thank you for your help, Preben Alsholm.
you are right,the solution for that pde doesn't exist.

BTW, I use the method you provided to another question and meet some problem.
the code is below:
> restart;
> pde := diff(u(x, t), `$`(t, 2)) = diff(u(x, t), `$`(x, 2));
> ibc := {u(0, t) = 0, u(1, t) = 0, u(x, 0) = sin(Pi*x), (D[2](u))(x, 0) = 3*x+1};
> res := pdsolve(pde);
> eq1 := eval(rhs(res), t = 0) = sin(Pi*x);
> eq2 := eval(diff(rhs(res), t), t = 0) = 3*x+1;
> eq1a := convert(diff(eq1, x), D);
> solve({eq2, eq1a}, {(D(_F1))(x), (D(_F2))(-x)});
> tt := `~`[map](int, %, x);
> select(has, %, _F2);
> eval(op(%), x = -x);
> -%;
> f2 := unapply(subs(%, _F2(x)), x);
> f1 := unapply(subs(op(select(has, tt, _F1)), _F1(x)), x);
> eval(res, [_F1 = f1, _F2 = f2]);
> sol := expand(%);
> pdetest(sol, [pde, op(ibc)]);
> plot3d(rhs(sol), x = 0 .. 1, t = 0 .. 2, axes = boxed);

the final result is wrong absolutely,
would you like give me some help?

one more question, in Maple, for the pde with analytic solution, Is there a general method to solve it?

thank you in advance.

thank you for your help, Preben Alsholm.
you are right,the solution for that pde doesn't exist.

BTW, I use the method you provided to another question and meet some problem.
the code is below:
> restart;
> pde := diff(u(x, t), `$`(t, 2)) = diff(u(x, t), `$`(x, 2));
> ibc := {u(0, t) = 0, u(1, t) = 0, u(x, 0) = sin(Pi*x), (D[2](u))(x, 0) = 3*x+1};
> res := pdsolve(pde);
> eq1 := eval(rhs(res), t = 0) = sin(Pi*x);
> eq2 := eval(diff(rhs(res), t), t = 0) = 3*x+1;
> eq1a := convert(diff(eq1, x), D);
> solve({eq2, eq1a}, {(D(_F1))(x), (D(_F2))(-x)});
> tt := `~`[map](int, %, x);
> select(has, %, _F2);
> eval(op(%), x = -x);
> -%;
> f2 := unapply(subs(%, _F2(x)), x);
> f1 := unapply(subs(op(select(has, tt, _F1)), _F1(x)), x);
> eval(res, [_F1 = f1, _F2 = f2]);
> sol := expand(%);
> pdetest(sol, [pde, op(ibc)]);
> plot3d(rhs(sol), x = 0 .. 1, t = 0 .. 2, axes = boxed);

the final result is wrong absolutely,
would you like give me some help?

one more question, in Maple, for the pde with analytic solution, Is there a general method to solve it?

thank you in advance.

thank you very much, Markiyan Hirnyk.

thank you very much, Markiyan Hirnyk.

@Alejandro Jakubi Then, that's the difference between Maple and Mathematica.

@Alejandro Jakubi 

Ok, so, that's a problem of software engineering planning.

1. The quality of commands is more important than the number of commands. Simple and useful is better.

2. Experience is the second point. Enhance the performance of basic commands is the first thing for maple, I think.

@acer Thank you very much, acer, I appreciate your answers.

@acer Thank you very much, acer, I appreciate your answers.

thank you for your method, acer,

on my computer, maple displays "Error, (in HWcall) object too large" under the command EigenConditionNumbers, how to solve it?

one more question, how to use Student[NumericalAnalysis][SpectralRadius] command to estimate the root range?

thank you for your method, acer,

on my computer, maple displays "Error, (in HWcall) object too large" under the command EigenConditionNumbers, how to solve it?

one more question, how to use Student[NumericalAnalysis][SpectralRadius] command to estimate the root range?

Dear Markiyan Hirnyk, all roots means that some roots are complex numbers.

thank you all the same.

Dear Markiyan Hirnyk, all roots means that some roots are complex numbers.

thank you all the same.

the more competition, the better software.

Thank you very much for your reply, rlopez.

I have two questions.

1.The reason I want eight functions is that I can use it for other computations later. So, can we make the F1~F8 continous, means no jump between branches for v in the range [3405..5054], with the new index rule for RootOf function? If it could be realized, I think that's the best method.

2.If not, the better idea is compute the discrete values of l2 with the corresponding v values, and then eliminate the jump between branches by hand. After that, we get the discrete function l2 values at discrete v values, and the interpolation can be further done. The key is how to distinguish different branches. From the figure derived with fsolve command, there are four branches which have no intersections with the real axis, so it's easy to pull them out. How about the other four? I have no idea.

So, would you like to give me some methods or advices to resolve the two problems above?
I will be very appreciated. Thanks in advance.