@Kitonum 

The version I used is Maple 2015.2.

@Carl Love 

Dear Carl,

I tried your sentences, but the result is still wrong, and it should be -infinity rather than 0. see eq5
 

Download test.mw

.

I don't know how to input the mathematical expression in this box, and I used the way by uploading  the maple files.

Would you like to tell me how to input the mathematical expression here?

Thanks.

@acer 

The function simplify(expr, size) is what I need.

Thanks for your help.

@Ramakrishnan and @Kitonum 9027 

I absorbed your ideas and methods, and have gotten the appropriate form.

But I sitll don't know how to take a term or a factor from an expression which should be dealt with alone.

I have done this step by 'copy' rather than by functions.

Thanks for your valuable tips and continue to wait for your further guide.

Have a nice weekend.

eq21 := collect(eq20, [exp(-sqrt(s)*x/sqrt(Dp)), exp((-2*lh+x)*sqrt(s)/sqrt(Dp)), exp((lh-x)*sqrt(s+thetac)/sqrt(Dc)), exp((-lh+x)*sqrt(s)/sqrt(Dp))], simplify);

q(x, s) = _F1(s)*exp(-sqrt(s)*x/sqrt(Dp))+(sqrt(Dp)*alpha1+sqrt(s))*_F1(s)*exp(sqrt(s)*(-2*lh+x)/sqrt(Dp))/(-sqrt(Dp)*alpha1+sqrt(s))+A1*Dc*exp((lh-x)*sqrt(s+thetac)/sqrt(Dc))/(sqrt(s+thetac)*(Dc*s-Dp*s-Dp*thetac))+sqrt(Dp)*(-sqrt(s+thetac)*Dp*alpha1*pinf*s^2-2*sqrt(s+thetac)*Dp*alpha1*pinf*s*thetac-sqrt(s+thetac)*Dp*alpha1*pinf*thetac^2+A2*sqrt(s+thetac)*Dp*alpha1*s+A2*sqrt(s+thetac)*Dp*alpha1*thetac+Dc*sqrt(s+thetac)*alpha1*pinf*s^2+Dc*sqrt(s+thetac)*alpha1*pinf*s*thetac+A1*Dc*alpha1*s^2+A1*Dc*alpha1*s*thetac+A1*sqrt(s+thetac)*sqrt(Dc)*s^2+A1*sqrt(s+thetac)*sqrt(Dc)*s*thetac-A2*Dc*sqrt(s+thetac)*alpha1*s)*exp((-lh+x)*sqrt(s)/sqrt(Dp))/((s+thetac)^(3/2)*(-sqrt(Dp)*alpha1+sqrt(s))*(Dc*s-Dp*s-Dp*thetac)*s)-(-pinf*s-pinf*thetac+A2)/(s*(s+thetac))

 
# Take the fourth term from eq21, whose fraction should be simplified.
eq22 := sqrt(Dp)*(-sqrt(s+thetac)*Dp*alpha1*pinf*s^2-2*sqrt(s+thetac)*Dp*alpha1*pinf*s*thetac-sqrt(s+thetac)*Dp*alpha1*pinf*thetac^2+A2*sqrt(s+thetac)*Dp*alpha1*s+A2*sqrt(s+thetac)*Dp*alpha1*thetac+Dc*sqrt(s+thetac)*alpha1*pinf*s^2+Dc*sqrt(s+thetac)*alpha1*pinf*s*thetac+A1*Dc*alpha1*s^2+A1*Dc*alpha1*s*thetac+A1*sqrt(s+thetac)*sqrt(Dc)*s^2+A1*sqrt(s+thetac)*sqrt(Dc)*s*thetac-A2*Dc*sqrt(s+thetac)*alpha1*s)*exp((-lh+x)*sqrt(s)/sqrt(Dp))/((s+thetac)^(3/2)*(Dc*s-Dp*s-Dp*thetac)*s*(-sqrt(Dp)*alpha1+sqrt(s)));

# eq22 should be separated into two parts, one of which contains sqrt(s+thetac), and the other otherwise.

eq23 := simplify(sqrt(Dp)*(-sqrt(s+thetac)*Dp*alpha1*pinf*s^2-2*sqrt(s+thetac)*Dp*alpha1*pinf*s*thetac-sqrt(s+thetac)*Dp*alpha1*pinf*thetac^2+A2*sqrt(s+thetac)*Dp*alpha1*s+A2*sqrt(s+thetac)*Dp*alpha1*thetac+Dc*sqrt(s+thetac)*alpha1*pinf*s^2+Dc*sqrt(s+thetac)*alpha1*pinf*s*thetac+A1*sqrt(s+thetac)*sqrt(Dc)*s^2+A1*sqrt(s+thetac)*sqrt(Dc)*s*thetac-A2*Dc*sqrt(s+thetac)*alpha1*s)*exp((-lh+x)*sqrt(s)/sqrt(Dp))/((s+thetac)^(3/2)*(Dc*s-Dp*s-Dp*thetac)*s*(-sqrt(Dp)*alpha1+sqrt(s))))+simplify(sqrt(Dp)*(A1*Dc*alpha1*s^2+A1*Dc*alpha1*s*thetac)*exp((-lh+x)*sqrt(s)/sqrt(Dp))/((s+thetac)^(3/2)*(Dc*s-Dp*s-Dp*thetac)*s*(-sqrt(Dp)*alpha1+sqrt(s))));

# The rational function of the first term in eq23 can be aparted into three proper fractions.

eq24 := convert(sqrt(Dp)*(Dc*alpha1*pinf*s^2+Dc*alpha1*pinf*s*thetac-Dp*alpha1*pinf*s^2-2*Dp*alpha1*pinf*s*thetac-Dp*alpha1*pinf*thetac^2+A1*sqrt(Dc)*s^2+A1*sqrt(Dc)*s*thetac-A2*Dc*alpha1*s+A2*Dp*alpha1*s+A2*Dp*alpha1*thetac)/((s+thetac)*(Dc*s-Dp*s-Dp*thetac)*s), parfrac, s);

# Now eq21 can be written into the acceptable form, eq25.

eq25 := _F1(s)*exp(-sqrt(s)*x/sqrt(Dp))+(sqrt(Dp)*alpha1+sqrt(s))*_F1(s)*exp(sqrt(s)*(-2*lh+x)/sqrt(Dp))/(-sqrt(Dp)*alpha1+sqrt(s))+A1*Dc*exp((lh-x)*sqrt(s+thetac)/sqrt(Dc))/(sqrt(s+thetac)*(Dc*s-Dp*s-Dp*thetac))-(-pinf*s-pinf*thetac+A2)/(s*(s+thetac))-sqrt(Dp)*exp(-(lh-x)*sqrt(s)/sqrt(Dp))*eq24/(sqrt(Dp)*alpha1-sqrt(s))-sqrt(Dp)*A1*Dc*alpha1*exp(-(lh-x)*sqrt(s)/sqrt(Dp))/(sqrt(s+thetac)*(sqrt(Dp)*alpha1-sqrt(s))*(Dc*s-Dp*s-Dp*thetac));

@Carl Love 

I am sorry for the inappropriate words.

I just want to seprate the whole expression into much simpler parts and each part is the simplest.

@Ramakrishnan 

Hello, Mr. Ramakrishnan.

I tired your way, but got a different result.

 eq22 := simplify(sqrt(Dp)*(-sqrt(s+thetac)*Dp*alpha1*pinf*s^2-2*sqrt(s+thetac)*Dp*alpha1*pinf*s*thetac-sqrt(s+thetac)*Dp*alpha1*pinf*thetac^2+A2*sqrt(s+thetac)*Dp*alpha1*s+A2*sqrt(s+thetac)*Dp*alpha1*thetac+Dc*sqrt(s+thetac)*alpha1*pinf*s^2+Dc*sqrt(s+thetac)*alpha1*pinf*s*thetac+A1*Dc*alpha1*s^2+A1*Dc*alpha1*s*thetac+A1*sqrt(s+thetac)*sqrt(Dc)*s^2+A1*sqrt(s+thetac)*sqrt(Dc)*s*thetac-A2*Dc*sqrt(s+thetac)*alpha1*s)*exp(sqrt(s)*(-lh+x)/sqrt(Dp))/((s+thetac)^(3/2)*(-sqrt(Dp)*alpha1+sqrt(s))*s*(Dc*s-Dp*s-Dp*thetac)));

sqrt(Dp)*(-sqrt(s+thetac)*Dp*alpha1*pinf*s^2-2*sqrt(s+thetac)*Dp*alpha1*pinf*s*thetac-sqrt(s+thetac)*Dp*alpha1*pinf*thetac^2+A2*sqrt(s+thetac)*Dp*alpha1*s+A2*sqrt(s+thetac)*Dp*alpha1*thetac+Dc*sqrt(s+thetac)*alpha1*pinf*s^2+Dc*sqrt(s+thetac)*alpha1*pinf*s*thetac+A1*Dc*alpha1*s^2+A1*Dc*alpha1*s*thetac+A1*sqrt(s+thetac)*sqrt(Dc)*s^2+A1*sqrt(s+thetac)*sqrt(Dc)*s*thetac-A2*Dc*sqrt(s+thetac)*alpha1*s)*exp(-(lh-x)*sqrt(s)/sqrt(Dp))/((s+thetac)^(3/2)*(-sqrt(Dp)*alpha1+sqrt(s))*s*(Dc*s-Dp*s-Dp*thetac))

@acer 

Dear Acer, Thanks for your advice.

eq20 is listed below.

In eq21, only the coefficient of exp((-lh+x)*sqrt(s)/sqrt(Dp)) is complicated.

The coefficient is a rational function and needed to be taken apart into the sum of several proper frations.

eq20 := `assuming`([pdsolve([eq13, eq14], q(x, s))], [alpha1 > 0, Dp > 0, Dc > 0, s > 0, s+thetac > 0, lh-x < 0]);

q(x, s) = -(A1*(thetac*s^(3/2)+s^(5/2)+(-s^2-s*thetac)*alpha1*sqrt(Dp))*Dc*exp((lh-x)*sqrt(s+thetac)/sqrt(Dc))+((-alpha1*(s+thetac)*(pinf*s+pinf*thetac-A2)*Dp^(3/2)+sqrt(Dp)*(A1*(s+thetac)*sqrt(Dc)+Dc*alpha1*(pinf*s+pinf*thetac-A2))*s)*sqrt(s+thetac)+A1*sqrt(Dp)*s*Dc*alpha1*(s+thetac))*exp((-lh+x)*sqrt(s)/sqrt(Dp))-(_F1(s)*(s*alpha1*(s+thetac)^2*Dp^(3/2)-thetac*(Dc-2*Dp)*s^(5/2)+(-Dc+Dp)*s^(7/2)+s^(3/2)*Dp*thetac^2-sqrt(Dp)*s^2*Dc*alpha1*(s+thetac))*exp(sqrt(s)*(-2*lh+x)/sqrt(Dp))+(-s*alpha1*(s+thetac)^2*Dp^(3/2)-thetac*(Dc-2*Dp)*s^(5/2)+(-Dc+Dp)*s^(7/2)+s^(3/2)*Dp*thetac^2+sqrt(Dp)*s^2*Dc*alpha1*(s+thetac))*_F1(s)*exp(-sqrt(s)*x/sqrt(Dp))-alpha1*(s+thetac)*(pinf*s+pinf*thetac-A2)*Dp^(3/2)+(-pinf*(Dc-2*Dp)*thetac+A2*(Dc-Dp))*s^(3/2)-pinf*(Dc-Dp)*s^(5/2)+s*Dc*alpha1*(pinf*s+pinf*thetac-A2)*sqrt(Dp)+thetac*sqrt(s)*Dp*(pinf*thetac-A2))*sqrt(s+thetac))/((s+thetac)^(3/2)*s*((-Dc+Dp)*s+Dp*thetac)*(-sqrt(Dp)*alpha1+sqrt(s)))

eq21 := collect(eq20, [exp(-sqrt(s)*x/sqrt(Dp)), exp(sqrt(s)*(-2*lh+x)/sqrt(Dp)), exp((lh-x)*sqrt(s+thetac)/sqrt(Dc)), exp((-lh+x)*sqrt(s)/sqrt(Dp))], simplify);

q(x, s) = exp(-sqrt(s)*x/sqrt(Dp))*_F1(s)+(sqrt(Dp)*alpha1+sqrt(s))*_F1(s)*exp(sqrt(s)*(-2*lh+x)/sqrt(Dp))/(-sqrt(Dp)*alpha1+sqrt(s))+A1*Dc*exp((lh-x)*sqrt(s+thetac)/sqrt(Dc))/(sqrt(s+thetac)*(Dc*s-Dp*s-Dp*thetac))+sqrt(Dp)*(-Dp*sqrt(s+thetac)*alpha1*pinf*s^2-2*Dp*sqrt(s+thetac)*alpha1*pinf*s*thetac-Dp*sqrt(s+thetac)*alpha1*pinf*thetac^2+A2*Dp*sqrt(s+thetac)*alpha1*s+A2*Dp*sqrt(s+thetac)*alpha1*thetac+Dc*sqrt(s+thetac)*alpha1*pinf*s^2+Dc*sqrt(s+thetac)*alpha1*pinf*s*thetac+A1*Dc*alpha1*s^2+A1*Dc*alpha1*s*thetac+A1*sqrt(Dc)*sqrt(s+thetac)*s^2+A1*sqrt(Dc)*sqrt(s+thetac)*s*thetac-A2*Dc*sqrt(s+thetac)*alpha1*s)*exp((-lh+x)*sqrt(s)/sqrt(Dp))/((s+thetac)^(3/2)*s*(Dc*s-Dp*s-Dp*thetac)*(-sqrt(Dp)*alpha1+sqrt(s)))-(-pinf*s-pinf*thetac+A2)/((s+thetac)*s)

@acer 

It works. Thanks for your tip.

@Kitonum 

It works. Thanks for your help.

@Preben Alsholm 

@ThU  Yes, I have loaded the inttrans package.