Solve the equation

∇^{2} V + 4 π^{2} (x^{2} + y^{2}) V = 4π cos {π (x^{2} +y^{2})}

over the square region 0≤x≤0.5 and 0≤y≤0.5. The boundary conditions are

V(x,0)=sin(πx^{2}), V(x,0.5)=sin{π(x^{2}+0.25)}

V(0,y)=sin(πy^{2}), V(0.5,y)=sin{π(y^{2}+0.25)}

Use the function ellipgen to solve this equation with 15 divisions of x and y. Plot your results and compare with a plot of the exact solution, which is given by V =sin{π(x2+y2)}.Solve the eigenvalue problem∇^{2}z+λG(x,y)z=0 over a rectangular region bounded by 0≤ x ≤1 and 0≤y ≤1.5, z=0 at all boundaries. Use the function ellipgen with six divisions in x and nine divisions in y. The function G(x,y) over this grid is given by the MATLAB statements G = ones(10,7); G(4:7,3:5) = 3*ones(4,3);. This represents a membrane with a central area thicker than its periphery. The eigenvalues are related to the natural frequencies of this membrane.

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