may not provide an accurate solution if the discretization of the domain is not fine enough. It also cannot handle partial differential equations with complex boundary conditions.

How is finite difference different from finite element method?

Finite difference divides the domain into equally spaced intervals, while finite element method divides the domain into smaller subdomains. Finite element method also uses basis functions to approximate the solution, while finite difference uses difference equations.

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