vagnhestagn

2022-09-07

I know the difference between the forward (explicit) Euler method and backward (implicit) Euler method in the context of approaching solutions to ODEs.

However, I have a doubt regarding the following method:

${y}_{n+1}={y}_{n}+\frac{h}{2}(f({t}_{n},{y}_{n})+f({t}_{n+1},{y}_{n+1}))$

I am not sure whether it is an implicit or an explicit method.

I tend to say it is implicit one step methode because we have to solve an equation to calculate ${y}_{n+1}.$

Thank you for your help.

However, I have a doubt regarding the following method:

${y}_{n+1}={y}_{n}+\frac{h}{2}(f({t}_{n},{y}_{n})+f({t}_{n+1},{y}_{n+1}))$

I am not sure whether it is an implicit or an explicit method.

I tend to say it is implicit one step methode because we have to solve an equation to calculate ${y}_{n+1}.$

Thank you for your help.

Jase Powell

Beginner2022-09-08Added 11 answers

You're quite right.

An explicit method is given by ${y}_{n+1}=F({t}_{n},{y}_{n})$, with no mention of ${y}_{n+1}$ on the RHS, and you can compute ${y}_{n+1}$ directly.

An implicit method is given by $F({t}_{n},{y}_{n},{y}_{n+1})=0$, and you have to solve an equation to find ${y}_{n+1}$.

Both these definitions are for one-step methods: ${y}_{n+1}$ depends directly or indirectly just on ${t}_{n}$ and ${y}_{n}$, not on any ${y}_{k}$ for $k<n$.

An explicit method is given by ${y}_{n+1}=F({t}_{n},{y}_{n})$, with no mention of ${y}_{n+1}$ on the RHS, and you can compute ${y}_{n+1}$ directly.

An implicit method is given by $F({t}_{n},{y}_{n},{y}_{n+1})=0$, and you have to solve an equation to find ${y}_{n+1}$.

Both these definitions are for one-step methods: ${y}_{n+1}$ depends directly or indirectly just on ${t}_{n}$ and ${y}_{n}$, not on any ${y}_{k}$ for $k<n$.

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1)?

How to expand and simplify $2(3x+4)-3(4x-5)$?

Find an equation equivalent to ${x}^{2}-{y}^{2}=4$ in polar coordinates.

How to graph $r=5\mathrm{sin}\theta$?

How to find the length of a curve in calculus?

When two straight lines are parallel their slopes are equal.

A)True;

B)FalseIntegration of 1/sinx-sin2x dx

Converting percentage into a decimal. $8.5\%$

Arrange the following in the correct order of increasing density.

Air

Oil

Water

BrickWhat is the exact length of the spiraling polar curve $r=5{e}^{2\theta}$ from 0 to $2\pi$?

What is $\frac{\sqrt{7}}{\sqrt{11}}$ in simplest radical form?

What is the slope of the tangent line of $r=-2\mathrm{sin}\left(3\theta \right)-12\mathrm{cos}\left(\frac{\theta}{2}\right)$ at $\theta =\frac{-\pi}{3}$?

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?

Use the summation formulas to rewrite the expression $\Sigma \frac{2i+1}{{n}^{2}}$ as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000.

How to calculate the right hand and left hand riemann sum using 4 sub intervals of f(x)= 3x on the interval [1,5]?