Finite-difference methods

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Problem 3.2.13 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 3.2 "Second Order Equations"

(1)(2)(3)

Let $h > 0$ be given. Consider the linear difference equation

\frac{y((n+2)h) - 2y((n+1)h) + y(nh)}{h^2} = -y(nh), \quad n = 0, 1, 2, \ldots .

(Note the analogy with the differential equation $y'' = -y$ .)

(1)

Find the general solution of the equation by trying suitable exponential substitutions.

(2)

Find the solution with $y(0) = 0$ and $y(h) = h$ . Denote it by $S_h(nh)$ , $n = 1, 2, \ldots$ .

(3)

Let $x$ be fixed and $h = \frac{x}{n}$ . Show that

\lim _{n \to \infty } S_{\frac{x}{n}}\left(\frac{nx}{n}\right) = \sin x.

Problem 1.4.1 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 1.4 "Two-Point Boundary Value Problems"

Divide the interval $[0, 1]$ into five equal subintervals, and apply the finite difference method in order to approximate the solution of the two-point boundary value problem

y''(t) = 125 t, \quad y(0) = y(1) = 0

at the four interior grid points. Compare your approximate values at the grid points with the exact solution at the grid points. Note: You should not expect very accurate approximations with only four interior grid points.

Problem 1.4.2 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 1.4 "Two-Point Boundary Value Problems"

Divide $[0, 1]$ into $n+1$ equal subintervals, and apply the finite difference approximation method to derive the linear system associated with the two-point boundary value problem

y''(t) - y'(t) = f(t), \quad y(0) = y(1) = 0.

Problem 1.4.3 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 1.4 "Two-Point Boundary Value Problems"

Divide $[0, 1]$ into five equal subintervals, and approximate the solution to

y''(t) - y'(t) = 125 t, \quad y(0) = y(1) = 0

at the four interior grid points. Compare the approximations with the exact values at the grid points.

Problem 2.1.30 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 2.1 "Chapter Problems"

Consider the partial differential equation (Laplace equation)

\frac{\partial ^{2} u}{\partial x^{2}}+\frac{\partial ^{2} u}{\partial y^{2}} = 0 \quad \text{on} \quad [0,1] \times [0,1]

with the boundary conditions

u(x,0) = 1, \qquad u(x,1) = 2, \qquad u(0,y) = 1, \qquad u(1,y) = 2.

Apply the central difference scheme

\left( \frac{\partial ^{2} u}{\partial x^{2}} \right)_{j,k} \approx \frac{u_{j-1,k}-2 u_{j,k}+u_{j+1,k}}{(\Delta x)^{2}}, \qquad \left( \frac{\partial ^{2} u}{\partial y^{2}} \right)_{j,k} \approx \frac{u_{j,k-1}-2 u_{j,k}+u_{j,k+1}}{(\Delta y)^{2}}

and then solve the linear equation. Consider the cases $\Delta x = \Delta y = 1/3$ and $\Delta x = \Delta y = 1/4$ .

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.

[Mey00] Meyer C.D. Matrix Analysis and Applied Linear Algebra. SIAM, 2000.

[StHa17] Steeb W., Hardy Y. Problems and Solutions in Introductory and Advanced Matrix Calculus. 2nd ed., World Scientific Publishing, 2017.