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I need your helps about solving the problem below with MATLAB. I am trying to solve 2D Stress Wave Propagation problem by using FDTD(Finite difference time domain) method on the cylindrical coord. I have discretize the domain and now ı am trying to apply boundary conditions. I have the boundary condition equation below for i=1 and j=2,...,ns-1(AB boundary).

This is my discretization; enter image description here

The equation (AB Boundary Condition); enter image description here The right-hand side of the equation is known. But the left side is unknown.

Linear equation system for the equation; for i=1 and j=2,...,4(ı choose the j = 4 for the last term to make simpler explanation) enter image description here

  • u_(i,j)^(k+1) is the radial displacement and v_(i,j)^(k+1) is theangular displacement for the (k+1). time step.
  • A1,B1,C1,A2,B2,C2,A3,B3,C3 are coefficients.
  • D1,D2,D3 are constants.

I want to find all u_(i,j)^(k+1) and v_(i,j)^(k+1). But there are just 3 equations and 8 unknowns. I dont know the way how to solve this with matrices like Ax=B. It is not make sense.

Unknowns : u_1,2^(k+1), u_1,3^(k+1),u_1,4^(k+1) , v_1,1^(k+1), v_1,2^(k+1), v_1,3^(k+1), v_1,4^(k+1) and v_1,5^(k+1)

enter image description here

Matlab code for the equations:

AB1 = (lambda(i,j)+2*nu(i,j))*(-3*u(i,j,k+1)/(2*d_r))+(lambda(i,j)/R(i,j))*(((v(i,j+1,k+1)-v(i,j-1,k+1))/(2*d_s))+u(i,j,k+1));
AB2 = -Pi-(lambda(i,j)+2*nu(i,j))*((4*u(i+1,j,k+1)-u(i+2,j,k+1))/(2*d_r));
AB1 = AB2; % AB1 IS UNKNOWN,  AB2 IS KNOWN
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