Fractional Sensitivity Equation Method: Application to Fractional Model Construction

Abstract

Fractional differential equations provide a tractable mathematical framework to describe anomalous behavior in complex physical systems, yet they introduce new sensitive model parameters, i.e. derivative orders, in addition to model coefficients. We formulate a sensitivity analysis of fractional models by developing a fractional sensitivity equation method. We obtain the adjoint fractional sensitivity equations, in which we present a fractional operator associated with logarithmic-power law kernel. We further construct a gradient-based optimization algorithm to compute an accurate parameter estimation in fractional model construction. We develop a fast, stable, and convergent Petrov–Galerkin spectral method to numerically solve the coupled system of original fractional model and its corresponding adjoint fractional sensitivity equations.

Appendices

Proof of Lemma 1

Part A: $\sigma \in (0,1)$. We start from the $RL-PL$ definition, given in (8).

$$\begin{aligned} {}_{a}^{RL-LP}{\mathcal {D}}_{x}^{\sigma } u&= \frac{1}{{\varGamma }(1-\sigma )} \frac{d}{dx} \int _{a}^{x} \, (x - s)^{-\sigma } \, \log (x-s) \, u(s) \, ds, \, \text { (integrate by parts)} \nonumber \\&= \frac{1}{{\varGamma }(1-\sigma )} \frac{d}{dx} \left\{ \frac{u(s) \, (x-s)^{1-\sigma }}{(-\sigma +1)^2} (1-(-\sigma +1) \, \log (x-s)) \Bigg |_{s=a}^{s=x}\right. \nonumber \\&\left. \quad - \int _{a}^{x} \, \frac{(x - s)^{-\sigma +1}}{(-\sigma +1)^2} \, (1-(-\sigma +1) \, \log (x-s)) \, u'(s) \, ds \right\} ,\nonumber \\&= \frac{1}{{\varGamma }(1-\sigma )} \frac{d}{dx} \left\{ \frac{u(a) \, (x-a)^{1-\sigma }}{(-\sigma +1)^2} (1-(-\sigma +1) \, \log (x-a))\right. \nonumber \\&\left. \quad -\,\int _{a}^{x} \, \frac{(x - s)^{-\sigma +1}}{(-\sigma +1)^2} \, (1-(-\sigma +1) \, \log (x-s)) \, u'(s) \, ds \right\} ,\nonumber \\&=\frac{u(a)}{{\varGamma }(1-\sigma )} \, \frac{\log (x-a)}{(x-a)^{\sigma }} \nonumber \\&\quad +\, \frac{1}{{\varGamma }(1-\sigma )} \int _{a}^{x} \, \frac{\log (x-s)}{(x - s)^{-\sigma }} \, u'(s) \, ds, \, \text { (by Leibnitz rule)} \nonumber \\&= \frac{u(a)}{{\varGamma }(1-\sigma )} \, \frac{\log (x-a)}{(x-a)^{\sigma }} + {}_{a}^{C-LP}{\mathcal {D}}_{x}^{\sigma } u \end{aligned}$$

(78)

Part B: $\sigma \in (1,2)$. Similarly, we start from the $RL-PL$ definition, given in (8).

$$\begin{aligned}&{}_{a}^{RL-LP}{\mathcal {D}}_{x}^{\sigma } u = \frac{1}{{\varGamma }(2-\sigma )} \frac{d^2}{dx^2} \int _{a}^{x} \, (x - s)^{-\sigma +1} \, \log (x-s) \, u(s) \, ds, \, \text { (integrate by parts twice)} \nonumber \\&\quad = \frac{1}{{\varGamma }(2-\sigma )} \frac{d^2}{dx^2} \left\{ \frac{u(s) \, (x-s)^{-\sigma +2}}{(-\sigma +2)^2} (1-(-\sigma +2)\log (x-s)) \Bigg |_{s=a}^{s=x}\right. \nonumber \\&\left. \qquad - \frac{u'(s) \, (x-s)^{-\sigma +3}}{(-\sigma +2)^2(-\sigma +3)^2} \left( 1-2(-\sigma +3)+(-\sigma +3)(-\sigma +2)\log (x-s) \right) \Bigg |_{s=a}^{s=x}\right. \nonumber \\&\left. \qquad + \int _{a}^{x} \, \frac{(x-s)^{-\sigma +3}}{(-\sigma +2)^2(-\sigma +3)^2}\right. \left. \left( 1-2(-\sigma +3)+(-\sigma +3)(-\sigma +2)\log (x-s) \right) \, u''(s) \, ds \phantom {\left\{ \frac{u(s) \, (x-s)^{-\sigma +2}}{(-\sigma +2)^2} (1-(-\sigma +2)\log (x-s)) \Bigg |_{s=a}^{s=x}\right. } \right\} , \nonumber \\&\quad = \frac{1}{{\varGamma }(2-\sigma )} \frac{d^2}{dx^2} \left\{ \frac{u(a) \, (x-a)^{-\sigma +2}}{(-\sigma +2)^2} (1-(-\sigma +2)\log (x-a))\right. \nonumber \\&\left. \qquad - \frac{u'(a) \, (x-a)^{-\sigma +3}}{(-\sigma +2)^2(-\sigma +3)^2} \left( 1-2(-\sigma +3)+(-\sigma +3)(-\sigma +2)\log (x-a) \right) \right. \nonumber \\&\left. \qquad + \int _{a}^{x} \, \frac{(x-s)^{-\sigma +3}}{(-\sigma +2)^2(-\sigma +3)^2} \right. \left. \left( 1-2(-\sigma +3)+(-\sigma +3)(-\sigma +2)\log (x-s) \right) \, u''(s) \, ds \phantom {\left\{ \frac{u(s) \, (x-s)^{-\sigma +2}}{(-\sigma +2)^2} (1-(-\sigma +2)\log (x-s)) \Bigg |_{s=a}^{s=x}\right. } \right\} ,\nonumber \\&\quad = \frac{u(a)}{{\varGamma }(2-\sigma )} \frac{1+(-\sigma +1)\log (x-a)}{(x-a)^{\sigma }} + \frac{u'(a)}{{\varGamma }(2-\sigma )} \frac{\log (x-a)}{(x-a)^{\sigma -1}} \nonumber \\&\qquad + \frac{1}{{\varGamma }(2-\sigma )} \int _{a}^{x} \, (x-s)^{-\sigma +1} \, \log (x-s) \, u''(s) \, ds , \, \text { (by Leibnitz rule)} \nonumber \\&\quad = \frac{u(a)}{{\varGamma }(1-\sigma )} \frac{1+(-\sigma +1)\log (x-a)}{(x-a)^{\sigma }} + \frac{u'(a)}{{\varGamma }(1-\sigma )} \frac{\log (x-a)}{(x-a)^{\sigma -1}} + {}_{a}^{C-PL}{\mathcal {D}}_{x}^{\sigma } u. \end{aligned}$$

(79)

Proof of Lemma 2

In Lemma 2.1 in [49] and also in [64], it is shown that $\Vert \cdot \Vert _{{^l}H^{\sigma }_{}({\varLambda })}$ and $\Vert \cdot \Vert _{{^r}H^{\sigma }_{}({\varLambda })}$ are equivalent. Therefore, for $u \in H^{\sigma }_{}({\varLambda })$, there exist positive constants $C_1$ and $C_2$ such that

$$\begin{aligned} \Vert u \Vert _{{}H^{\sigma }_{}({\varLambda })} \le C_1 \Vert u \Vert _{{^l}H^{\sigma }_{}({\varLambda })}, \quad \Vert u \Vert _{{}H^{\sigma }_{}({\varLambda })} \le C_2 \Vert u \Vert _{{^r}H^{\sigma }_{}({\varLambda })}, \end{aligned}$$

(80)

which leads to

$$\begin{aligned} \Vert u \Vert _{{}H^{\sigma }_{}({\varLambda })}^2&\le C_1^2 \Vert u \Vert _{{^l}H^{\sigma }_{}({\varLambda })}^2 + C_2^2 \Vert u \Vert _{{^r}H^{\sigma }_{}({\varLambda })}^2 , \nonumber \\&= C_1^2 \,\Vert {}_{a}^{}{\mathcal {D}}_{x}^{\sigma }\, (u)\Vert _{L^2({\varLambda })}^2+ C_2^2 \,\Vert {}_{x}^{}{\mathcal {D}}_{b}^{\sigma }\, (u)\Vert _{L^2({\varLambda })}^2+(C_1^2+C_2^2)\, \Vert u \Vert _{L^2({\varLambda })}^2 , \nonumber \\&\le \tilde{C}_1 \,\Vert u \Vert _{{^c}H^{\sigma }_{}({\varLambda })}^2, \end{aligned}$$

(81)

where $\tilde{C}_1$ is a positive constant. Similarly, we can show that $\Vert u \Vert _{{^c}H^{\sigma }_{}({\varLambda })}^2 \le \tilde{C}_2 \, \Vert u \Vert _{{}H^{\sigma }_{}({\varLambda })}$, where $\tilde{C}_2$ is a positive constant.

Proof of Lemma 4

$\mathcal {X}_1$ is endowed with the norm $\Vert \cdot \Vert _{\mathcal {X}_1}$, where $\Vert \cdot \Vert _{\mathcal {X}_1}\equiv \Vert \cdot \Vert _{{^c}H^{\beta _1/2}_{}({\varLambda }_1)}$ by Lemma 2. Moreover, $\mathcal {X}_2$ is associated with the norm

$$\begin{aligned} \Vert \cdot \Vert _{\mathcal {X}_2} \equiv \left\{ \Vert \cdot \Vert _{{^c}H^{\beta _2/2}_0 ((a_2,b_2); L^2({\varLambda }_{1}))}^2 + \Vert \cdot \Vert _{ L^2((a_2,b_2); \mathcal {X}_{1})}^2 \right\} ^{\frac{1}{2}}, \end{aligned}$$

(82)

where

$$\begin{aligned}&\Vert u \Vert _{{^c}H^{\beta _2/2}_0 ((a_2,b_2); L^2({\varLambda }_{1}))}^2 \nonumber \\ {}&\quad = \int _{a_1}^{b_1}\left( \int _{a_2}^{b_2}\, \vert {}_{a_2}^{}{\mathcal {D}}_{x_2}^{\beta _2/2} u \vert ^2 \, dx_2 + \int _{a_2}^{b_2}\, \vert {}_{x_2}^{}{\mathcal {D}}_{b_2}^{\beta _2/2} u \vert ^2 \, dx_2 + \int _{a_2}^{b_2}\, \vert u \vert ^2 \, dx_2 \right) \,dx_1 \nonumber \\&\quad = \int _{a_1}^{b_1}\int _{a_2}^{b_2}\, \vert {}_{a_2}^{}{\mathcal {D}}_{x_2}^{\beta _2/2} u \vert ^2 \, dx_2 dx_1 {+} \int _{a_1}^{b_1}\int _{a_2}^{b_2}\, \vert {}_{x_2}^{}{\mathcal {D}}_{b_2}^{\beta _2/2} u \vert ^2 \, dx_2 dx_1 + \int _{a_1}^{b_1} \int _{a_2}^{b_2}\, \vert u \vert ^2 \, dx_2 dx_1 \nonumber \\&\quad = \Vert {}_{x_2}^{}{\mathcal {D}}_{b_2}^{\beta _2/2}\, (u)\Vert _{L^2({\varLambda }_d)}^2+\Vert {}_{a_2}^{}{\mathcal {D}}_{x_2}^{\beta _2/2}\, (u)\Vert _{L^2({\varLambda }_d)}^2+\Vert u \Vert _{L^2({\varLambda }_d)}^2, \end{aligned}$$

(83)

and

$$\begin{aligned}&\Vert u \Vert _{L^2((a_2,b_2); \mathcal {X}_{1})}^2\nonumber \\&\quad = \int _{a_2}^{b_2}\, \left( \int _{a_1}^{b_1}\, \vert {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2} u \vert ^2 \, dx_1 + \int _{a_1}^{b_1}\, \vert {}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2} u \vert ^2 \, dx_1 + \int _{a_1}^{b_1}\, \vert u \vert ^2 \, dx_1 \right) \,dx_2 \nonumber \\&\quad = \int _{a_2}^{b_2}\int _{a_1}^{b_1} \vert {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2} u \vert ^2 dx_1 dx_2 + \int _{a_2}^{b_2}\int _{a_1}^{b_1} \vert {}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2} u \vert ^2 dx_1 dx_2 + \int _{a_2}^{b_2}\int _{a_1}^{b_1} \vert u \vert ^2 dx_1 dx_2\nonumber \\&\quad = \Vert {}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2}\, u\Vert _{L^2({\varLambda }_2)}^2+\Vert {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2}\, u\Vert _{L^2({\varLambda }_2)}^2+\Vert u \Vert _{L^2({\varLambda }_2)}^2. \end{aligned}$$

(84)

We use the mathematical induction to carry out the proof. Therefore, we assume the following equality holds

$$\begin{aligned} \Vert \cdot \Vert _{\mathcal {X}_{k-1}} \equiv \left\{ \sum _{i=1}^{k-1} \left( \Vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (\cdot )\Vert _{L^2({\varLambda }_{k-1})}^2+\Vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (\cdot )\Vert _{L^2({\varLambda }_{k-1})}^2 \right) + \Vert \cdot \Vert _{L^2({\varLambda }_{k-1})}^2 \right\} ^{\frac{1}{2}}. \end{aligned}$$

(85)

Since,

$$\begin{aligned}&\Vert u \Vert _{{^c}H^{\beta _k/2}_0 ((a_k,b_k); L^2({\varLambda }_{k-1}))}^2 \\&\quad = \int _{{\varLambda }_{k-1}}^{}\, \left( \int _{a_k}^{b_k}\, \vert {}_{a_k}^{}{\mathcal {D}}_{x_k}^{\beta _k/2} u \vert ^2 \, dx_k + \int _{a_k}^{b_k}\, \vert {}_{x_k}^{}{\mathcal {D}}_{b_k}^{\beta _k/2} u \vert ^2 \, dx_k + \int _{a_k}^{b_k}\, \vert u \vert ^2 \, dx_k \right) \,d{\varLambda }_{k-1} \\&\quad = \int _{{\varLambda }_{k-1}}^{}\int _{a_k}^{b_k}\, \vert {}_{a_k}^{}{\mathcal {D}}_{x_k}^{\beta _k/2} u \vert ^2 \, dx_k d{\varLambda }_{k-1} + \int _{{\varLambda }_{k-1}}^{}\int _{a_k}^{b_k}\, \vert {}_{x_k}^{}{\mathcal {D}}_{b_k}^{\beta _k/2} u \vert ^2 \, dx_k d{\varLambda }_{k-1} \\&\qquad + \int _{{\varLambda }_{k-1}}^{}\int _{a_k}^{b_k}\, \vert u \vert ^2 \, dx_k d{\varLambda }_{k-1} \nonumber \\&\quad = \Vert {}_{x_k}^{}{\mathcal {D}}_{b_k}^{\beta _k/2}\, (u)\Vert _{L^2({\varLambda }_k)}^2+\Vert {}_{a_k}^{}{\mathcal {D}}_{x_k}^{\beta _k/2}\, (u)\Vert _{L^2({\varLambda }_k)}^2+\Vert u \Vert _{L^2({\varLambda }_k)}^2, \end{aligned}$$

and

$$\begin{aligned} \Vert u \Vert _{L^2((a_k,b_k); \mathcal {X}_{k-1})}^2&= \int _{a_k}^{b_k} \left( \sum _{i=1}^{k-1} \left( \int _{{\varLambda }_{k-1}}^{} \vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} u \vert ^2 d{\varLambda }_{k-1} + \int _{{\varLambda }_{k-1}}^{} \vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2} u \vert ^2 d{\varLambda }_{k-1} \right) \right. \\&\quad \left. +\, \int _{{\varLambda }_{k-1}}^{} \vert u \vert ^2 d{\varLambda }_{k-1}\phantom {\left( \sum _{i=1}^{k-1} \left( \int _{{\varLambda }_{k-1}}^{} \vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} u \vert ^2 d{\varLambda }_{k-1} + \int _{{\varLambda }_{k-1}}^{} \vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2} u \vert ^2 d{\varLambda }_{k-1} \right) \right. } \right) dx_k \\&= \sum _{i=1}^{k-1} \left( \int _{{\varLambda }_k}^{} \vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} u \vert ^2 d{\varLambda }_k + \int _{{\varLambda }_k}^{} \vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2} u \vert ^2 d{\varLambda }_k \right) + \int _{{\varLambda }_k}^{} \vert u \vert ^2 d{\varLambda }_k\\&= \sum _{i=1}^{k-1} \left( \Vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, u\Vert _{L^2({\varLambda }_k)}^2+\Vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, u\Vert _{L^2({\varLambda }_k)}^2 \right) +\Vert u \Vert _{L^2({\varLambda }_k)}^2, \end{aligned}$$

we can show that

$$\begin{aligned} \Vert \cdot \Vert _{\mathcal {X}_{k}} \equiv \left\{ \sum _{i=1}^{k} \left( \Vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (\cdot )\Vert _{L^2({\varLambda }_{k})}^2+\Vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (\cdot )\Vert _{L^2({\varLambda }_{k})}^2 \right) + \Vert \cdot \Vert _{L^2({\varLambda }_{k})}^2 \right\} ^{\frac{1}{2}}. \end{aligned}$$

(86)

Proof of Lemma 7

According to [57], we have ${}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i} u={}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} ({}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} u)$ and ${}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2} u={}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}({}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2} u)$. Let $\bar{u}={}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} u$. Then,

$$\begin{aligned}&\left( {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i} u,v\right) _{{\varLambda }_d}= \left( {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} \bar{u},v\right) _{{\varLambda }_d}=\int _{{\varLambda }_d}^{} \frac{1}{{\varGamma }(1-\beta _i/2)} \left[ \frac{d}{dx_i}\, \int _{a_i}^{x_i}\frac{\bar{u}(s)\,ds}{(x_i-s)^\beta _i/2} \right] v\,d{\varLambda }_d \nonumber \\&\quad = \left\{ \frac{v}{{\varGamma }(1-\beta _i/2)\int _{a_i}^{x_i} \frac{\bar{u}ds}{(x_i-s)^{\beta _i/2}}} \right\} ^{b_i}_{x_i=a_i} - \int _{{\varLambda }_d}^{} \frac{1}{{\varGamma }(1-\beta _i/2)}\int _{a_i}^{x_i}\frac{\bar{u}(s)\,ds}{(x_i-s)^{\beta _i/2}} \frac{dv}{dx_i}\, d{\varLambda }_d.\nonumber \\ \end{aligned}$$

(87)

Based on the homogeneous boundary conditions, $\left\{ \frac{v}{{\varGamma }(1-\beta _i/2)\int _{a_i}^{x_i} \frac{\bar{u}ds}{(x_i-s)^{\beta _i/2}}} \right\} ^{b_i}_{x_i=a_i}=0.$ Therefore,

$$\begin{aligned} \left( {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i} u,v\right) _{{\varLambda }_d}= & {} -\int _{{\varLambda }_i}^{}\frac{1}{{\varGamma }(1-\beta _i/2)}\int _{a_i}^{x_i}\frac{\bar{u}(s)\,ds}{(x_i-s)^{\beta _i/2}} \frac{dv}{dx_i}\, d{\varLambda }_i. \end{aligned}$$

(88)

Moreover, we find that

$$\begin{aligned}&\frac{d}{ds}\, \int _{a_i}^{b_i}\frac{u}{(x_i-s)^{\beta _i/2}}dx_i\nonumber \\&\quad = \frac{d}{ds} \left\{ \left\{ \frac{v\, (x_i-s)^{1-\beta _i/2}}{1-\beta _i/2}\right\} _{x_i=s_i}^{b_i}-\frac{1}{1-\beta _i/2}\int _{s}^{b_i}\frac{dv}{dx_i}(x_i-s)^{1-\beta _i/2}dx_i \right\} \nonumber \\&\quad = -\frac{1}{1-\beta _i/2} \int _{s}^{b_i} \frac{dv}{dx_i}(x_i-s)^{1-\beta _i/2}\, dx_i = \int _{s}^{b_i} \frac{\frac{dv}{dx_i}}{(x_i-s)^{\beta _i/2}}\, dx_i. \end{aligned}$$

(89)

Therefore, we get

$$\begin{aligned}&\left( {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} \bar{u},v\right) _{{\varLambda }_d}\nonumber \\&\quad =-\int _{{\varLambda }_d}^{} \frac{1}{{\varGamma }(1-\nu )_i}\, \bar{u}(s) \left( -\frac{d}{ds} \int _{s}^{b_i}\frac{v}{(x_i-s)^{\beta _i/2}}dx_i\right) \,ds=\left( \bar{u},{}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}v\right) _{{\varLambda }_d}. \quad \end{aligned}$$

Proof of Lemma 8

We know that

$$\begin{aligned} \left| \left( {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u, {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \right) _{{\varOmega }} \right| = \left( \int _{{\varLambda }_d}^{} \int _{0}^{T} \left| {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u \, {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \right| ^2\, dt d{\varLambda }_d \right) ^{\frac{1}{2}}. \end{aligned}$$

Therefore, by Hölder inequality

$$\begin{aligned}&\left| \left( {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u, {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \right) _{{\varOmega }} \right| \\&\quad \le \left( \int _{{\varLambda }_d}^{} \int _{0}^{T} \vert {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u \vert ^2\, dt d{\varLambda }_d \right) ^{\frac{1}{2}} \, \left( \int _{{\varLambda }_d}^{} \int _{0}^{T} \vert {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \vert ^2\, dt d{\varLambda }_d \right) ^{\frac{1}{2}} \\&\quad \le \left( \int _{{\varLambda }_d}^{} \int _{0}^{T} \vert {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u \vert ^2\, dt d{\varLambda }_d + \int _{{\varLambda }_d}^{} \int _{0}^{T} \vert u \vert ^2\, dt d{\varLambda }_d \right) ^{\frac{1}{2}} \\&\qquad \times \left( \int _{{\varLambda }_d}^{} \int _{0}^{T} \vert {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \vert ^2\, dt d{\varLambda }_d + \int _{{\varLambda }_d}^{} \int _{0}^{T} \vert v \vert ^2\, dt d{\varLambda }_d \right) ^{\frac{1}{2}} \\&\quad = \Vert {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u \Vert _{L^2({\varOmega })} \, \Vert {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \Vert _{L^2({\varOmega })} = \Vert u \Vert _{{}_{}^{l}H^{\alpha /2}(I; L^2({\varLambda }_d))} \, \Vert v \Vert _{{}_{}^{r}H^{\alpha /2}(I; L^2({\varLambda }_d))}. \end{aligned}$$

Moreover, by equivalence of $\vert \cdot \vert _{H^{s}(I)} \equiv \vert \cdot \vert ^{*}_{H^{s}(I)} = \vert \cdot \vert ^{1/2}_{{^l}H^{s}(I)} \vert \cdot \vert ^{1/2}_{{^r}H^{s}(I)}$ we have

$$\begin{aligned} \left| ({}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u, {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v)_{I}\right|= & {} \int _{0}^{T} \left| {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u \, {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \right| ^2\, dt \nonumber \\\ge & {} \int _{0}^{T} \left| {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u \right| ^2 dt \, \int _{0}^{T} \left| {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \right| ^2\, dt \ge \tilde{\beta }_1 \Vert u \Vert _{{^l}H^{s}(I)} \Vert v \Vert _{{^r}H^{s}(I)},\nonumber \\ \end{aligned}$$

(90)

where $0<\tilde{\beta }_1\le 1$. Therefore,

$$\begin{aligned} \left| ({}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u, {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v)_{{\varOmega }} \right| ^2= & {} \int _{{\varLambda }_d}^{} \int _{0}^{T} \left| {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u \, {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \right| ^2\, dt \, d{\varLambda }_d \nonumber \\\ge & {} \int _{{\varLambda }_d}^{} \left( \int _{0}^{T} \vert {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u \vert ^2 dt \, \int _{0}^{T} \vert {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \vert ^2\, dt\right) \, d{\varLambda }_d \nonumber \\\ge & {} \bar{\beta } \int _{{\varLambda }_d}^{} \int _{0}^{T} \vert {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u \vert ^2 dt d{\varLambda }_d\, \int _{{\varLambda }_d}^{} \int _{0}^{T} \vert {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v \vert ^2\, dt \, {\varLambda }_d \nonumber \\\ge & {} \bar{\beta } \tilde{\beta }_2 \Vert u \Vert _{{^l}H^{s}(I)} \Vert v \Vert _{{^r}H^{s}(I)}, \end{aligned}$$

(91)

where $0<\tilde{\beta }_2\le 1$ and $0<\bar{\beta }$.

Proof of the Stability Theorem 5

Part A: $d=1$. It is evident that u and v are in Hilbert spaces (see [49, 64]). For $0< \tilde{\beta } \le 1$, we have

$$\begin{aligned} \vert a(u,v)\vert= & {} \left| \left( {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2}\, (u),{}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2}\, (v)\right) _{{\varOmega }} + \left( {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2}\, (u),{}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2}\, (v)\right) _{{\varOmega }} \right. \\&\left. \quad + \left( {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2}\, (u),{}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2}\, (v)\right) _{{\varOmega }}+(u,v)_{{\varOmega }}\right| \\\ge & {} \tilde{\beta } \left( \left| \left( {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2}\, (u),{}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2}\, (v)\right) _{{\varOmega }}\right| + \left| \left( {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2}\, (u),{}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2}\, (v)\right) _{{\varOmega }}\right| \right. \\&\left. \quad +\left| \left( {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2}\, (u),{}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2}\, (v)\right) _{{\varOmega }}\right| +\vert (u,v)_{{\varOmega }}\vert \right) , \end{aligned}$$

since ${\sup }_{{u \in U}} \vert a(u , v)\vert >0$. Next, by equivalence of spaces and their associated norms, (63), and (64), we obtain

$$\begin{aligned} \left| \left( {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2}\, (u),{}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2}\, (v)\right) _{{\varOmega }}\right|\ge & {} C_1 \Vert {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u\Vert _{L^2({\varOmega })} \, \Vert {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v\Vert _{L^2({\varOmega })}, \\ \left| \left( {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2}\, (u),{}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2}\, (v)\right) _{{\varOmega }} \right|\ge & {} C_2 \Vert {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2} u \Vert _{L^2({\varOmega })}\, \Vert {}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2} v \Vert _{L^2({\varOmega })}, \quad \end{aligned}$$

and

$$\begin{aligned} \left| \left( {}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2}\, (u),{}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2}\, (v)\right) _{{\varOmega }}\right| \ge C_3 \Vert {}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2} u \Vert _{L^2({\varOmega })} \, \Vert {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2} v \Vert _{L^2({\varOmega })}, \end{aligned}$$

(92)

where $C_1$, $C_2$, and $C_3$ are positive constants. Therefore,

$$\begin{aligned} \vert a(u,v)\vert\ge & {} \tilde{C} \tilde{\beta } \left\{ \Vert {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} u\Vert _{L^2({\varOmega })} \, \Vert {}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} v\Vert _{L^2({\varOmega })} + \Vert {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2} u \Vert _{L^2({\varOmega })}\, \Vert {}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2} v \Vert _{L^2({\varOmega })} \right. \nonumber \\&\quad \left. + \Vert {}_{a_1}^{}{\mathcal {D}}_{x_1}^{\beta _1/2} u \Vert _{L^2({\varOmega })} \, \Vert {}_{x_1}^{}{\mathcal {D}}_{b_1}^{\beta _1/2} v \Vert _{L^2({\varOmega })} \right\} , \end{aligned}$$

(93)

where $\tilde{C}$ is $min\{C_1, \, C_2, \, C_3 \}$. Also, the norm $\Vert u \Vert _{U} \, \Vert v \Vert _{V}$ is equivalent to the right hand side of inequality (93). Therefore, $\vert a(u,v)\vert \ge C \, \Vert u \Vert _{U}\Vert v \Vert _{V}$.

Part B: $d > 1$. Similarly, we have

$$\begin{aligned} \vert a(u,v)\vert&\ge \beta \left( \left| \left( {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2} (u),{}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2} (v)\right) _{{\varOmega }}\right| \right. \nonumber \\&\left. \quad + \sum _{i=1}^{d} \left( \left| \left( {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} (u),{}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2} (v)\right) _{{\varOmega }}\right| +\left| \left( {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2} (u),{}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2} (v)\right) _{{\varOmega }}\right| \right) \right) , \end{aligned}$$

(94)

where $0< \beta \le 1$. Recalling that as the direct consequences of (63), we obtain

$$\begin{aligned} \left| \left( {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (u),{}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (v)\right) _{{\varOmega }} \right|&\equiv \Vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (u) \Vert _{L^2({\varOmega })} \, \Vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (v)\Vert _{L^2({\varOmega })}, \\ \left| \left( {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (u),{}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (v)\right) _{{\varOmega }} \right|&\equiv \Vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (u) \Vert _{L^2({\varOmega })} \, \Vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (v)\Vert _{L^2({\varOmega })}. \end{aligned}$$

Thus,

$$\begin{aligned}&\sum _{i=1}^{d} \left( \left| \left( {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (u),{}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (v)\right) _{{\varOmega }} \right| +\left| \left( {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (u),{}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (v)\right) _{{\varOmega }} \right| \right) , \nonumber \\&\quad \ge \tilde{C} \sum _{i=1}^{d} \left( \Vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (u) \Vert _{L^2({\varOmega })} \, \Vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (v)\Vert _{L^2({\varOmega })} + \Vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (u) \Vert _{L^2({\varOmega })} \, \Vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (v)\Vert _{L^2({\varOmega })}\right) , \nonumber \\&\quad \ge \tilde{C}_1 \, \tilde{\beta } \sum _{i=1}^{d} \left( \Vert {}_{a_i}^{}{\mathcal {D}}_{x_i}^{\beta _i/2}\, (u) \Vert _{L^2({\varOmega })} + \Vert {}_{x_i}^{}{\mathcal {D}}_{b_i}^{\beta _i/2}\, (u) \Vert _{L^2({\varOmega })} \right) \nonumber \\&\qquad \times \sum _{j=1}^{d} \left( \Vert {}_{x_j}^{}{\mathcal {D}}_{b_j}^{\nu _j}\, (v)\Vert _{L^2({\varOmega })}, + \Vert {}_{a_j}^{}{\mathcal {D}}_{x_j}^{\nu _j}\, (v)\Vert _{L^2({\varOmega })}\right) , \end{aligned}$$

(95)

for $u,\, v \in L^2(I; \mathcal {X}_d)$, where $0<\tilde{C}$ and $0<\tilde{\beta }\le 1$. Furthermore, Lemma 8 yields

$$\begin{aligned} \left| \left( {}_{0}^{}{\mathcal {D}}_{t}^{\alpha /2}(u),{}_{t}^{}{\mathcal {D}}_{T}^{\alpha /2}(v)\right) _{{\varOmega }}\right| \equiv \Vert u \Vert _{{}_{}^{r}H^{\alpha /2}(I;L^2({\varLambda }_d))} \, \, \Vert v \Vert _{{}_{}^{l}H^{\alpha /2}(I;L^2({\varLambda }_d))}. \end{aligned}$$

(96)

Therefore, from (95) and (96) we have

$$\begin{aligned} \vert a(u,v)\vert \ge \beta \left( \Vert u \Vert _{{}_{}^{r}H^{\alpha /2}(I; L^2({\varLambda }_d))} \, \, \Vert v \Vert _{{}_{}^{l}H^{\alpha /2}(I;L^2({\varLambda }_d))} + \Vert u \Vert _{L^2(I; \mathcal {X}_d)} \, \Vert v \Vert _{L^2(I; \mathcal {X}_d)}\right) , \end{aligned}$$

(97)

where

$$\begin{aligned}&\Vert u \Vert _{{}_{}^{r}H^{\alpha /2}(I; L^2({\varLambda }_d))} \, \, \Vert v \Vert _{{}_{}^{l}H^{\alpha /2}(I;L^2({\varLambda }_d))} + \Vert u \Vert _{L^2(I; \mathcal {X}_d)} \, \Vert v \Vert _{L^2(I; \mathcal {X}_d)} \nonumber \\&\quad \ge \tilde{C}_2 \left( \Vert u \Vert _{{}_{}^{r}H^{\alpha /2}(I; L^2({\varLambda }_d))} + \Vert u \Vert _{L^2(I; \mathcal {X}_d)} \right) \left( \Vert v \Vert _{{}_{}^{l}H^{\alpha /2}(I;L^2({\varLambda }_d))}+\Vert v \Vert _{L^2(I; \mathcal {X}_d)}\right) \nonumber \\ \end{aligned}$$

(98)

for $u \in U$, $v \in U$ and $0<\tilde{C}_2\le 1$. By considering (97) and (98), we get

$$\begin{aligned} \vert a(u,v)\vert \ge C \, \Vert u \Vert _{U}\Vert v \Vert _{V}. \end{aligned}$$

(99)