
Citation: Jörg Fehr, Jan Heiland, Christian Himpe, Jens Saak. Best practices for replicability, reproducibility and reusability of computer-based experiments exemplified by model reduction software[J]. AIMS Mathematics, 2016, 1(3): 261-281. doi: 10.3934/Math.2016.3.261
[1] | Rajagopalan Ramaswamy, Gunaseelan Mani . Application of fixed point result to solve integral equation in the setting of graphical Branciari ℵ-metric spaces. AIMS Mathematics, 2024, 9(11): 32945-32961. doi: 10.3934/math.20241576 |
[2] | Muhammad Suhail Aslam, Mohammad Showkat Rahim Chowdhury, Liliana Guran, Isra Manzoor, Thabet Abdeljawad, Dania Santina, Nabil Mlaiki . Complex-valued double controlled metric like spaces with applications to fixed point theorems and Fredholm type integral equations. AIMS Mathematics, 2023, 8(2): 4944-4963. doi: 10.3934/math.2023247 |
[3] | Afrah Ahmad Noman Abdou . Chatterjea type theorems for complex valued extended b-metric spaces with applications. AIMS Mathematics, 2023, 8(8): 19142-19160. doi: 10.3934/math.2023977 |
[4] | Saif Ur Rehman, Iqra Shamas, Shamoona Jabeen, Hassen Aydi, Manuel De La Sen . A novel approach of multi-valued contraction results on cone metric spaces with an application. AIMS Mathematics, 2023, 8(5): 12540-12558. doi: 10.3934/math.2023630 |
[5] | Awais Asif, Sami Ullah Khan, Thabet Abdeljawad, Muhammad Arshad, Ekrem Savas . 3D analysis of modified F-contractions in convex b-metric spaces with application to Fredholm integral equations. AIMS Mathematics, 2020, 5(6): 6929-6948. doi: 10.3934/math.2020444 |
[6] | Yunpeng Zhao, Fei He, Shumin Lu . Several fixed-point theorems for generalized Ćirić-type contraction in Gb-metric spaces. AIMS Mathematics, 2024, 9(8): 22393-22413. doi: 10.3934/math.20241089 |
[7] | Abdullah Shoaib, Poom Kumam, Shaif Saleh Alshoraify, Muhammad Arshad . Fixed point results in double controlled quasi metric type spaces. AIMS Mathematics, 2021, 6(2): 1851-1864. doi: 10.3934/math.2021112 |
[8] | Monairah Alansari, Mohammed Shehu Shagari, Akbar Azam, Nawab Hussain . Admissible multivalued hybrid Z-contractions with applications. AIMS Mathematics, 2021, 6(1): 420-441. doi: 10.3934/math.2021026 |
[9] | Ahmed Alamer, Faizan Ahmad Khan . Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems. AIMS Mathematics, 2024, 9(3): 6266-6280. doi: 10.3934/math.2024305 |
[10] | Afrah Ahmad Noman Abdou . A fixed point approach to predator-prey dynamics via nonlinear mixed Volterra–Fredholm integral equations in complex-valued suprametric spaces. AIMS Mathematics, 2025, 10(3): 6002-6024. doi: 10.3934/math.2025274 |
In this paper, we are concerned with the motion of the supersonic potential flow in a two-dimensional expanding duct denoted by
f(0)>0,f′(x)>0,f′′(x)≥0,f′∞=limx→+∞f′(x)exists, | (1) |
and
φ(−y)=φ(y),φ(±f(0))=0,φ′(±f(0))=∓f′(0),φ′′≤0. | (2) |
At the inlet, the flow velocity is assumed to be along the normal direction of the inlet and its speed is given by
c∗<c1<q0(y)<c2<ˆq, | (3) |
where
vu=tanθ=±f′(x),ony=±f(x), | (4) |
where
The supersonic flow in the duct is described by the 2-D steady isentropic compressible Euler equations:
{∂∂x(ρu)+∂∂y(ρv)=0,∂∂x(ρu2+p)+∂∂y(ρuv)=0,∂∂x(ρuv)+∂∂y(ρv2+p)=0, | (5) |
where
∂u∂y=∂v∂x, | (6) |
then for polytropic gas, the following Bernoulli law holds
12q2+c2γ−1=12ˆq2, | (7) |
where
{(c2−u2)∂u∂x−uv(∂u∂y+∂v∂x)+(c2−v2)∂v∂y=0,∂u∂y−∂v∂x=0, | (8) |
Such problem (8) with initial data (3) and boundary condition (4) has already been studied by Wang and Xin in [25]. In their paper, by introducing the velocity potential
For better stating our results, we firstly give the description of domains
Our main results in the paper are:
Theorem 1.1. Assume that
|q′0(y)|≤−φ′′(y)1+(φ′(y))2q0c√q20−c2onΓin, | (9) |
then two alternative cases will happen in the duct, one contains vacuum, and the other does not. More concretely, the two cases are as follows:
(i) When vacuum actually appears in the duct, there exists a global solution
∂→nc2=0,onlup∪llow∖{M,N}, | (10) |
here
(ii) If vacuum is absent in the duct, the problem (8) with (3) and (4) has a global solution
Theorem 1.2. Assume that
f(0)>0,f′(0)>0,f′∞≥0,f′′(x)≤0. | (11) |
If
|q′0(y)|>−φ′′(y)1+(φ′(y))2q0c√q20−c2onΓin, | (12) |
then the
Remark 1. The condition (9) is equivalent to that the two Riemann invariants are both monotonically increasing along
{∂tu+u∂xu=0,u(0,x)=u0(x). | (13) |
As well known, if the initial data satisfies
minx∈Ru′0(x)<0, | (14) |
then the solution
{(∂x+λ−∂y)R+=0,(∂x+λ+∂y)R−=0,R±(0,y)=R0±(y). | (15) |
By the results of [16] and [30], there exists a global solution of the Cauchy problem (15) if and only if the two Riemann invariants
Remark 2. By the effect of expanding duct, as far as we know, it is hard to give a necessary and sufficient condition to ensure that vacuum must form at finite location in the duct. In Proposition 4.1, we will give a sufficient condition such that the vacuum will appear in finite place.
Remark 3. For the M-D compressible Euler equations, if the gases are assumed irrotational, by introducing the velocity potential
Remark 4. If the initial data contains a vacuum, especially for physical vacuum, the local well-posedness results of the compressible Euler equations have been studied in [8], [9], [12], [13], [18] and so on. But in general, such a local classical solution will blow up in finite time as shown in [3], [26] and the references therein.
Remark 5. For the case that Riemann invariants are both constants on
Remark 6. The symmetry of
The paper is organized as follows. In Section 2, we give some basic structures of the steady plane isentropic flow and discuss the monotonicity of two Riemann invariants
In this section, we start with some basic structures of the steady plane isentropic flow, which can be characterized by the Euler system (5).
Firstly, equation (8) can be written as the matrix form
(c2−u2−uv0−1)∂∂x(uv)+(−uvc2−v210)∂∂y(uv)=0. | (16) |
The characteristic equation of (16) is
(c2−u2)λ2+2uvλ+(c2−v2)=0. |
So for the supersonic flow, (8) is hyperbolic and we can get two eigenvalues
λ+=uv+c√u2+v2−c2u2−c2,λ−=uv−c√u2+v2−c2u2−c2. | (17) |
Correspondingly, the two families of characteristics in the
dy±dx=λ±. | (18) |
By standard computation, the Riemann invariants
R±=θ±F(q), | (19) |
where
{(∂x+λ−∂y)R+=0,(∂x+λ+∂y)R−=0, | (20) |
which can also be written as
∂±R∓=0 | (21) |
if we set
Let A be the Mach angle defined by
λ±=tan(θ±A). | (22) |
Under a proper coordinate rotation, the eigenvalues
From now on, we will discuss the monotonicity of the Riemann invariants
Lemma 2.1. For
|q′0(y)|≤−φ′′(y)1+(φ′(y))2q0c√q20−c2onΓin. | (23) |
Proof. Since
R′±(y)=−φ′′(y)1+(φ′(y))2±√q20−c2q0cq′0(y)onΓin. | (24) |
Thus
Due to Lemma 2.1, the following analysis will be focused on these two cases.
Case I.
Case II. At least one of
The next lemma shows the partial derivatives
Lemma 2.2. As the
∂λ+∂R−>0and∂λ−∂R+>0. | (25) |
Proof. By using (22) and the chain rule, it follows from direct computation that
∂λ+∂R−=(γ+1)q24(q2−c2)sec2(θ+A),∂λ−∂R+=(γ+1)q24(q2−c2)sec2(θ−A), | (26) |
thus (25) holds true for the supersonic flow.
In this section, the existence of the
Next we solve the solution in the border region
Let the
Lemma 3.1. If (9) holds true, then for the
Proof. We will prove the conclusion between these inner and border regions separately.
Step 1. Inner region
For any fixed two points
∂−R−=ddx(R−(x,y−(x)))≤0 | (27) |
along l in region
Step 2. Border region
Next we consider the solution in border region
R++R−=2arctanf′(x),f′′(x)≥0 | (28) |
on
Repeating this process in inner region
We can ensure that the hyperbolic direction is always in the x-direction by a proper rotation of coordinates if necessary. Without loss of generality, we may assume that
Lemma 3.2. If (9) holds true, then we have
Proof. In terms of the definition of
∂x=λ+∂−−λ−∂+λ+−λ−,∂y=∂+−∂−λ+−λ−. | (29) |
Thus we have
∂yR+=∂+R+λ+−λ−,∂yR−=−∂−R−λ+−λ−. | (30) |
Combining this with Lemma 3.1 yields the result.
Corollary 1. Under the assumption that
|∂+R+|=λ+−λ−√λ2−+1|∇R+|and|∂−R−|=λ+−λ−√λ2++1|∇R−|. | (31) |
Lemma 3.3. Let
Proof. It follows from (29) that
∂l=v−uλ−λ+−λ−∂++uλ+−vλ+−λ−∂−. | (32) |
which implies that
∂lq=v−λ−uλ+−λ−∂+q+λ+u−vλ+−λ−∂−q=u(vu−λ−)λ+−λ−∂+q+u(λ+−vu)λ+−λ−∂−q. | (33) |
Since
Corollary 2. Combining Lemma 3.3 and the Bernoulli law yields that the sound speed is monotonically decreasing along the streamline, that is to say
We now focus on the solution in the inner regions
Set
{(∂x+λ−∂y)R+=0,(∂x+λ+∂y)R−=0,inD2i−1R+(x,y)=W+(x,y)onAiCi−1,R−(x,y)=W−(x,y)onBiCi−1. | (34) |
By standard iteration method, the local existence of the problem (34) can be achieved, one can also see [4] and [15] for more details. In order to get the global solution to the problem (34) in the region
Lemma 3.4. If (9) holds true, then
Proof. It comes from the definition of
∂−∂+R+=∂−∂+R+−∂+∂−R+=(∂−λ+−∂+λ−)∂yR+. | (35) |
Due to
∂−λ+−∂+λ−=∂λ+∂R−∂−R−−∂λ−∂R+∂+R+≤0 | (36) |
and
In terms of Lemma 3.1, one has
Lemma 3.5. Suppose that
Proof. Firstly, we give the estimates of
For any point
|R+(x,y)|≤‖W+‖C(AiCi−1),|R−(x,y)|≤‖W−‖C(BiCi−1) | (37) |
in the region
Fix
x0>x1>...>xN−1>xN,xN=φ(yN). | (38) |
If
W+(x0,y0)=2arctanf′(x1)−R−(x1,y1)=2arctanf′(x1)−R−(x2,y2)=2arctanf′(x1)+2arctanf′(x2)+R+(x2,y2)=N∑i=12arctanf′(xi)−R−(xN,yN). | (39) |
Similarly, if
W+(x0,y0)=N∑i=12arctanf′(xi)+R+(xN,yN). | (40) |
Combining (39) and (40) yields that
‖W+‖C(AiCi−1)≤2Narctanf′∞+max{‖R+‖C(Γin),‖R−‖C(Γin)}. | (41) |
Analogously, the estimate of
Next, we give the estimates of
It follows from Lemma 3.4 that
|∂+R+(x,y)|≤‖∂+W+‖C(AiCi−1),|∂−R−(x,y)|≤‖∂−W−‖C(BiCi−1). | (42) |
As the process shown in the estimates of
v−uλ−λ+−λ−∂+R++uλ+−vλ+−λ−∂−R−=±2uf′′(x)1+(f′(x))2,ony=±f(x). | (43) |
By the process of the reflection shown in the estimates of
∂+W+(x0,y0)≤∂+R+(x1,y1)=(2uλ+−λ−v−uλ−f′′1+(f′)2−uλ+−vv−uλ−∂−R−)(x1,y1)≤(2uλ+−λ−v−uλ−f′′1+(f′)2)(x1,y1)−(uλ+−vv−uλ−)(x1,y1)×∂−R−(x2,y2)=(2uλ+−λ−v−uλ−f′′1+(f′)2)(x1,y1)+(uλ+−vv−uλ−)(x1,y1)×(2uλ+−λ−uλ+−vf′′1+(f′)2)(x2,y2)+(uλ+−vv−uλ−)(x1,y1)×(v−uλ−uλ+−v)(x2,y2)×∂+R+(x2,y2) | (44) |
Since
2uf′′1+(f′)2,uλ+−vv−uλ−,λ+−λ−v−uλ−,λ+−λ−uλ+−v |
are uniformly bounded. Denote the maximum one of these bounds as the constant G, thus (44) can be expressed as
∂+W+(x0,y0)≤∂+R+(x1,y1)≤G2−G∂−R−(x1,y1)≤G2−G∂−R−(x2,y2)≤G2+G3+G2∂+R+(x2,y2)≤N∑i=1Gi+1+GN∂+R+(xN,yN). | (45) |
Similarly, if
∂+W+(x0,y0)≤N∑i=1Gi+1−GN∂−R−(xN,yN). | (46) |
Combining (45), (46) and (31) yields that
‖∂+W+‖C(AiCi−1)≤N∑i=1Gi+1+GNmax{‖∂+R+‖C(Γin),‖∂−R−‖C(Γin)}. | (47) |
Analogously, the same estimate of
Thus, we complete the proof of Lemma 3.5.
Corollary 3. Based on Lemma 3.5, we have that
Proof. Substituting (31) into (47) gives that
\begin{equation} |\nabla R_{\pm}| = \frac{\sqrt{{\lambda}_{\mp}^2+1}}{{\lambda}_+-{\lambda}_-}|{\partial}_{\pm}R_{\pm}|\lesssim \frac{1}{c}\left(\sum\limits_{i = 1}^{N}G^{i+1}+G^{N} \max\{\|R_{+}\|_{C^{1}(\Gamma_{\text{in}})}, \|R_{-}\|_{C^{1}(\Gamma_{\text{in}})}\}\right). \end{equation} | (48) |
Combining the local existence and the Corollary 3 induces the global
We now turn to consider the border region
\begin{equation} \left\{\begin{aligned} &\begin{aligned} &(\partial_{x}+\lambda_{-}\partial_{y})R_{+} = 0, \\ &(\partial_{x}+\lambda_{+}\partial_{y})R_{-} = 0, \\ \end{aligned} \quad{\quad} {\quad}\text{in}\; D_{2i}\\ &R_{-}(x, y) = Z_{-}(x, y)\quad\quad\quad\; \text{on}\;A_{i}C_{i}, \\ &R_{+}+R_{-} = 2\arctan f^{\prime}(x)\quad \text{on}\;A_{i}A_{i+1}, \end{aligned} \right. \end{equation} | (49) |
where
Lemma 3.6. Suppose that
Proof. For any point
\begin{equation} |R_{-}(x, y)| \leq \| Z_{-}\|_{C(A_{i}C_{i})}, \; |R_{+}(x, y)| \leq 2\arctan f^{\prime}_{\infty}+\| Z_{-}\|_{C(A_{i}C_{i})}. \end{equation} | (50) |
in the region
\begin{equation} |\partial_{-}R_{-}(x, y)|\leq \| \partial_{-}Z_{-}\|_{C(A_{i}C_{i})} , \; |\partial_{+}R_{+}(x, y)| \leq \|\partial_{+}R_{+}\|_{C(A_{i}A_{i+1})}. \end{equation} | (51) |
Moreover, by (43), we have that
\begin{equation} \|\partial_{+}R_{+}\|_{C(A_{i}A_{i+1})}\leq G^2+G \|{\partial}_-R_-\|_{C(A_{i}A_{i+1})}\leq G^2+G\|\partial_{-}Z_{-}\|_{C(A_{i}C_{i})}. \end{equation} | (52) |
By the same process shown in Lemma 3.5, we know that
Corollary 4. Based on Lemma 3.6, we have that
Combining the local existence and the Corollary 4 induces the global
Theorem 3.7. For case I, there exists a global
\begin{equation} {\partial}_+R_+\geq 0, \quad {\partial}_-R_-\leq 0 \quad \mathit{\text{in}}\; (\overline{\Omega}_{non}\setminus\{M, N\}). \end{equation} | (53) |
Moreover, there exists a constant
\begin{equation} |{\partial}_+R_+|+|{\partial}_-R_-|\le C \quad \mathit{\text{in}}\; (\overline{\Omega}_{non}\setminus\{M, N\}). \end{equation} | (54) |
So far we have established the global
Now we begin to consider the problem in case Ⅱ. We will prove Theorem 1.2.
Proof of Theorem 1.2. Firstly, we consider the special case
Suppose that
Let
\begin{equation} \left\{ \begin{aligned} &\frac{d y_{-}^{(0)}(x)}{d x} = \lambda_{-}, \\ &y_{-}^{(0)}(x_{0}) = \varphi^{-1}(x_{0}). \end{aligned} \right. \end{equation} | (55) |
Then
\frac{{\partial}}{{\partial} x}\bigg(\frac{{\partial} R_+}{{\partial} y}\bigg)+{\lambda}_-\frac{{\partial}}{{\partial} y}\bigg(\frac{{\partial} R_+}{{\partial} y}\bigg)+\frac{{\partial} {\lambda}_-}{{\partial} R_-}\frac{{\partial} R_-}{{\partial} y}\frac{{\partial} R_+}{{\partial} y} = -\frac{{\partial} {\lambda}_-}{{\partial} R_+}\bigg(\frac{{\partial} R_+}{{\partial} y}\bigg)^2. |
Use the second equation in (20), we can get
\frac{\partial R_{-}}{\partial y} = \frac{\partial_{x}R_{-}+\lambda_{-}\partial_{y}R_{-}}{\lambda_{-}-\lambda_{+}}. |
Substitute it into the above identity to obtain that
\begin{equation} \frac{\partial}{\partial x}\bigg(\frac{\partial R_{+}}{\partial y}\bigg)+\lambda_{-}\frac{\partial}{\partial y}\bigg(\frac{\partial R_{+}}{\partial y}\bigg)+\frac{\partial \lambda_{-}}{\partial R_{-}}\frac{\partial_{x}R_{-}+\lambda_{-}\partial_{y}R_{-}}{\lambda_{-}-\lambda_{+}}\frac{\partial R_{+}}{\partial y} = -\frac{\partial \lambda_{-}}{\partial R_{+}}\bigg(\frac{\partial R_{+}}{\partial y}\bigg)^{2}. \end{equation} | (56) |
Define
\begin{equation} \frac{d}{d x}({H_{0}(x)\partial_{y}R_{+}})^{-1} = \frac{\partial \lambda_{-}}{\partial R_{+}}H_{0}^{-1}, \end{equation} | (57) |
where
If
\begin{equation} \frac{\partial R_{+}}{\partial y}(x_{1}, y_{1}) = \frac{\partial_{y} R_{+}(x_{0}, y_{0})}{H_{0}(x)(1+\partial_{y} R_{+}(x_{0}, y_{0})\int^{x_{1}}_{x_{0}}\frac{\partial \lambda_{-}}{\partial R^{+}}H_{0}^{-1}(s, y_{-}^{(0)}(s))ds)}. \end{equation} | (58) |
Since
\begin{equation} H_{0}(x) = \exp\{h_{0}(R_{-}(x, y^{(0)}_{-}(x))-h_{0}(R_{-}(x_{0}, y_{0})\}, \end{equation} | (59) |
and
Note that
\begin{equation} \frac{\partial R_{+}}{\partial y}(x_{1}, y_{1})\leq\frac{\partial_{y} R_{+}(x_{0}, y_{0})}{H_{0}(x)(1+\partial_{y} R_{+}(x_{0}, y_{0})K_{0}(x_{1}-x_{0}))}. \end{equation} | (60) |
By the boundary condition (43), one has that
\begin{equation} \partial_{y}R_{-}(x_{1}, y_{1}) = \bigg(\frac{v-u\lambda_{-}}{u\lambda_{+}-v}\partial_{y}R_{+}\bigg)(x_{1}, y_{1}) = \bigg(\frac{\cos(\theta+A)}{\cos(\theta-A)}\partial_{y}R_{+}\bigg)(x_{1}, y_{1}). \end{equation} | (61) |
The coefficient
\begin{equation} \partial_{y}R_{-}(x_{1}, y_{1}) < \partial_{y}R_{+}(x_{1}, y_{1}). \end{equation} | (62) |
Analogously, let
\begin{equation} \left\{ \begin{aligned} &\frac{d y_{+}^{(1)}(x)}{d x} = \lambda_{+}, \\ &y_{+}^{(1)}(x_{1}) = y_{1}. \end{aligned} \right. \end{equation} | (63) |
one can obtain the similar equation of
\begin{equation} \frac{d}{d x}({H_{1}(x)\partial_{y}R_{+}})^{-1} = \frac{\partial \lambda_{+}}{\partial R_{-}}H_{1}^{-1}, \end{equation} | (64) |
where
\begin{equation} \frac{\partial R_{-}}{\partial y}(x_{2}, y_{2}) = \frac{\partial_{y} R_{-}(x_{1}, y_{1})}{H_{1}(x)(1+\partial_{y} R_{-}(x_{1}, y_{1})\int^{x_{2}}_{x_{1}}\frac{\partial \lambda_{+}}{\partial R^{-}}H_{1}^{-1}(s, y_{+}^{(1)}(s))ds)}. \end{equation} | (65) |
Note that
\begin{equation} \frac{\partial R_{-}}{\partial y}(x_{2}, y_{2})\leq\frac{\partial_{y} R_{-}(x_{1}, y_{1})}{H_{1}(x)(1+\partial_{y} R_{-}(x_{1}, y_{1})K_{1}^{*}(x_{2}-x_{1}))}. \end{equation} | (66) |
By the same process and after series of reflection if necessary, the denominator of (66) will tend to
Next, we consider the case
The proof is very similar to the case
\begin{eqnarray} \big((v-u\lambda_{-})\partial_{y}R_{+}\big)(x_1, y_1)-\big((u\lambda_{+}-v)\partial_{y}R_{-}\big)(x_1, y_1){} \\ = -2u(x_1, y_1)\frac{f^{\prime\prime}(x_1)}{1+(f^{\prime}(x_1))^{2}}. \end{eqnarray} | (67) |
Thus we have
\begin{eqnarray} {\partial}_y R_-(x_1, y_1)& = &\bigg(\frac{v-u{\lambda}_-}{u{\lambda}_+-v}{\partial}_y R_+\bigg)(x_1, y_1){} \\ &+&\bigg(\frac{2u}{u{\lambda}_+-v}\bigg)(x_1, y_1)\times\frac{f''(x_1)}{1+(f'(x_1))^2}{} \\ & < &{\partial}_y R_+(x_1, y_1). \end{eqnarray} | (68) |
This is same as (62). So we omit the details of proof for the case
Therefore, we complete the proof of Theorem 1.2.
Due to the conservation of mass, vacuum can not appear in the inner regions of the duct. So if vacuum exists, it must locate on the two walls. On the other hand, since the two walls are the streamlines, Corollary 2 shows that the sound speed is monotonically decreasing along the two walls. When the sound speed decreases to zero, it corresponds to the formation of vacuum.
First, we list some properties of the vacuum boundary, one can also see [4] and [25] for more details.
Lemma 4.1. When the vacuum area actually appears, the first vacuum point M and N must form at
Based on Theorem 3.7, we have obtained the
\begin{equation} \begin{aligned} &\tilde{R}^{n}_{\pm}(y) = \tilde{R}_{\pm}(y), \; y\in [-y_{n}, y_{n}], \; f(x_{M})-y_{n} = \frac{1}{n}, \; n = 1, 2, ...\\ &|q^{\prime}(y)|\leq -\frac{\psi^{\prime\prime}(y)}{1+(\psi^{\prime}(y))^{2}}\frac{qc}{\sqrt{q^{2}-c^{2}}}, \quad y\in [-f(x_{M}), -y_{n}]\cup[y_{n}, f(x_{M})]. \end{aligned} \end{equation} | (69) |
Consider the problem in the region
Let
\begin{equation} R_{\pm}(x, y) = \lim\limits_{n\rightarrow\infty}R^{n}_{\pm}(x, y), \quad (x, y)\in {\Omega}_{vac}. \end{equation} | (70) |
To sum up, we get that
Theorem 4.2. Assume that
\begin{equation} \partial_{+}R_{+}\geq 0, \partial_{-}R_{-}\leq 0 {\quad}in{\quad}(\overline{{\Omega}_{vac}}\backslash\{l_{up}\cup l_{low}\}). \end{equation} | (71) |
Next, we consider the regularity near vacuum boundary. In fact, there are many results about physical vacuum singularity. A vacuum boundary (or singularity) is called physical if
\begin{equation} 0 < \left|\frac{{\partial} c^2}{{\partial} \vec{n}}\right| < +\infty \end{equation} | (72) |
in a small neighborhood of the boundary, where
Theorem 4.3. Let
Proof. For convenience, we only consider the case near the vacuum line
\begin{equation} \partial_{\vec{n}}c^{2} = -(\gamma-1)q\partial_{\vec{n}}q. \end{equation} | (73) |
By the relation
\begin{equation} \partial_{x}q = \frac{qc}{2\sqrt{q^{2}-c^{2}}}(\partial_{x}R_{+}-\partial_{x}R_{-}), \; \partial_{y}q = \frac{qc}{2\sqrt{q^{2}-c^{2}}}(\partial_{y}R_{+}-\partial_{y}R_{-}). \end{equation} | (74) |
Substituting it into (73) gives that
\begin{equation} \partial_{\vec{n}}c^{2} = -\frac{(\gamma-1)q^{2}c}{2\sqrt{q^{2}-c^{2}}}[(n_{2}-n_{1}\lambda_{-})\partial_{y}R_{+}+(n_{1}\lambda_{+}-n_{2})\partial_{y}R_{-}]. \end{equation} | (75) |
Combining this with (30) yields that
\begin{equation} \partial_{\vec{n}}c^{2} = -\frac{(\gamma-1)q^{2}(u^{2}-c^{2})}{4(q^{2}-c^{2})}[(n_{2}-n_{1}\lambda_{-})\partial_{+}R_{+}-(n_{1}\lambda_{+}-n_{2})\partial_{-}R_{-}]. \end{equation} | (76) |
Construct a line
\begin{equation} \partial_{-}\partial_{+}R_{+} = \frac{\partial_{-}\lambda_{+}-\partial_{+}\lambda_{-}}{\lambda_{+}-\lambda_{-}}\partial_{+}R_{+}. \end{equation} | (77) |
Integrating (77) from
\begin{equation} \partial_{+}R_{+}(S_{h}) = \partial_{+}R_{+}(P_{h})\exp\left \{\int^{S_{h}}_{P_{h}}\frac{\partial_{-}\lambda_{+}-\partial_{+}\lambda_{-}}{\lambda_{+}-\lambda_{-}}ds \right \}. \end{equation} | (78) |
This together with (26) yields that
\begin{equation} \partial_{+}R_{+}(S_{h}) = \partial_{+}R_{+}(P_{h})\exp\left \{\int^{S_{h}}_{P_{h}}\frac{\gamma+1}{8c}(-T_{1}\partial_{+}R_{+}+T_{2}\partial_{-}R_{-})ds \right \}, \end{equation} | (79) |
where
T_{1} = \frac{q^{2}(u^{2}-c^{2})\sec^{2}(\theta-A)}{(q^{2}-c^{2})^{\frac{3}{2}}}, \quad T_{2} = \frac{q^{2}(u^{2}-c^{2})\sec^{2}(\theta+A)}{(q^{2}-c^{2})^{\frac{3}{2}}}. |
Since
We claim that
We will use the proof by contradiction to obtain above assertions. Assume that
\begin{equation} \partial_{+}R_{+}(x, y)\geq \partial_{+}R_{+}(S_{\bar{g}_{n}}) \geq \frac{\bar{g}}{2}\quad \text{on} {\quad} (x, y)\in P_{\bar{g}_n}S_{\bar{g}_n}. \end{equation} | (80) |
By the fact that
\begin{eqnarray} \int^{S_{\bar{g}_{n}}}_{P_{\bar{g}_{n}}}\frac{\gamma+1}{8c}(-T_{1}\partial_{+}R_{+}+T_{2}\partial_{-}R_{-})ds &\leq& \int^{S_{\bar{g}_{n}}}_{N_{\bar{g}_{n}}}-\frac{\gamma+1}{8c}T_{1}\partial_{+}R_{+}ds{} \\ &\leq& -\frac{\gamma+1}{16c(N_{\bar{g}_{n}})}s_{0}g_{0}\bar{g}, \end{eqnarray} | (81) |
where
\begin{equation} \partial_{+}R_{+}(S_{\bar{g}_{n}})\leq \partial_{+}R_{+}(P_{\bar{g}_{n}})\exp\left \{-\frac{\gamma+1}{16c(N_{\bar{g}_{n}})}s_{0}g_{0}\bar{g}\right\}. \end{equation} | (82) |
Note that
\begin{equation} \overline{\lim\limits_{n\rightarrow \infty}}\partial_{+}R_{+}(S_{\bar{g}_{n}})\leqslant \partial_{+}R_{+}(P)\lim\limits_{n\rightarrow\infty}\exp\left \{-\frac{\gamma+1}{16c(N_{\bar{g}_{n}})}s_{0}g_{0}\bar{g}\right\} = 0, \end{equation} | (83) |
which is a contradiction. Thus, it follows
\begin{equation} \overline{\lim\limits_{h\rightarrow 0}}\partial_{+}R_{+}(S_{h}) = 0, {\quad} \varliminf\limits_{h\rightarrow 0}\partial_{-}R_{-}(S_{h}) = 0. \end{equation} | (84) |
This together with (71) gives that
\begin{equation} \lim\limits_{h\rightarrow 0}\partial_{+}R_{+}(S_{h}) = 0, {\quad} \lim\limits_{h\rightarrow 0}\partial_{-}R_{-}(S_{h}) = 0. \end{equation} | (85) |
Combining it with (76) yields that
\begin{equation} \lim\limits_{h\rightarrow 0}\frac{{\partial} c^2}{{\partial} \vec{n}}(S_h) = 0. \end{equation} | (86) |
Note that
\begin{equation} \frac{{\partial} c^2}{{\partial} \vec{n}} = 0, {\quad} \text{on}{\quad} l_{low}\setminus\{N\}. \end{equation} | (87) |
Similarly, one has that
\begin{equation} \frac{{\partial} c^2}{{\partial} \vec{n}} = 0, {\quad} \text{on}{\quad} l_{up}\setminus\{M\}. \end{equation} | (88) |
Thus, we prove the Theorem 4.2.
So far we have established the solution to the problem when vacuum actually appears. When it comes to the case that there is no vacuum in the duct, by the same prior estimates in Lemma 3.5 and Lemma 3.6, one only needs to check whether the duct can be covered by characteristics in finite steps.
Theorem 4.4. Assume that
Proof. It remains to check whether the duct can be covered by characteristics in finite steps. Since the duct is convex, then there exists
\begin{equation} |\tilde{C}_{i_{0}}\tilde{A}_{i_{0}+1}| > 0, \; |\tilde{B}_{i_{0}+1}\tilde{C}_{i_{0}+1}| > 0. \end{equation} | (89) |
Similarly, we can get that
\begin{equation} |\tilde{C}_{i_{0}+1}\tilde{B}_{i_{0}+2}| > 0, \; |\tilde{A}_{i_{0}+2}\tilde{C}_{i_{0}+2}| > 0. \end{equation} | (90) |
By the induction method, we have that
\begin{eqnarray} |\tilde{C}_{i_{0}+2(k-1)}\tilde{A}_{i_{0}+2k-1}| > 0, \; |\tilde{B}_{i_{0}+2k-1}\tilde{C}_{i_{0}+2k-1}| > 0, {} \\ |\tilde{C}_{i_{0}+2(k-1)}\tilde{B}_{i_{0}+2k}| > 0, \; |\tilde{A}_{i_{0}+2k}\tilde{C}_{i_{0}+2k}| > 0, \quad k = 1, 2, ... \end{eqnarray} | (91) |
Suppose that the image of the
\begin{equation} \angle C^{0}_{i_{0}}OC^{1}_{i_{0}} = \frac{\pi}{2\mu}, \quad \theta_{\tilde{B}_{i_{0}+1}} = \angle \tilde{C}_{i_{0}}O\tilde{B}_{i_{0}+1} > 0, \end{equation} | (92) |
where
\begin{equation} i\leq \lfloor\frac{\pi}{2\mu\theta_{\tilde{B}_{i_{0}+1}}}\rfloor, \end{equation} | (93) |
where
Proof of Theorem 1.1. Under the assumptions of Theorem 1.1, Theorem 1.1 comes from Theorem 3.1 and Theorem 4.1-4.3.
Finally, we give a sufficient condition to ensure that vacuum must form at finite location in the duct.
Proposition 1. If
Proof. We will use the proof by contradiction. Assume that there is no vacuum on
\begin{equation} \frac{dq}{ds} = \frac{d(R_{+}-R_{-})}{ds}\frac{1}{2F^{\prime}(q)} = \frac{qc}{2\sqrt{q^{2}-c^{2}}}(\partial_{x}(R_{+}-R_{-})+\tan\theta \partial_{y}(R_{+}-R_{-})). \end{equation} | (94) |
This together with (29) and (43) yields that
\begin{equation} \frac{dq}{ds} = \frac{qc}{\sqrt{q^{2}-c^{2}}}\left(\frac{\tan\theta-\lambda_{-}}{\lambda_{+}-\lambda_{-}}\partial_{+}R_{+}+\frac{f^{\prime\prime}(x)}{1+(f^{\prime}(x))^{2}}\right). \end{equation} | (95) |
By (95) and the Bernoulli law (7), we get that
\begin{equation} \frac{dc}{ds} = -\frac{(\gamma-1)q^{2}}{2\sqrt{q^{2}-c^{2}}} \left(\frac{\tan\theta-\lambda_{-}}{\lambda_{+}-\lambda_{-}}\partial_{+}R_{+}+\frac{f^{\prime\prime}(x)}{1+(f^{\prime}(x))^{2}}\right). \end{equation} | (96) |
Since
c_{B_{1}} = -\int^{+\infty}_{0}\frac{dc}{ds}\sqrt{1+(f^{\prime}(x))^{2}}dx, |
then
\begin{eqnarray} c_{B_{1}}&\geq& \int^{+\infty}_{0}\frac{(\gamma-1)q^{2}}{2\sqrt{q^{2}-c^{2}}}\frac{f^{\prime\prime}(x)}{1+(f^{\prime}(x))^{2}} \sqrt{1+(f^{\prime}(0))^{2}}dx{}\\ &\geq&\frac{q_{B_{1}}(\gamma-1)}{2}\sqrt{1+(f^{\prime}(0))^{2}}\int^{+\infty}_{0}-\frac{f^{\prime\prime}(x)}{1+(f^{\prime}(x))^{2}}dx{}\\ & = &\frac{q_{B_{1}}(\gamma-1)}{2}\sqrt{1+(f^{\prime}(0))^{2}}(\arctan f^{\prime}_{\infty}-\arctan f^{\prime}(0)){} \\ & > &c_{B_{1}}, \end{eqnarray} | (97) |
which is a contradiction. Thus, we complete the proof.
The authors would like to thank the referee for his (or her) very helpful suggestions and comments that lead to an improvement of the presentation.
[1] | D. H. Bailey, J. M. Borwein, and V. Stodden. Set the default to “open”. Notices of the AMS, 60(6): 679–680, 2013. URL http://dx:doi:org/10:1090/noti1014. |
[2] | D. H. Bailey, J. M. Borwein, and V. Stodden. Facilitating reproducibility in scientific computing:Principles and practice. In H. Atmanspacher and S. Maasen, editors, Reproducibility: Principles,Problems, Practices, and Prospects, pages 205–232. Wiley, 2016. ISBN 9781118864975. URL http://dx:doi:org/10:1002/9781118865064:ch9. |
[3] | W. Bangerth and T. Heister. Quo vadis, scientific software? SIAM News, 47(1), 2014. URL http://citeseerx:ist:psu:edu/viewdoc/summary?doi=10:1:1:636:5594. Accessed: |
[4] | 2016-09-22. |
[5] | 4. L. A. Barba. Why should i believe your supercomputing research? http://medium:com/@lorenaabarba/why-should-i-believe-your-supercomputing-researcha7cbf4cbc6b4#:anqkh3s, 2016. URL http://archive:is/2Zgx4. |
[6] | 5. N. Barnes. Publish your computer code: it is good enough. Nature, 4:753, 2010. URL http://dx:doi:org/10:1038/467753a. |
[7] | 6. P. Bourque and R. E. Fairley, editors. Guide to the Software Engineering Body of Knowledge(SWEBOK), Version 3.0. IEEE Computer Society, 2014. URL http://swebok:org. |
[8] | 7. B. Brembs. Earning credibility in post-factual science? bjoern:brembs:net/2016/02/earning-credibility-in-post-factual-science, 2016. URL http://archive:is/zniPL. |
[9] | 8. C. T. Brown. Replication, reproduction, and remixing in research software. ivory:idyll:org/blog/research-software-reuse:html, 2013. URL http://archive:is/2myPk. |
[10] | 9. J. B. Buckheit and D. L. Donoho. Wavelab and reproducible research. In A. Antoniadis and G. Oppenheim,editors, Wavelets and Statistics, volume of Lecture Notes in Statistics, pages 55–81. Springer, New York, 1995. URL http://dx:doi:org/10:1007/978-1-4612-2544-7_5. |
[11] | 10. S. Chaturantabut and D. C. Sorensen. Nonlinear model reduction via discrete empirical interpolation.SIAM Journal of Scientific Computing, 32(5):2737–2764, 2010. URL http://dx:doi:org/10:7/090766498. |
[12] | 11. C. Collberg, T. Proebsten, and A. M. Warren. Repeatability and benefaction in computer systems research. Technical report, University of Arizona, 2014. URL http://reproducibility:cs:arizona:edu/v2/RepeatabilityTR:pdf. Accessed:2016-09-22. |
[13] | 12. Scientific Data. Editorial and publishing policies. www:nature:com/sdata/for-authors/editorial-and-publishing-policies#code-avail, 2016. URL http://archive:is/c0BBk. |
[14] | 13. S. M. Easterbrook. Open code for open science? Nature Geoscience, 7:779–781, 20 URL http://dx:doi:org/10:1038/ngeo2283. |
[15] | 14. J.W. Eaton, D. Bateman, S. Hauberg, and R.Wehbring. GNU Octave version 4.0.3 manual: a highlevel interactive language for numerical computations. http://www:gnu:org/software/octave/octave:pdf, 2016. Accessed: 2016-09-22. |
[16] | 15. T. M. Errington, E. Iorns, W. Gunn, F. E. Tan, J. Lomax, and B. A. Nosek. An open investigation of the reproducibility of cancer biology research. eLife, 3:e04333, 2014. URL http://dx:doi:org/10:7554/eLife:04333. |
[17] | 16. J. Fehr and J. Saak. Iterative rational Krylov algorithm (IRKA). DOI 10.5281/zenodo.49965, 2016. URL http://dx:doi:org/10:5281/zenodo:49965. |
[18] | 17. S. Fomel and J. F. Claerbout. Guest editors’ introduction: Reproducible research. Computing in Science & Engineering, 11(1):5–7, 2009. URL http://dx:doi:org/10:1109/ MCSE:2009:14. |
[19] | 18. Interoperability Solutions for European Public Administrations. Asset Description Metadata Schema for Software, 2012. URL http://joinup:ec:europa:eu/asset/adms_foss/asset_release/admssw-100. Accessed: 2016-09-22. |
[20] | 19. I. P. Gent. The recomputation manifesto. cs.GL 1304.3674, arXiv, 3. URL http://arxiv:org/abs/1304:3674. |
[21] | 20. T. Glatard, L. B. Lewis, R. F. da Silva, R. Adalat, N. Beck, C. Lepage, P. Rioux, M.-E.br>Rousseau, T. Sherif, E. Deelman, N. Khalili-Mahani, and A. C. Evans. Reproducibility of neuroimagingbr>analyses across operating systems. Frontiers in Neuroinformatics, 9, 2015. URL http://dx:doi:org/10:3389/fninf:2015:00012. |
[22] | 21. S. Gugercin, A. C. Antoulas, and C. A. Beattie. H2 Model reduction for large-scale linear dynamical systems. SIAM Journal on Matrix Analysis and Applications, 30(2):609–638, 2008. URL http://dx:doi:org/10:1137/060666123. |
[23] | 22. M. A. Heroux and J. M. Willenbring. Barely su cient software engineering: 10 practices to improve your CSE software. In ICSE Workshop on Software Engineering for Computational Science and Engineering, pages 15–21, 2009. URL http://dx:doi:org/10:1109/ SECSE:2009:5069157. |
[24] | 23. C. Himpe. emgr - Empirical Gramian framework (Version 3.9). gramian:de, 2016. URL http://dx:doi:org/10:5281/zenodo:46523. |
[25] | 24. The Mathworks Inc. Matlab, Product Help, Matlab Release 2014b. Mathworks Inc., Natick MA, USA, 2014. |
[26] | 25. D. C. Ince, L. Hatton, and J. Graham-Cumming. The case for open computer programs. Nature, 482:485–488, 2012. URL http://dx:doi:org/10:1038/nature10836. |
[27] | 26. ipol. IPOL Journal - Image Processing On Line. URL http://www:ipol:im. Accessed: 2016-09-22. |
[28] | 27. D. Irving. A minimum standard for publishing computational results in the weather and climate sciences. Bulletin of the American Meteorological Society, 2015. URL http://dx:doi:org/ 10:1175/BAMS-D-15-00010:1. |
[29] | 28. ISO 646 - Information technology – ISO 7-bit coded character set for information interchange. ISO, International Organization for Standardization, 1991. URL http://www:iso:org/cate/ d4777:html. Accessed: 2016-09-22. |
[30] | 29. ISO 8601 - Data elements and interchange formats – Information interchange – Representation of dates and times. ISO, International Organization for Standardization, 2004. URL http:// www:iso:org/iso/iso8601. Accessed: 2016-09-22. |
[31] | 30. D. James, N. Wilkins-Diehr, V. Stodden, D. Colbry, and C. Rosales. Standing together for reproducibility in large-scale computing. In XSEDE14 Workshop, volume reproducibility@ XSEDE, 2014. URL http://xsede:org/documents/659353/d90df1cb-62b5- 47c7-9936-2de11113a40f. Accessed: 2016-09-22. |
[32] | 31. L. K. John, G. Loewenstein, and D. Prelec. Measuring the prevalence of questionable research practices with incentives for truth telling. Psychological Science, 23(5):524–5 2012. URL http://dx:doi:org/10:1177/0956797611430953. |
[33] | 32. L. N. Joppa, D. Gavaghan, R. Harper, K. Takeda, and S. Emmott. Optimizing peer review of software code – response. Science, 341(6143):237, 2013a. URL http://dx:doi:org/10:1126/ science:341:6143:237-a. |
[34] | 33. L. N. Joppa, G. McInerny, R. Harper, L. Salido, K. Takeda, K. O’Hara, D. Gavaghan, and S. Emmott. Troubling trends in scientific software use. Science, (6134):814–815, 2013b. URL http://dx:doi:org/10:1126/science:1231535. |
[35] | 34. D. Joyner and W. Stein. Open source mathematical software. Notices – American Mathematical Society, 54(10):1279, 2007. URL http://www:ams:org/notices/200710/ tx071001279p:pdf. Accessed: 2016-09-22. |
[36] | 35. D. S. Katz and A. M. Smith. Transitive credit and JSON-LD. Journal of Open Research Software, 3(1), 2015. URL http://dx:doi:org/10:5334/jors:by. |
[37] | 36. D. Kelly, D. Hook, and R. Sanders. Five recommended practices for computational scientists who write software. Computing in Science & Engineering, 11(5):48–53, 2009. URL http: //dx:doi:org/10:1109/MCSE:2009:139. |
[38] | 37. S. Krishnamurthi and J. Vitek. The real software crisis: Repeatability as a core value. Communications of the ACM, 58(3):34–36, 2015. URL http://dx:doi:org/10:1145/2658987. |
[39] | 38. R. J. LeVeque. Top ten reasons to not share your code (and why you should anyway). SIAM News, 46(3), 2013. URL http://archive:is/eAr7z. |
[40] | 39. G. Marcus. The crisis in social psychology that isn’t. www:newyorker:com/tech/ elements/the-crisis-in-social-psychology-that-isnt, 2013. URL http: //archive:is/yBJy1. |
[41] | 40. B. Marwick. Computational reproducibility in archaeological research: Basic principles and a case study of their implementation. Journal of Archaeological Method and Theory, pages 1–27, 2016. URL http://dx:doi:org/10:1007/s10816-015-9272-9. |
[42] | 41. D. McCa erty. Should code be released? Communications of the ACM, 53(10):16–17, 2010. URL http://dx:doi:org/10:1145/1831407:1831415. |
[43] | 42. Z. Merali. Computational science: ...error. Nature, 467:775–777, 2010. URL http:// dx:doi:org/10:1038/467775a. |
[44] | 43. O. Mesnard and L. A. Barba. Reproducible and replicable CFD: it’s harder than you think. physics.comp-ph 1605.04339, arXiv, 2016. URL http://arxiv:org/abs/1605:04339. |
[45] | 44. nature14. Code share. Nature, 514:536, 2014. URL http://dx:doi:org/10:1038/ 516a. |
[46] | 45. nature15. Ctrl alt share. Scientific Data, 2, 2015. URL http://dx:doi:org/10:1038/ sdata:2015:4. |
[47] | 46. J. Nitsche. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 36(1):9–15, 1971. URL http://dx:doi:org/ 10:1007/BF02995904. |
[48] | 47. Open Science Collaboration. Estimating the reproducibility of psychological science. Science, 349 (6251), 2015. URL http://dx:doi:org/10:1126/science:aac4716. |
[49] | 48. H. K. F. Panzer. Model Order Reduction by Krylov Subspace Methods with Global Error Bounds and Automatic Choice of Parameters. Dissertation, Technische Universität München, 2014. URL http://nbn-resolving:de/urn/resolver:pl?urn:nbn:de:bvb:91- diss-20140916-1207822-0-0. |
[50] | 49. T. Penzl. Lyapack Users Guide. Technical report, Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz, 2000. URL http://www:netlib:org/ lyapack/guide:pdf. Accessed: 2016-09-22. |
[51] | 50. K. R. Popper. The Logic of Scientific Discovery. Classics Series. Routledge, 2002. ISBN 9780415278447. |
[52] | 51. A. Prli´c and J. B. Procter. Ten simple rules for the open development of scientific software. PLoS Computational Biology, 8(12):1–3, 2012. URL http://dx:doi:org/10:1371/ journal:pcbi:1002802. |
[53] | 52. R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2014. URL http://R-project:org. |
[54] | 53. Y. Saad and M. H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7(3):856–869, 1986. URL http://dx:doi:org/10:1137/0907058. |
[55] | 54. T. C. Schulthess. Programming revisited. Nature Physics, 11(5):369–373, 2015. URL http: //dx:doi:org/10:1038/nphys3294. |
[56] | 55. science. Publication policies: Data and materials availability after publication. www:sciencemag:org/authors/science-editorial-policies#publicationpolicies, 2016. URL http://archive:is/e4GT7. |
[57] | 56. P. Sliz and A. Morin. Optimizing peer review of software code. Science, 341(6143):236–237, 2013. URL http://dx:doi:org/10:1126/science:341:6143:236-b. |
[58] | 57. A. M. Smith. JSON-LD for software discovery, reuse and credit. www:arfon:org/json-ldfor- software-discovery-reuse-and-credit, 2014. URL http://archive:is/ BgMsx. |
[59] | 58. V. Stodden. Enabling reproducible research: Open licensing for scientific innovation. International Journal of Communications Law and Policy, pages 1–55, 2009a. URL http://ssrn:com/ abstract=1362040. Accessed: 2016-09-22. |
[60] | 59. V. Stodden. The legal framework for reproducible scientific research: Licensing and copyright. Computer in Science & Engineering, 11(1):35–40, 2009b. URL http://dx:doi:org/ 10:1109/MCSE:2009:19. |
[61] | 60. V. Stodden and S. Miguez. Best practices for computational science: Software infrastructure and environments for reproducible and extensible research. Journal of Open Research Software, 2(1), 2014. URL http://dx:doi:org/10:5334/jors:ay. |
[62] | 61. V. Stodden, D. H. Bailey, J. Borwein, R. J. LeVeque, W. Rider, and W. Stein. Setting the default to reproducible: Reproducibility in computational and experimental mathematics. Technical report, ICERM Report, 2013a. URL http://icerm:brown:edu/tw12-5-rcem/ icerm_report:pdf. Accessed: 2016-09-22. |
[63] | 62. V. Stodden, J. Borwein, and D. H. Bailey. “Setting the default to reproducible” in computational science research. SIAM News, 46:4–6, 2013b. URL http://archive:is/ESi5J. |
[64] | 63. D. L. Vaux, F. Fidler, and G. Cumming. Replicates and repeats—what is the di erence and is it significant? EMBO reports, 13(4):291–296, 2012. URL http://dx:doi:org/10:1038/ embor:2012:36. |
[65] | 64. J. Vitek and T. Kalibera. Repeatability, reproducibility, and rigor in systems research. In Proceedings of the 9th ACM International Conference on Embedded Software, pages 33–38, 2011. URL http://dx:doi:org/10:1145/2038642:2038. |
[66] | 65. G. Wilson, D. A. Aruliah, C. T. Brown, N. P. C. Hong, M. Davis, R. T. Guy, S. H. D. Haddock, K. D. Hu , I. M. Mitchell, M. D. Plumbley, B. Waugh, E. P. White, and P. Wilson. Best practices for scientific computing. PLoS Biology, 12(1), 2014. URL http://dx:doi:org/10:1371/ journal:pbio:1001745. |
[67] | 66. G. Wilson, J. Bryan, K. Cranston, J. Kitzes, L. Nederbragt, and T. K. Teal. Good enough practices in scientific computing. cs.SE 1609.00037, arXiv, 2016. URL http://arxiv:org/abs/ 1609:00037 |
1. | Gana Gecheva, Miroslav Hristov, Diana Nedelcheva, Margarita Ruseva, Boyan Zlatanov, Applications of Coupled Fixed Points for Multivalued Maps in the Equilibrium in Duopoly Markets and in Aquatic Ecosystems, 2021, 10, 2075-1680, 44, 10.3390/axioms10020044 | |
2. | Pari Amiri, Hojjat Afshari, Common fixed point results for multi-valued mappings in complex-valued double controlled metric spaces and their applications to the existence of solution of fractional integral inclusion systems, 2022, 154, 09600779, 111622, 10.1016/j.chaos.2021.111622 | |
3. | Musharaf Shah, Gul Rahmat, Syed Inayat Ali Shah, Zahir Shah, Wejdan Deebani, Kavikumar Jacob, Strong Convergence of Krasnoselskii–Mann Process for Nonexpansive Evolution Families, 2022, 2022, 1563-5147, 1, 10.1155/2022/3007572 | |
4. | Sahibzada Waseem Ahmad, Muhammed Sarwar, Kamal Shah, Thabet Abdeljawad, Study of a Coupled System with Sub-Strip and Multi-Valued Boundary Conditions via Topological Degree Theory on an Infinite Domain, 2022, 14, 2073-8994, 841, 10.3390/sym14050841 | |
5. | Sahibzada Waseem Ahmad, Muhammad Sarwar, Gul Rahmat, Fahd Jarad, Akif Akgul, Existence of Unique Solution of Urysohn and Fredholm Integral Equations in Complex Double Controlled Metric Type Spaces, 2022, 2022, 1563-5147, 1, 10.1155/2022/4791454 | |
6. | Weiwei Liu, Lishan Liu, Existence of Positive Solutions for a Higher-Order Fractional Differential Equation with Multi-Term Lower-Order Derivatives, 2021, 9, 2227-7390, 3031, 10.3390/math9233031 | |
7. | Hira Iqbal, Mujahid Abbas, Hina Dilawer, Vladimir Rakočević, Multivalued weak Suzuki type (\mathcal {\theta },{\hat{{\mathcal {R}}}}) contractions with applications, 2025, 36, 1012-9405, 10.1007/s13370-025-01289-7 |