In [1]:
set(0, "defaultlinelinewidth", 5);
set (0, "defaulttextfontname", "TimesNewRoman")
set (0, "defaulttextfontsize", 20)
set (0, "DefaultAxesFontName", "TimesNewRoman")
set(0, 'DefaultAxesFontSize', 20)
In [2]:
xe = linspace(0,2, 6); 
A = 1; B = -sinh(2)/cosh(2);
ye = A*cosh(xe) + B*sinh(xe);
plot(xe, ye); legend("Exact soln");
In [3]:
N = 10;
x = linspace(0,2,N); dx = x(2) - x(1);
A1 = zeros(N,N); b = zeros(N,1);
A2 = zeros(N,N); 
alpha = -2 - dx.^2;

dv = alpha*ones(N,1);
dv1 = ones(N-1,1);
A1 = diag(dv) + diag(dv1,1) + diag(dv1,-1);
A1(1,:) = 0; A1(N,:) = 0; 
A2 = diag(dv) + diag(dv1,1) + diag(dv1,-1);
A2(1,:) = 0; A2(N,:) = 0; 
A1(1,1) = 1; A1(N,N-1) = -1; A1(N,N) = 1;
A2(N,N) = alpha; A2(N,N-1) = 2;  A2(1,1) = 1;
b(1) = 1;
f = (inv(A1)*b)';
g = (inv(A2)*b)';
plot(x, f, x, g, xe, ye,'ok');
xlabel("x"); ylabel("f(x)");
h=legend ("First order BC", "Second Order", "Exact Soln");
In [4]:
N = 10;
x = linspace(0,2,N); dx = x(2) - x(1);
A1 = zeros(N, N); b = zeros(N,1);

alpha = -2 - dx^2;

dv = alpha*ones(N,1);
dv1 = ones(N-1,1);
A1 = diag(dv) + diag(dv1,1) + diag(dv1,-1);
A1(1,:) = 0; A1(N,:) = 0; 


A1(1,1) = 1; A2(1,1) = 1;
A1(end,end-1) = -1; A1(end,end) = 1;

b(1) = 1;
In [5]:
As = sparse(A1); 
f =As*b

f =

   1
   1
   0
   0
   0
   0
   0
   0
   0
   0
In [6]:
As = sparse(A1)
f = As\b

As =

Compressed Column Sparse (rows = 10, cols = 10, nnz = 27 [27%])

  (1, 1) ->  1
  (2, 1) ->  1
  (2, 2) -> -2.0494
  (3, 2) ->  1
  (2, 3) ->  1
  (3, 3) -> -2.0494
  (4, 3) ->  1
  (3, 4) ->  1
  (4, 4) -> -2.0494
  (5, 4) ->  1
  (4, 5) ->  1
  (5, 5) -> -2.0494
  (6, 5) ->  1
  (5, 6) ->  1
  (6, 6) -> -2.0494
  (7, 6) ->  1
  (6, 7) ->  1
  (7, 7) -> -2.0494
  (8, 7) ->  1
  (7, 8) ->  1
  (8, 8) -> -2.0494
  (9, 8) ->  1
  (8, 9) ->  1
  (9, 9) -> -2.0494
  (10, 9) -> -1
  (9, 10) ->  1
  (10, 10) ->  1

f =

   1.00000
   0.81118
   0.66241
   0.54636
   0.45729
   0.39080
   0.34361
   0.31338
   0.29864
   0.29864
In [7]:
plot(x, f, "-;First order BC;");
hold on;
plot(xe, ye, "-;Exact;",'Marker',"o","linestyle","none");
xlabel("x"); ylabel("f(x)");
legend;
hold off;
In [ ]: