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MATLAB: How to solve a system of Nonlinear Differential Equations?

  1. Apr 9, 2012 #1
    Hello to everybody,
    I'm very new to solving ODES and equations with MATLAB. I have been asked to solve a system of nonlinear equations for simulating the growth of a solid tumor.

    Assuming that we have the 5 unknowns which are dxd arrays: f,g,m,p and n.
    f(x,t) is the volume fraction of tumor (the fraction of tumor tissue against water and healthy tissue)
    g(x,t) is the volume fraction of dead tumor tissue (the fraction of dead tumor cells, i.e. necrotic cells)
    m(x,t) is a variable call "cell chemical potential"
    p(x,t) is the cell-to-cell solid pressure
    n(x,t) is the concentration of nutrients

    Then Wise et al (An adaptive multigrid algorithm for simulating solid tumor growth using mixture models, doi: 10.1016/j.mcm.2010.07.007) have derived that the tumor can be simulated by solving the following hard-to-solve system:

    (1) df/dt=M*div(f*grad (m))+n*(f-g)-lamdaL*g-div(-grad(p)+m*grad(f))
    (2) m=f^3-1.5*f^2+0.5*f-div(grad(f))
    (3) dg/dt=M*div(g*grad (m))+heaviside(nN-n)*(f-g)-lamdaL*g-div(-grad(p)+m*grad(f))
    (4) -div(grad(p))=n*(f-g)-lamdaL-div(m*grad(f))
    (5) 0=div(D*grad(n))+(vp)*(nc-n)-n(f-g)

    For outer boundary conditions m=p=q=0, n=1, z*grad(f)=0, where z is the outward pointing unit normal.
    (The rest parameters that have not been mentioned are scalars).

    Could someone please show me the way how to solve that with Matlab?
  2. jcsd
  3. Apr 10, 2012 #2
    Any idea?
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