Best way to solve this non-linear equation?

I have a set of equations I need to solve, the first equation (not the one shown below) is a PDE solved through pdepe(), but for the form of the following equation I am not sure which approach to take to solve it:

I have tried solving it symbolically through solve() and dsolve(), but that did not work, with solve() returning an empty sym and dsolve returning the following:

syms a(t) c(x) d(x)

b = 2*10^(-7);
eqn = diff(a,t) == -b*(diff(c,x)*diff((d/c),x) + rho_a*diff((d/c),2,x));
sol = dsolve(eqn,a);

Error using mupadengine/feval_internal

No differential equations found. Specify differential equations by using symbolic functions.

Error in dsolve>mupadDsolve (line 334)

T = feval_internal(symengine,'symobj::dsolve',sys,x,options);

Error in dsolve (line 203)

sol = mupadDsolve(args, options);

Error in odetest (line 6)

sol = dsolve(eqn,a);

Answer 1

[John D'Errico]

dsolve is not used to solve a PDE. It applies ONLY to an ODE or a system of ODEs.

help dsolve

 DSOLVE Symbolic solution of ordinary differential equations.
    DSOLVE will not accept equations as strings in a future release.
    Use symbolic expressions or sym objects instead.
    For example, use syms y(t); dsolve(diff(y)==y) instead of  dsolve('Dy=y').
 
    DSOLVE(eqn1,eqn2, ...) accepts symbolic equations representing
    ordinary differential equations and initial conditions. 
 
    By default, the independent variable is 't'. The independent variable
    may be changed from 't' to some other symbolic variable by including
    that variable as the last input argument.
 
    The DIFF function constructs derivatives of symbolic functions (see sym/symfun).
    Initial conditions involving derivatives must use an intermediate
    variable. For example,
      syms x(t)
      Dx = diff(x);
      dsolve(diff(Dx) == -x, Dx(0) == 1)
 
    If the number of initial conditions given is less than the
    number of dependent variables, the resulting solutions will obtain
    arbitrary constants, C1, C2, etc.
    
    Three different types of output are possible.  For one equation and one
    output, the resulting solution is returned, with multiple solutions to
    a nonlinear equation in a symbolic vector.  For several equations and
    an equal number of outputs, the results are sorted in lexicographic
    order and assigned to the outputs.  For several equations and a single
    output, a structure containing the solutions is returned.
 
    If no closed-form (explicit) solution is found, then a
    warning is given and the empty sym is returned.  
  
    DSOLVE(...,'IgnoreAnalyticConstraints',VAL) controls the level of 
    mathematical rigor to use on the analytical constraints of the solution 
    (branch cuts, division by zero, etc). The options for VAL are TRUE or 
    FALSE. Specify FALSE to use the highest level of mathematical rigor
    in finding any solutions. The default is TRUE.
 
    DSOLVE(...,'MaxDegree',n) controls the maximum degree of polynomials
    for which explicit formulas will be used in SOLVE calls during the
    computation. n must be a positive integer smaller than 5. 
    The default is 2.
 
    DSOLVE(...,'Implicit',true) returns the solution as a vector of
    equations, relating the dependent and the independent variable. This
    option is not allowed for systems of differential equations.
 
    DSOLVE(...,'ExpansionPoint',a) returns the solution as a series around
    the expansion point a.
 
    DSOLVE(...,'Order',n) returns the solution as a series with order n-1.
 
    Examples:
 
       % Example 1
       syms x(t) a
       dsolve(diff(x) == -a*x) returns
 
         ans = C1/exp(a*t)
 
       % Example 2: changing the independent variable 
       x = dsolve(diff(x) == -a*x, x(0) == 1, 's') returns
 
         x = 1/exp(a*s)
 
       syms x(s) a
       x = dsolve(diff(x) == -a*x, x(0) == 1) returns
 
         x = 1/exp(a*s)
 
       % Example 3: solving systems of ODEs
       syms f(t) g(t)
       S = dsolve(diff(f) == f + g, diff(g) == -f + g,f(0) == 1,g(0) == 2)
       returns a structure S with fields
 
         S.f = (i + 1/2)/exp(t*(i - 1)) - exp(t*(i + 1))*(i - 1/2)
         S.g = exp(t*(i + 1))*(i/2 + 1) - (i/2 - 1)/exp(t*(i - 1))
 
       syms f(t) g(t)
       v = [f;g];
       A = [1 1; -1 1];
       S = dsolve(diff(v) == A*v, v(0) == [1;2])
       returns a structure S with fields
 
         S.f = exp(t)*cos(t) + 2*exp(t)*sin(t)
         S.g = 2*exp(t)*cos(t) - exp(t)*sin(t)
 
       % Example 3: using options
       syms y(t)
       dsolve(sqrt(diff(y))==y) returns
       
         ans = 0
       
       syms y(t)
       dsolve(sqrt(diff(y))==y, 'IgnoreAnalyticConstraints', false) warns
         Warning: The solutions are subject to the following conditions:
         (C67 + t)*(1/(C67 + t)^2)^(1/2) = -1 
       
       and returns
        
         ans = -1/(C67 + t)
 
       % Example 4: Higher order systems
       syms y(t) a
       Dy = diff(y);
       D2y = diff(y,2);
       dsolve(D2y == -a^2*y, y(0) == 1, Dy(pi/a) == 0)
       syms w(t)
       Dw = diff(w); 
       D2w = diff(w,2);
       w = dsolve(diff(D2w) == -w, w(0)==1, Dw(0)==0, D2w(0)==0)
 
    See also SOLVE, SUBS, SYM/DIFF, odeToVectorfield.

    Documentation for dsolve
       doc dsolve

There are no symbolic solvers in MATLAB that apply to a PDE. Sorry. In SOME (moderately rare, and usually pretty simple) cases, an analytical solution can be found. For example, sometimes, separation of variables can be used to derive a solution.

As it is though, you ask for the best way to solve the PDE. That would be by the use of PDEPE (which you claim to have already done) or similar tools. Or you could write your own code, using a variety of methods for the numerical solution of a PDE.

[HVdB]

Thank you for your reply. How would one go about formulating the flux and source terms for pdepe in this case though? Since the equation lacks a spatial derivative of a(t). Or is pdepe not suitable for this equation?

[Torsten]

And what kind of equations do you have for c(t,x) and d(t,x) ?

[HVdB]

The complete system of equations is:

With rho_a (which is c in this case) and rho_v (which is d) dependent on the temperature T, which in turn depends on (t,x).

[Torsten]

This system cannot be solved by a standard MATLAB solver (e.g. pdepe).

You will have to discretize on your own in space and solve the resulting system of ordinary differential equations using ODE15S, e.g.

Look up "method-of-lines" for more details.

[HVdB]

Thank you! That sounds like a good approach!