Solve the Laplace equation

Matlab, 84, 81.2 79.1 bytes = 113 - 30%

function u=l(f,N,a,b);A=toeplitz([2,-1,(3:N)*0]);A([1,2,end-[1,0]])=eye(2);u=[a,f((1:N-2)/N)*(N-1)^2,b]/A;plot(u)

Note that in this example the I use row vectors, this means that the matrix A is transposed. f is taken as a function handle, a,b are the left/right side Dirichlet contraints.

function u=l(f,N,a,b);
A=toeplitz([2,-1,(3:N)*0]);       % use the "toeplitz" builtin to generate the matrix
A([1,2,end-[1,0]])=eye(2);        % adjust first and last column of matrix
u=[a,f((1:N-2)/N)*(N-1)^2,b]/A;   % build right hand side (as row vector) and right mu
plot(u)                           % plot the solution

For the example f = 10*x, u_L = 0 u_R = 1, N = 15 this results in:

Mathematica, 52.5 bytes (= 75 * (1 - 30%))

+0.7 bytes per @flawr 's comment.

ListPlot[{#,u@#}&/@Subdivide@#4/.NDSolve[-u''@x==#&&u@0==#2&&u@1==#3,u,x]]&

This plots the output.

e.g.

ListPlot[ ... ]&[10 x, 0, 1, 15]

Explanation

NDSolve[-u''@x==#&&u@0==#2&&u@1==#3,u,x]

Solve for the function u.

Subdivide@#4

Subdivide the interval [0,1] into N (4th input) parts.

{#,u@#}&/@ ...

Map u to the output of Subdivide.

ListPlot[ ... ]

Plot the final result.

Non-graphing solution: 58 bytes

u/@Subdivide@#4/.NDSolve[-u''@x==#&&u@0==#2&&u@1==#3,u,x]&

R, 123.2 102.9 98.7 bytes (141-30%)

Edit: Saved a handful of bytes thanks to @Angs!

If someone wants to edit the pictures feel free to do so. This is basically an R adaptation of both the matlab and python versions posted.

function(f,a,b,N){n=N-1;x=1:N/n;A=toeplitz(c(2,-1,rep(0,N-2)));A[1,1:2]=1:0;A[N,n:N]=0:1;y=n^-2*sapply(x,f);y[1]=a;y[N]=b;plot(x,solve(A,y))}

Ungolfed & explained:

u=function(f,a,b,N){
    n=N-1;                                              # Alias for N-1
    x=0:n/n;                                            # Generate the x-axis
    A=toeplitz(c(2,-1,rep(0,N-2)));                     # Generate the A-matrix
    A[1,1:2]=1:0;                                       # Replace first row--
    A[N,n:N]=0:1;                                       # Replace last row
    y=n^-2*sapply(x,f)                                  # Generate h^2*f(x)
    y[1]=a;y[N]=b;                                      # Replace first and last elements with uL(a) and uR(b)
    plot(x,solve(A,y))                                  # Solve the matrix system A*x=y for x and plot against x 
}

Example & test cases:

The named and ungolfed function can be called using:

u(function(x)-2,0,0,10)
u(function(x)x*10,0,1,15)
u(function(x)sin(2*pi*x),1,1,20)
u(function(x)x^2,0,0,30)

Note that the f argument is an R-function.

To run the golfed version simply use (function(...){...})(args)