Question

please show excel formulas for both explicit and implicut methods

please solve using excel and show formulas You are given the following system of differential equations: 99x, +2999x 2000x1 - 3000x2 If x1(0)=x2(0)-1, obtain a solution from t=0 to 0.3 using a step size of 0.03 with a. The explicit Euler's method b. The implicit Euler's method (note that this problem can be solved via a set of simultaneous linear equations for each time step) C. Plot all results on the same graph d. Compare the two different numerical solutions in terms of stability

Answer 1

Solving Ordinary Differential Equations in Excel

There are 2 ways so below is the formula

1] Make one column with tntn values, and one with vnvn values.

Fill the first column with increments the size of hh, adding on from the t0t0 value.

For the vnvn values, after adding the initial v0v0 value, use this:

vn+1=vn+h(dvdt)vn+1=vn+h(dvdt)

Of course, you need to convert it into a form that Excel understands, which depends on what columns you are using and the nature of your excel document.

2]

=PDSOLVE(eqns, vars, lbc, rbc, xdom, tdom, [options])

Column B: Time (heading in B1)

Initial value: 0 (in B2)

Formula (in B3): "=B2+A$2"

Use PDSOLVE to solve a system of partial differential equations the following form:

∂u1∂t=f1(t,x,u,ux,uxx)∂u1∂t=f1t,x,u,ux,uxx

∂u2∂t=f2(t,x,u,ux,uxx)∂u2∂t=f2t,x,u,ux,uxx

where u=[u1,u2]u=[u1,u2], ux=[u1,x,u2,x]ux=[u1,x,u2,x], uxx=[u1,xx,u2,xx]

Initial Conditions

Initial value for each variable ui is required.

Boundary Conditions

So the General non-linear boundary conditions can be specified at left and right ends of the system spatial domain [xs,xe].
Generally, for each equation fi in the system: none, one or two boundary conditions are required depending on the order of the equation.

• If fi does not depend on ui,xui,x and ui,xxui,xx then no boundary conditions are required for equation i.
• If fi depends only on ui,xui,x then only one boundary condition is required for equation i.
• If fi depends depends on ui,xxui,xx then two boundary conditions are required for equation