Compact approximation stencils based on integrated flat radial basis functions.Compact local integrated RBF stencils.

clearvars

clc

f=@(x)-exp(-5*x)*(9975*sin(100*x)+1000*cos(100*x));

% parameters

nx=91;

Lx=1;

dx=Lx/(nx-1);

h=dx;

x=linspace(0,Lx,nx);

%Boundary conditions

u(1,:)=0;

u(nx,:)=0;

% Multiquadric RBF

beta=20;

a=beta*h;

% Define G(x)

G=zeros(nx,nx+2);

for j=1:nx

for i=1:nx

G(i,j)=sqrt((x(i)-x(j))^2+a^2);

end

end

% Extract the first and last rows of the matrix G

G1=G(1,:);

G2=G(end,:);

G_bar=[G1;G2];

G_a=G(2:end-1,:);

%H

H=zeros(nx,nx);

for i=1:nx

for j=1:nx

H(i,j)=(((x(i)-x(j))*sqrt((x(i)-x(j)).^2+a.^2))/2)+((a.^2)/2)*...

log((x(i)-x(j))+sqrt((x(i)-x(j)).^2+a.^2));

end

end

H_a=[H,ones(nx,1),zeros(nx,1)];

% Plot H

% surf(x, x, H); % 3D plot of H

% xlabel('x');

% ylabel('x');

% zlabel('H');

% title('Plot of H');

% colorbar;

% hold on

%H_bar

H1=zeros(nx,nx);

for i=1:nx

for j=1:nx

H1(i,j)=((sqrt((x(i)-x(j)).^2+a.^2))/6)+((a.^2)/2)*(x(i)-x(j))*....

log((x(i)-x(j))+sqrt((x(i)-x(j)).^2+a.^2))-((a.^2)/2)*...

sqrt((x(i)-x(j)).^2+a.^2);

end

end

% 3D plot of H1

% surf(x, x, H1);

% xlabel('x');

% ylabel('x');

% zlabel('H1');

% title('Plot of H1');

% colorbar;

% Add the new column x and 1

H_bar=[H1,x',ones(size(H,1),1)];

H_inv=pinv(H_bar);

% Conversion matrix

C=[H_bar;G_bar];

C_inv=pinv(C);

%f

f_1=f(0);

f_nx=f(end);

D=[u;f_1;f_nx];

E=C_inv*D;

f_interior=G_a*E;

% Populate the tri-diagonal structure of A

% Construct the system matrix A

A=zeros(nx-2,nx-2);

for i=2:nx-1

A(i-1,i-1)=f_interior(i-1);

if i>2

A(i-1,i-1)=f_interior(i-1);

end

if i<nx-1

A(i-1,i-1)=f_interior(i);

end

end

% Construct the vector b

b=zeros(nx-2,1);

for i=1:nx-1

b(i)=-exp(-5*x(i))*(9975*sin(100*x(i))+1000*cos(100*x(i)));

end

%LU decompose

N=length(A);

L=zeros(N,N);

U=zeros(N,N);

for i=1:N

L(i,i)=1;

end

U(1,:)=A(1,:);

L(:,1)=A(:,1)/U(1,1);

for i=2:N

for j=i:N

U(i,j)=A(i,j)-L(i,1:i-1)*U(1:i-1,j);

end

for k=i+1:N

L(k,i)=(A(k,i)-L(k,1:i-1)*U(1:i-1,i))/U(i,i);

end

end

%Solve Ly = B for y using forward substitution

y=zeros(N,1);

y(1)=b(1)/L(1,1);

for k=2:N

y(k)=(b(k)-L(k,1:k-1)*y(1:k-1))/L(k,k);

end

% Solve Ux = y for x using backward substitution

z=zeros(N,1);

z(N)=y(N)/U(N,N);

for k=N-1:-1:1

z(k)=(y(k)-U(k,k+1:N)*z(k+1:N))/U(k,k);

end

% Plot numerical solution

% Adjust the length of x and z for plotting

x_plot = x(2:end-1); % Adjust x to match the length of z

z_plot = z; % Use z as is

% Plot numerical solution

figure;

plot(x_plot, z_plot, 'r-', 'LineWidth', 1.5);

hold on;

% Define the exact solution function

uexact=@(x)sin(100*x).*exp(-5*x);

% Evaluate the exact solution

u_exact_values=uexact(x);

% Plot the exact solution

figure;

plot(x,u_exact_values,'b','LineWidth',1.5);

xlabel('x');

ylabel('u(x)');

title('Exact Solution');

% Compute the RMS error

u_interior_exact = uexact(x(2:end-1));

RMS_error = sqrt(sum((z - u_interior_exact).^2)/nx);

disp(['RMS Error: ', num2str(RMS_error)]);

[AKennedy]

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