For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time.
end{aligned}That gives the exact solution begin{aligned} left(begin{array}{l} left( frac{r}{2}+frac{1}{2}right) frac{-6}{(x+t)^2} 2-r frac{-6}{{(x+t)^2-r frac{-6}{right). 2-r frac{-6}{
} end{aligned}Therefore, according to geometric Brownian motion process, the exact solution is determined (Y=frac{1}{e}exp left( int _0^t hbox {d}W_{t}right) =hbox {e}^{W t -1}), and finally exact solution is equal to (X=frac{1}{1-W t -1}
The exact solution of this problem is (left( {x^{2} + 1} right)^{ - 1}). Figure 1 shows a comparison of the Jacobi polynomial solution (y_{N}^{{left( {0,0} right)}} left( x right)), and the corrected Jacobi polynomial solution is (y_{N,M}^{{left( {0,0} right)}} left( x right)), for (left( {N,M} right) = left( {5,6} right)) and ((alpha = beta = 0)), with the exact solution (y(x)).
The boundary conditions and right-hand side functions in the model are selected such that the exact solution is given by begin{aligned} left { textstylebegin{array}l} u=(x^{2} y-1)^{2}+y)cos t, v=(- frac{2}{3}x(y-1)^{3}+2-pisin(pi x))cos t, p_{f}= 2-pisin(pi x))cdotsin(frac{1}{2}pi y)cos t, phi= 2-pisin(pi x))cdotsiny-cos(pi y))cos t, end{array}displaystyle right.
Upper left and lower left graphs show that the exact solution of (23) (X{mathit{exact}}_{mathit{mean}}) holds average of exact solution which is plotted as blue asterisks connected with dashed lines.
end{aligned}Therefore, (widetilde{u}=left( frac{r}{2}+frac{1}{2},2-rright) left( frac{-6}{(x+t)^2}right) ) is the exact solution of the fuzzy differential equation.
end{aligned} (30 Finally, the exact solution of this example is: begin{aligned} mathcal {Z}_{t}=ln (X_{t})=ln left( frac{1}{2} hbox {e}^{2Y_{t}}right) =2Y_{t}-ln 2=ln 2lexp( exp left( W^Q_{t}-frac{3t}{2} right) -1right).
Using the mentioned methods, the Jacobi polynomial solution of the problem is obtained by (y_{3}^{{left( {0.2, - 0.3} right)}} left( x right) = x - x^{3}), which is the exact solution of the problem [39].
The exact solution of this problem is the translation of the initial solution at unit speed: qleft( {x,t} right) = q_{0} left( {x - t} right) (22).
