You didn't mention values of a and b. So I am passing that as an argument here.
u_xt = @(x,t,a,b) exp(a*t+b*x); % Create a function handle
x=[0:0.1:1]; %NOTE: I chose the step-size here as 0.1
t=[0:0.1:1]; %NOTE: Since your boundary conditions for x and t are identical [0 1] the step size has to be same for both
a = 5; % (say)
b = 2; % (say)
result = u_xt(x,t,a,b);
plot(u_xt,x);
% If you want to plot both on the same figure
hold on;
plot(u_xt,t);
