The meshfree strong form methods for solving one dimensional inverse Cauchy-Stefan problem

Abstract

In this paper, we extend the application of meshfree node based schemes for solving one-dimensional inverse Cauchy-Stefan problem. The aim is devoted to recover the initial and boundary conditions from some Cauchy data lying on the admissible curve s(t) as the extra overspecifications. To keep matters simple, the problem has been considered in one dimensional, however the physical domain of the problem is supposed as an irregular bounded domain in $\mathbb {R}^2$. The methods provide the space-time approximations for the heat temperature derived by expanding the required approximate solutions using collocation scheme based on radial point interpolation method (RPIM). The proposed method makes appropriate shape functions which possess the important Delta function property to satisfy the essential conditions automatically. In addition, to conquer the ill-posedness of the problem, particular optimization technique has been applied for solving the system of equations $Ax=b$ in which A is a nonsymmetric stiffness matrix. As the consequences, reliable approximate solutions are obtained which continuously depend on input data.

An iterative scheme based on the Galperin-Zheng minimization for solving the inverse Cauchy-Stefan problem

First of all, consider the set of free parameters $\mathbf {Q}=\{\mathbf {X},\mathbf {Y},u_j,\epsilon _j\}$ with the fixed values $0<\eta ,\{\omega _i\}_{i=1}^3<1,~\sum _{i=1}^3\omega _i=1$, where $\mathbf {X},\mathbf {Y}$ are the center and collocation points belong to $\Omega$. Also, $u_j$’s are the approximation coefficients introduced in equation (2.13) and $\epsilon _j$’s are the fuzzy shape parameters generated by using the formula

$$\begin{aligned} \epsilon _j=\epsilon _{\min }^2\big (\frac{\epsilon _{\max }}{\epsilon _{\min }}\big )^{\frac{2(j-1)}{N-1}},\quad j=\overline{1,N}. \end{aligned}$$

(6.1)

The space-time RBF’s find the fuzzy form:

$$\begin{aligned} R_j(x,t)=(1-e^{-\epsilon _j^2(x-x_j)^2})(1-e^{-\epsilon _j^2(t-t_j)^2}),\quad j=\overline{1,N}. \end{aligned}$$

(6.2)

Second, based on the Galperin-Zheng minimization technique [89–93], define

$$\begin{aligned} \mathbf {L}_{\infty }^{weak}=\min _{Q} \mathcal{F}=\omega _1|a.u|+\omega _2|b.u-\beta |+\omega _3|b^{'}u-\gamma |, \end{aligned}$$

(6.3)

where

$$\begin{aligned} a.u= & {} \sum _{k=1}^na_ku_k,\quad a_k=\sum _{i=0}^nG_{i+1,k+1}^{-1}A_i+\alpha G_{n+4,k+1}^{-1},\\ b.u= & {} \sum _{k=1}^nb_ku_k,\quad a_k=\sum _{i=0}^nG_{i+1,k+1}^{-1}B_i+\sum _{j=0}^m G_{n+j+2,k+1}^{-1}C_j,\\ b^{'}.u= & {} \sum _{k=1}^nb_k^{'}u_k,\quad a_k=\sum _{i=0}^nG_{i+1,k+1}^{-1}B_i^{'}+\sum _{j=0}^m G_{n+j+2,k+1}^{-1}C_j^{'},\\ A_i= & {} \int \limits _0^T\int \limits _0^{s(t)}\{\frac{\partial R_i(x,t)}{\partial t}-\frac{\partial ^2 R_i(x,t)}{\partial x^2}\}dxdt,\\= & {} \int \limits _0^T\Bigg [s(t)-\frac{\sqrt{\pi }}{2\epsilon } \mathrm {erf}(\epsilon _i(s(t)-x_i))+\frac{\sqrt{\pi }}{2\epsilon } \mathrm {erf}(-\epsilon _ix_i)\Bigg ]\Bigg [2\epsilon _i^2(t-t_i)e^{-\epsilon _i^2(t-t_i)^2}\Bigg ]dt,\\ B_i= & {} \int \limits _0^TR_i(s(t),t)dt,\quad B_i^{'}=-\int \limits _0^T\frac{\partial R_i(s(t),t)}{\partial x}dt,\\ \alpha= & {} \int \limits _0^Ts(t)dt,\quad \beta =\int \limits _0^TA(t)dt,\quad \gamma =\int \limits _0^TB(t)dt,\\ C_j= & {} \int \limits _0^TP_j(s(t),t)dt= {\left\{ \begin{array}{ll} T,&{}j=0,\\ \alpha ,&{}j=1,\\ 1/2T^2,&{}j=2, \end{array}\right. } \\&~C_j^{'}=-\int \limits _0^T\frac{\partial P_j(s(t),t)}{\partial x}dt {\left\{ \begin{array}{ll} 0,&{}j=0,\\ -T,&{}j=1,\\ 0,&{}j=2, \end{array}\right. } \end{aligned}$$

and take

$$\begin{aligned} \mathbf {L}_{\infty }^{strong}=|\widehat{u}(x,t)-u_{exact}(x,t)|, \end{aligned}$$

(6.4)

where $u_{exact}(x,t),\widehat{u}(x,t)$ are the exact and approximate solutions for the problem obtained in Sect. 3. Finally, refer to the relations 6.3 and 6.4 as the weak and strong formulations, respectively. Now, we deal with the following steps to get the satisfactory results:

Step 1: Take the minimum number of center and collocation points $\{\mathbf {X},\mathbf {Y}\}$, use the fuzzy RBF’s $R_i(x,t)$ and solve the problem by employing the procedure given in section 3.

Step2: Insert the obtained $u_j$’s and apply them in the weak formulation 6.3.

Step 3: Report the value $\mathbf {L}_{\infty }^{weak}$ and stop the procedure if

$$\begin{aligned} \mathbf {L}_{\infty }^{weak}\le \eta . \end{aligned}$$

(6.5)

Otherwise, update the number of center and collocation points $\{\mathbf {X},\mathbf {Y}\}$, repeat the Step 1 and Step 2 until the inequality 6.5 is satisfied.

We implement the described technique and resolve the Example 1. The results are given in Table 3 show the applicability of the technique. It is worth mentioning that here we set $\omega _1=\omega _2=\omega _3=\frac{1}{3},~\eta =10^{-6}$ and utilize the numerical integration technique, namely, Gauss-Legendre with 40 nodes in the computations. It is seen that we obtain the acceptable results by taking the center points $Card(\mathbf X )=Card(\mathbf Y )\ge 50$.

Table 3 RMS errors of the Galperin-Zheng formulation.