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A new method based on polynomials equipped with a parameter to solve two parabolic inverse problems with a nonlocal boundary condition

J. HajishafieihaDepartment of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

S. AbbasbandyDepartment of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, IranCorrespondence[email protected] [email protected]

Figures & data

Figure 1. Relative error graphs of v in Example 1 at time $T = 1 / 2$ for N = 30 and $τ = 1 / 1000$ in CGL points: (a) without noisy data and (b) with noisy data.

Figure 2. Numerical and exact solutions of Example 1 for N = 40, $τ = 1 / 1000$ and $T = 1 / 2$ in CGL points.

Table 1. Relative errors of $v (x, t)$ for Example 1; $N = 40, τ = 1 / 1000$ and regular grid points

Display Table

Table 2. Relative errors of $v (x, t)$ , for Example 1; $N = 40, τ = 1 / 1000$ and CGL grid points

Display Table

Table 3. Relative errors of source parameter $r (t)$ for Example 1; $N = 40, τ = 1 / 1000$ and regular grid points

Display Table

Table 4. Relative errors of source parameter $r (t)$ for Example 1; $N = 40, τ = 1 / 1000$ and CGL grid points

Display Table

Figure 3. Absolute error graphs of v in Example 2 at time $T = 1 / 2$ for N = 15 and $τ = 1 / 5000$ in CGL points: (a) without noisy data and (b) with noisy data.

Figure 4. Numerical and exact solutions of Example 2 for $N = 15,$ $τ = 1 / 5000$ and $T = 1 / 2$ in CGL points.

Table 5. Relative errors of $v (x, t)$ for Example 2; $N = 15, τ = 1 / 5000$ and regular grid points

Display Table

Table 6. Relative errors of $v (x, t)$ for Example 2; $N = 15, τ = 1 / 5000$ and CGL grid points

Display Table

Table 7. Relative errors of source parameter $p (t)$ for Example 2; $N = 15, τ = 1 / 5000$ and regular grid points

Display Table

Table 8. Relative errors of source parameter $p (t)$ for Example 2; $N = 15, τ = 1 / 5000$ and CGL grid points

Display Table

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A new method based on polynomials equipped with a parameter to solve two parabolic inverse problems with a nonlocal boundary condition

Table 1. Relative errors of $v (x, t)$ for Example 1; $N = 40, τ = 1 / 1000$ and regular grid points

Table 2. Relative errors of $v (x, t)$ , for Example 1; $N = 40, τ = 1 / 1000$ and CGL grid points

Table 3. Relative errors of source parameter $r (t)$ for Example 1; $N = 40, τ = 1 / 1000$ and regular grid points

Table 4. Relative errors of source parameter $r (t)$ for Example 1; $N = 40, τ = 1 / 1000$ and CGL grid points

Table 5. Relative errors of $v (x, t)$ for Example 2; $N = 15, τ = 1 / 5000$ and regular grid points

Table 6. Relative errors of $v (x, t)$ for Example 2; $N = 15, τ = 1 / 5000$ and CGL grid points

Table 7. Relative errors of source parameter $p (t)$ for Example 2; $N = 15, τ = 1 / 5000$ and regular grid points

Table 8. Relative errors of source parameter $p (t)$ for Example 2; $N = 15, τ = 1 / 5000$ and CGL grid points

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A new method based on polynomials equipped with a parameter to solve two parabolic inverse problems with a nonlocal boundary condition

Figures & data

Table 1. Relative errors of v(x,t) for Example 1; N=40,τ=1/1000 and regular grid points

Table 2. Relative errors of v(x,t), for Example 1; N=40,τ=1/1000 and CGL grid points

Table 3. Relative errors of source parameter r(t) for Example 1; N=40,τ=1/1000 and regular grid points

Table 4. Relative errors of source parameter r(t) for Example 1; N=40,τ=1/1000 and CGL grid points

Table 5. Relative errors of v(x,t) for Example 2; N=15,τ=1/5000 and regular grid points

Table 6. Relative errors of v(x,t) for Example 2; N=15,τ=1/5000 and CGL grid points

Table 7. Relative errors of source parameter p(t) for Example 2; N=15,τ=1/5000 and regular grid points

Table 8. Relative errors of source parameter p(t) for Example 2; N=15,τ=1/5000 and CGL grid points

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Table 1. Relative errors of $v (x, t)$ for Example 1; $N = 40, τ = 1 / 1000$ and regular grid points

Table 2. Relative errors of $v (x, t)$ , for Example 1; $N = 40, τ = 1 / 1000$ and CGL grid points

Table 3. Relative errors of source parameter $r (t)$ for Example 1; $N = 40, τ = 1 / 1000$ and regular grid points

Table 4. Relative errors of source parameter $r (t)$ for Example 1; $N = 40, τ = 1 / 1000$ and CGL grid points

Table 5. Relative errors of $v (x, t)$ for Example 2; $N = 15, τ = 1 / 5000$ and regular grid points

Table 6. Relative errors of $v (x, t)$ for Example 2; $N = 15, τ = 1 / 5000$ and CGL grid points

Table 7. Relative errors of source parameter $p (t)$ for Example 2; $N = 15, τ = 1 / 5000$ and regular grid points

Table 8. Relative errors of source parameter $p (t)$ for Example 2; $N = 15, τ = 1 / 5000$ and CGL grid points