The part "Numerical methods in PDE"

Homework $1$

September $30,2024$

Euler difference scheme,

Numerical differentiation

For the equation $x^{}(t)=\frac{t+x}{sinx}+3,x(0)=x_0,$ write the Euler difference scheme for its numerical solution $($i$.$e$.,$ write a formula for one iteration: $x_{k+1}=$dots $).$

For the system of equations

$x^{}=e^{ty}+y^2$

$y^{}=x(y+2t)+1$

$x(t_0)=x_0,y(t_0)=y_0,$ write the Euler algorithm for its numerical solution $($i$.$e$.,$ write a formula for one iteration: $x_{k+1}=$dots,$y_{k+1}=$dots $).$

$3.$ Assume that we need to solve the equation $x^{}=x^2+x(sint+2),x(0)=1,$ in the segment $0,1$ on the domain $\{(t,x)|tin[0,1],$xin$[0,3]\}.$ Estimate the number $N$ of the partition of the segment $0,1$ to find the solution with the global precision at most $=0\mathrm{.}001.$

$4.$ Find a and $b$ such that the following approximation for the derivative has the highest possible precision:

$x^{}(t),$~~$,ax(t-2h)+bx(t+h)$

Write the corresponding Euler scheme for the equation $x^{}=f(t,x).$

$5.$ Find $a,b,c$ such that the following approximation for the derivative has the highest possible precision:

$x^{}(t)$~~$ax(t-2h)+bx(t)+cx(t+h)$

Write the corresponding Euler scheme for the equation $x^{}=f(t,x).$

$6.$ Find the rates of convergence $($local and global$)$ for the Euler difference scheme constructed with the approximations of derivatives from Problems $4$ and $5.$