
This study aimed to investigate the existence, uniqueness, and Ulam-Hyers stability of solutions in a nonlinear coupled system of Hilfer-Hadamard sequential fractional integrodifferential equations, which were further enhanced by nonlocal coupled Hadamard fractional integrodifferential multipoint boundary conditions. The desired conclusions were obtained by using well-known fixed-point theorems. It was emphasized that the fixed-point technique was useful in determining the existence and uniqueness of solutions to boundary value problems. In addition, we examined the solution's Ulam-Hyers stability for the suggested system. The resulting results were further demonstrated and validated using demonstration instances.
Citation: Subramanian Muthaiah, Manigandan Murugesan, Muath Awadalla, Bundit Unyong, Ria H. Egami. Ulam-Hyers stability and existence results for a coupled sequential Hilfer-Hadamard-type integrodifferential system[J]. AIMS Mathematics, 2024, 9(6): 16203-16233. doi: 10.3934/math.2024784
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This study aimed to investigate the existence, uniqueness, and Ulam-Hyers stability of solutions in a nonlinear coupled system of Hilfer-Hadamard sequential fractional integrodifferential equations, which were further enhanced by nonlocal coupled Hadamard fractional integrodifferential multipoint boundary conditions. The desired conclusions were obtained by using well-known fixed-point theorems. It was emphasized that the fixed-point technique was useful in determining the existence and uniqueness of solutions to boundary value problems. In addition, we examined the solution's Ulam-Hyers stability for the suggested system. The resulting results were further demonstrated and validated using demonstration instances.
For simplicity, we consider Poisson equation with a Dirichlet boundary condition as our model problem.
−Δu=f,inΩ, | (1) |
u=g,on∂Ω, | (2) |
where
Using integration by parts, we can get the variational form: find
(∇u,∇v)=(f,v),∀v∈H10(Ω). | (3) |
Various finite element methods have been introduced to solve the Poisson equations (1)-(2), such as the Galerkin finite element methods (FEMs)[2, 3], the mixed FEMs [15] and the finite volume methods (FVMs) [6], etc. The FVMs emphasis on the local conservation property and discretize equations by asking the solution satisfying the flux conservation on a dual mesh consisting of control volumes. The mixed FEMs is another category method that based on the variable
The classical conforming finite element method obtains numerical approximate results by constructing a finite-dimensional subspace of
(∇uh,∇vh)=(f,vh),∀vh∈V0h, | (4) |
where
One obvious disadvantage of discontinuous finite element methods is their rather complex formulations which are often necessary to ensure connections of discontinuous solutions across element boundaries. For example, the IPDG methods add parameter depending interior penalty terms. Besides additional programming complexity, one often has difficulties in finding optimal values for the penalty parameters and corresponding efficient solvers. Most recently, Zhang and Ye [21] developed a discontinuous finite element method that has an ultra simple weak formulation on triangular/tetrahedal meshes. The corresponding numerical scheme can be written as: find
(∇wuh,∇wvh)=(f,vh),∀vh∈V0h, | (5) |
where
Following the work in [21, 22], we propose a new conforming DG finite element method on rectangular partitions in this work. It can be obtained from the conforming formulation simply by replacing
In this paper, we keep the same finite element space as DG method, replace the boundary function with the average of the inner function, and use the weak gradient arising from local Raviart-Thomas (RT) elements [5] to approximate the classic gradient. Moreover, the derivation process in this paper is based on rectangular RT elements [16]. Error estimates of optimal order are established for the corresponding conforming DG approximation in both a discrete
The rest of this paper is organized as follows: In Section 2, we shall present the conforming DG finite element scheme for the Poisson equation on rectangular partitions. Section 3 is devoted to a discussion of the stability and solvability of the new method. In Section 4, we shall prepare ourselves for error estimates by deriving some identities. Error estimates of optimal order in
Throughout this paper, we adopt the standard definition of Sobolev space
H10(Ω)={v∈H1(Ω):v|∂Ω=0}, |
and the space
H(div,Ω)={q∈[L2(Ω)]d:∇⋅q∈L2(Ω)}. |
Assume that the domain
For any interior edge
{v}=12(v|∂T1+v|∂T2),[[v]]=v|∂T1n1+v|∂T2n2, | (6) |
where
{v}=v|eand[[v]]=v|en. | (7) |
We define a discontinuous finite element space
Vh={v∈L2(Ω):v|T∈Qk(T),∀T∈Th}, | (8) |
and its subspace
V0h={v∈Vh:v=0on∂Ω}, | (9) |
where
Definition 2.1. For a given
(∇dv,q)T:=−(v,∇⋅q)T+⟨{v},q⋅n⟩∂T,∀q∈RTk(T), | (10) |
where
The weak gradient operator
(∇dv)|T=∇d(v|T). |
We introduce the following bilinear form:
a(v,w)=(∇dv,∇dw), |
the conforming DG algorithm to solve the problems (1) - (2) is given by
Conforming DG algorithm 1. Find
a(uh,vh)=(f,vh),∀vh∈V0h, | (11) |
where
We will prove the existence and uniqueness of the solution of equation (11). Firstly, we present the following two useful inequalities to derive the forthcoming analysis.
Lemma 3.1 (trace inequality). Let
‖φ‖2e≤C(h−1T‖φ‖2T+hT‖∇φ‖2T), | (12) |
where
Lemma 3.2 (inverse inequality). Let
‖∇φ‖T≤C(n)h−1T‖φ‖T,∀T∈Th. | (13) |
Then, we define the following semi-norms in the discontinuous finite element space
|||v|||2=a(v,v)=∑T∈Th‖∇dv‖2T, | (14) |
‖v‖21,h=∑T∈Th‖∇v‖2T+∑e∈E0hh−1e‖[[v]]‖2e. | (15) |
We have the equivalence between the semi-norms
Lemma 3.3. For any
C1‖v‖1,h≤|||v|||≤C2‖v‖1,h, | (16) |
where
Proof. It follows from the definition of
‖∇dv‖2T1=(∇dv,∇dv)T1=−(v,∇⋅∇dv)T1+⟨{v}n,∇dv⟩∂T1=(∇v,∇dv)T1−⟨(v−{v})n,∇dv⟩∂T1≤‖∇v‖T1‖∇dv‖T1+‖(v−{v})n‖∂T1‖∇dv‖∂T1≤‖∇dv‖T1(‖∇v‖T1+h−12T1‖(v−{v})n‖∂T1). | (17) |
For any
(v−{v})|en1=v|∂T1n1−12(v|∂T1+v|∂T2)n1=12(v|∂T1n1+v|∂T2n2)=12[[v]]e. |
Then we can get
‖(v−{v})n‖2∂T1≤12∑e∈∂T1‖[[v]]‖2e. | (18) |
Substituting (18) into (17) gives
‖∇dv‖2T1≤C2‖∇dv‖T1(‖∇v‖T1+∑e∈∂T1h−12e‖[[v]]‖e), |
this completes the proof of the right-hand of (16).
To prove the left-hand of (16), we consider the subspace of
D(k,T):={q∈RTk(T):q⋅n=0on∂T}. |
Note that
‖∇v‖T=supq∈D(k,T)(∇v,q)T‖q‖T. | (19) |
Using the integration by parts, Cauchy-Schwarz inequality, the definition of
(∇v,q)T=−(v,∇⋅q)T+⟨v,q⋅n⟩∂T=(∇dv,q)T−⟨{v},q⋅n⟩∂T=(∇dv,q)T≤‖∇dv‖T⋅‖q‖T, |
where we have used the fact that
‖∇v‖T≤‖∇dv‖T. | (20) |
We define the space
‖[[v]]‖e=supq∈De(k,T)⟨[[v]],q⋅n⟩e‖q⋅n‖e. | (21) |
Following the integration by parts and the definition of
(∇dv,q)T=(∇v,q)T−⟨v,q⋅n⟩e+⟨{v},q⋅n⟩e. |
Together with (20), we obtain
|⟨[[v]],q⋅n⟩e|=2|(∇dv,q)T−(∇v,q)T|≤2|(∇dv,q)T|+2|(∇v,q)T|≤C(‖∇dv‖T‖q‖T+‖∇v‖T‖q‖T)≤C‖∇dv‖T‖q‖T. |
Substituting the above inequality into (21), by the scaling argument [13], for such
‖[[v]]‖e≤C‖∇dv‖T‖q‖T‖q⋅n‖e≤Ch12‖∇dv‖T. | (22) |
Combining (20) and (22) gives a proof of the left-hand of (16).
Lemma 3.4. The semi-norm
Proof. We shall only verify the positivity property for
The above two lemmas imply the well posedness of the scheme (11). We prove the existence and uniqueness of solution of the conforming DG method in Theorem 3.1.
Theorem 3.1. The conforming DG scheme (11) has and only has one solution.
Proof. To prove the scheme (11) is uniquely solvable, it suffices to verify that the homogeneous equation has zero as its unique solution. To this end, let
a(uh,uh)=0, |
which leads to
In this section, we will derive an error equation which will be used for the error estimates. For any
(∇⋅q,v)T=(∇⋅Πhq,v)T,∀v∈Qk(T). | (23) |
For any
‖Πh(∇w)−∇w‖≤Chk‖w‖1+k. | (24) |
Moreover, it is easy to verify the following property holds true.
Lemma 4.1. For any
∑T∈Th(−∇⋅q,v)T=∑T∈Th(Πhq,∇dv)T,∀v∈V0h. | (25) |
Proof.
∑T∈Th⟨{v},Πhq⋅n⟩∂T=0. | (26) |
By the definition of
∑T∈Th(−∇⋅q,v)T=∑T∈Th(−∇⋅Πhq,v)T=∑T∈Th(−∇⋅Πhq,v)T+∑T∈Th⟨{v},Πhq⋅n⟩∂T=∑T∈Th(Πhq,∇dv)T. |
This completes the proof of the lemma.
Before establishing the error equation, we define a continuous finite element subspace of
˜Vh={v∈H1(Ω):v|T∈Qk(T),∀T∈Th}. | (27) |
so as a subspace of
˜V0h:={v∈˜Vh:v|∂Ω=0}. | (28) |
Lemma 4.2. For any
∇dv=∇v. |
Proof. By the definition of
(∇dv,q)T=−(v,∇⋅q)T+⟨{v},q⋅n⟩∂T=−(v,∇⋅q)T+⟨v,q⋅n⟩∂T=(∇v,q)T, |
which gives
(∇dv−∇v,q)T=0,∀q∈RTk(T). |
Letting
Let
‖Ihu−u‖≤Chk+1‖u‖k+1, | (29) |
‖∇Ihu−∇u‖≤Chk‖u‖k+1. | (30) |
It is obvious that
Lemma 4.3. Denote
a(eh,vh)=lu(vh), | (31) |
where
lu(vh)=∑T∈Th(∇Ihu−Πh∇u,∇dvh). | (32) |
Proof. Since
∑T∈Th(∇dIhu,∇dvh)T=∑T∈Th(∇Ihu,∇dvh)T=∑T∈Th(∇Ihu−Πh∇u+Πh∇u,∇dvh)T=∑T∈Th(∇Ihu−Πh∇u,∇dvh)T+∑T∈Th(Πh∇u,∇dvh)T=lu(vh)−∑T∈Th(∇⋅∇u,vh)T=lu(vh)+(f,vh). |
By the definition of the scheme (11), we have
∑T∈Th(∇dIhu−∇duh,∇dvh)T=lu(vh). |
This completes the proof of the lemma.
The goal of this section is to derive the error estimates in
Theorem 5.1. Let
|||eh|||≤Chk|u|k+1. | (33) |
Proof. Letting
|||eh|||2=lu(eh). | (34) |
From the Cauchy-Schwarz inequality, the triangle inequality, the definition of
lu(vh)=∑T∈Th(∇Ihu−Πh(∇u),∇dvh)T≤∑T∈Th‖∇Ihu−Πh(∇u)‖T‖∇dvh‖T≤(∑T∈Th‖∇Ihu−Πh(∇u)‖2T)12(∑T∈Th‖∇dvh‖2T)12=(∑T∈Th‖∇Ihu−∇u+∇u−Πh(∇u)‖2T)12|||vh|||≤(∑T∈Th‖∇Ihu−∇u‖2T+‖∇u−Πh(∇u)‖2T)12|||vh|||≤Chk|u|k+1|||vh|||. |
Then, we have
lu(eh)≤Chk|u|k+1|||eh|||. | (35) |
Substituting (35) to (34), we obtain
|||eh|||2≤Chk|u|k+1|||eh|||, |
which completes the proof of the lemma.
It is obvious that
(∇˜uh,∇v)=(f,v),∀v∈˜V0h. | (36) |
For any
(∇duh−∇˜uh,∇v)=0,∀v∈˜V0h. | (37) |
In the rest of this section, we derive an optimal order error estimate for the conforming DG approximation (11) in
−∇⋅(∇Φ)=uh−˜uh,inΩ. | (38) |
Assume that the dual problem satisfies
‖Φ‖2≤C‖uh−˜uh‖. | (39) |
In the following of this paper, we note
Theorem 5.2. Assume
‖u−uh‖≤Chk+1|u|k+1. | (40) |
Proof. First, we shall derive the optimal order for
a(Φh,v)=(εh,v),∀v∈V0h. | (41) |
Since
(∇duh−∇˜uh,∇IhΦ)=0,∇dIhΦ=∇IhΦ, |
which gives
(∇duh−∇˜uh,∇dIhΦ)=0. | (42) |
Setting
‖εh‖2=a(Φh,εh)=∑T∈Th(∇dΦh,∇dεh)T=∑T∈Th(∇d(Φh−IhΦ),∇duh−∇˜uh)T≤|||Φh−IhΦ|||(|||uh−Ihu|||+‖∇(Ihu−˜uh)‖). |
Then, by the Cauchy-Schwarz inequality, (33) and (39), we obtain
‖εh‖2≤Ch|Φ|2hk|u|k+1≤Chk+1|u|k+1‖εh‖, |
which gives
‖εh‖≤Chk+1|u|k+1. | (43) |
Combining the error estimate of finite element solution, the triangle inequality and (43) yields (40), which completes the proof of the theorem.
In this section, we shall present some numerical results for the conforming discontinuous Galerkin method analyzed in the previous sections.
We solve the following Poisson equation on the unit square domain
−Δu=2π2sin(πx)sin(πy)in Ω | (44) |
u=0on ∂Ω. | (45) |
The exact solution of the above problem is
We first use the
level | rate | rate | |||
by |
|||||
6 | 0.1996E-02 | 1.97 | 0.8887E-02 | 1.98 | 1024 |
7 | 0.5013E-03 | 1.99 | 0.2228E-02 | 2.00 | 4096 |
8 | 0.1255E-03 | 2.00 | 0.5574E-03 | 2.00 | 16384 |
by |
|||||
6 | 0.2427E-02 | 1.97 | 0.1027E+00 | 1.02 | 3072 |
7 | 0.6100E-03 | 1.99 | 0.5105E-01 | 1.01 | 12288 |
8 | 0.1527E-03 | 2.00 | 0.2546E-01 | 1.00 | 49152 |
by |
|||||
5 | 0.1533E-03 | 3.00 | 0.2042E-01 | 2.03 | 1536 |
6 | 0.1915E-04 | 3.00 | 0.5061E-02 | 2.01 | 6144 |
7 | 0.2394E-05 | 3.00 | 0.1260E-02 | 2.01 | 24576 |
by |
|||||
5 | 0.7959E-05 | 4.00 | 0.1965E-02 | 3.00 | 2560 |
6 | 0.4971E-06 | 4.00 | 0.2451E-03 | 3.00 | 10240 |
7 | 0.3140E-07 | 3.98 | 0.3059E-04 | 3.00 | 40960 |
by |
|||||
4 | 0.1055E-04 | 4.97 | 0.1421E-02 | 4.05 | 960 |
5 | 0.3314E-06 | 4.99 | 0.8735E-04 | 4.02 | 3840 |
6 | 0.1057E-07 | 4.97 | 0.5417E-05 | 4.01 | 15360 |
by |
|||||
2 | 0.2835E-02 | 6.24 | 0.1450E+00 | 5.49 | 84 |
3 | 0.4532E-04 | 5.97 | 0.4718E-02 | 4.94 | 336 |
4 | 0.7115E-06 | 5.99 | 0.1478E-03 | 5.00 | 1344 |
The same test case is also computed using the
level | rate | rate | |||
by |
|||||
6 | 0.4006E-03 | 1.99 | 0.2389E-02 | 1.99 | 4096 |
7 | 0.1003E-03 | 2.00 | 0.5982E-03 | 2.00 | 16384 |
8 | 0.2510E-04 | 2.00 | 0.1496E-03 | 2.00 | 65536 |
by |
|||||
6 | 0.2360E-04 | 2.99 | 0.3186E-02 | 1.99 | 9216 |
7 | 0.2953E-05 | 3.00 | 0.7976E-03 | 2.00 | 36864 |
8 | 0.3692E-06 | 3.00 | 0.1995E-03 | 2.00 | 147456 |
by |
|||||
5 | 0.1413E-04 | 4.08 | 0.1650E-02 | 2.97 | 4096 |
6 | 0.8676E-06 | 4.03 | 0.2072E-03 | 2.99 | 16384 |
7 | 0.5398E-07 | 4.01 | 0.2593E-04 | 3.00 | 65536 |
by |
|||||
3 | 0.2226E-02 | 4.59 | 0.5414E-01 | 3.52 | 400 |
4 | 0.9610E-04 | 4.53 | 0.3723E-02 | 3.86 | 1600 |
5 | 0.3279E-05 | 4.87 | 0.2392E-03 | 3.96 | 6400 |
To test the superconvergence of
−Δu+u=fin Ωu=0on ∂Ω, |
where
u=(x−x2)(y−y3). | (46) |
Uniform square grids as shown in Figure 1 are used for numerical computation. The numerical results are listed in Table 3. Surprising, for this problem, the
level | rate | rate | |||
by |
|||||
3 | 0.8265E-02 | 1.06 | 0.4577E-01 | 1.14 | 16 |
4 | 0.2772E-02 | 1.58 | 0.1732E-01 | 1.40 | 64 |
5 | 0.7965E-03 | 1.80 | 0.6331E-02 | 1.45 | 256 |
6 | 0.2142E-03 | 1.90 | 0.2290E-02 | 1.47 | 1024 |
7 | 0.5564E-04 | 1.94 | 0.8213E-03 | 1.48 | 4096 |
8 | 0.1419E-04 | 1.97 | 0.2928E-03 | 1.49 | 16384 |
To test further the superconvergence of
−∇(a∇u)=fin Ωu=0on ∂Ω, |
where
u=(x−x3)(y2−y3). | (47) |
Uniform square grids as shown in Figure 1 are used for computation. The numerical results are listed in Table 4. Surprising, again, the
level | rate | rate | |||
by |
|||||
3 | 0.4929E-02 | 0.97 | 0.5371E-01 | 0.80 | 16 |
4 | 0.1917E-02 | 1.36 | 0.2401E-01 | 1.16 | 64 |
5 | 0.6004E-03 | 1.67 | 0.9407E-02 | 1.35 | 256 |
6 | 0.1682E-03 | 1.84 | 0.3507E-02 | 1.42 | 1024 |
7 | 0.4457E-04 | 1.92 | 0.1275E-02 | 1.46 | 4096 |
8 | 0.1148E-04 | 1.96 | 0.4576E-03 | 1.48 | 16384 |
In this paper, we establish a new numerical approximation scheme based on the rectangular partition to solve second order elliptic equation. We derived the numerical scheme and then proved the optimal order of convergence of the error estimates in
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level | rate | rate | |||
by |
|||||
6 | 0.1996E-02 | 1.97 | 0.8887E-02 | 1.98 | 1024 |
7 | 0.5013E-03 | 1.99 | 0.2228E-02 | 2.00 | 4096 |
8 | 0.1255E-03 | 2.00 | 0.5574E-03 | 2.00 | 16384 |
by |
|||||
6 | 0.2427E-02 | 1.97 | 0.1027E+00 | 1.02 | 3072 |
7 | 0.6100E-03 | 1.99 | 0.5105E-01 | 1.01 | 12288 |
8 | 0.1527E-03 | 2.00 | 0.2546E-01 | 1.00 | 49152 |
by |
|||||
5 | 0.1533E-03 | 3.00 | 0.2042E-01 | 2.03 | 1536 |
6 | 0.1915E-04 | 3.00 | 0.5061E-02 | 2.01 | 6144 |
7 | 0.2394E-05 | 3.00 | 0.1260E-02 | 2.01 | 24576 |
by |
|||||
5 | 0.7959E-05 | 4.00 | 0.1965E-02 | 3.00 | 2560 |
6 | 0.4971E-06 | 4.00 | 0.2451E-03 | 3.00 | 10240 |
7 | 0.3140E-07 | 3.98 | 0.3059E-04 | 3.00 | 40960 |
by |
|||||
4 | 0.1055E-04 | 4.97 | 0.1421E-02 | 4.05 | 960 |
5 | 0.3314E-06 | 4.99 | 0.8735E-04 | 4.02 | 3840 |
6 | 0.1057E-07 | 4.97 | 0.5417E-05 | 4.01 | 15360 |
by |
|||||
2 | 0.2835E-02 | 6.24 | 0.1450E+00 | 5.49 | 84 |
3 | 0.4532E-04 | 5.97 | 0.4718E-02 | 4.94 | 336 |
4 | 0.7115E-06 | 5.99 | 0.1478E-03 | 5.00 | 1344 |
level | rate | rate | |||
by |
|||||
6 | 0.4006E-03 | 1.99 | 0.2389E-02 | 1.99 | 4096 |
7 | 0.1003E-03 | 2.00 | 0.5982E-03 | 2.00 | 16384 |
8 | 0.2510E-04 | 2.00 | 0.1496E-03 | 2.00 | 65536 |
by |
|||||
6 | 0.2360E-04 | 2.99 | 0.3186E-02 | 1.99 | 9216 |
7 | 0.2953E-05 | 3.00 | 0.7976E-03 | 2.00 | 36864 |
8 | 0.3692E-06 | 3.00 | 0.1995E-03 | 2.00 | 147456 |
by |
|||||
5 | 0.1413E-04 | 4.08 | 0.1650E-02 | 2.97 | 4096 |
6 | 0.8676E-06 | 4.03 | 0.2072E-03 | 2.99 | 16384 |
7 | 0.5398E-07 | 4.01 | 0.2593E-04 | 3.00 | 65536 |
by |
|||||
3 | 0.2226E-02 | 4.59 | 0.5414E-01 | 3.52 | 400 |
4 | 0.9610E-04 | 4.53 | 0.3723E-02 | 3.86 | 1600 |
5 | 0.3279E-05 | 4.87 | 0.2392E-03 | 3.96 | 6400 |
level | rate | rate | |||
by |
|||||
3 | 0.8265E-02 | 1.06 | 0.4577E-01 | 1.14 | 16 |
4 | 0.2772E-02 | 1.58 | 0.1732E-01 | 1.40 | 64 |
5 | 0.7965E-03 | 1.80 | 0.6331E-02 | 1.45 | 256 |
6 | 0.2142E-03 | 1.90 | 0.2290E-02 | 1.47 | 1024 |
7 | 0.5564E-04 | 1.94 | 0.8213E-03 | 1.48 | 4096 |
8 | 0.1419E-04 | 1.97 | 0.2928E-03 | 1.49 | 16384 |
level | rate | rate | |||
by |
|||||
3 | 0.4929E-02 | 0.97 | 0.5371E-01 | 0.80 | 16 |
4 | 0.1917E-02 | 1.36 | 0.2401E-01 | 1.16 | 64 |
5 | 0.6004E-03 | 1.67 | 0.9407E-02 | 1.35 | 256 |
6 | 0.1682E-03 | 1.84 | 0.3507E-02 | 1.42 | 1024 |
7 | 0.4457E-04 | 1.92 | 0.1275E-02 | 1.46 | 4096 |
8 | 0.1148E-04 | 1.96 | 0.4576E-03 | 1.48 | 16384 |
level | rate | rate | |||
by |
|||||
6 | 0.1996E-02 | 1.97 | 0.8887E-02 | 1.98 | 1024 |
7 | 0.5013E-03 | 1.99 | 0.2228E-02 | 2.00 | 4096 |
8 | 0.1255E-03 | 2.00 | 0.5574E-03 | 2.00 | 16384 |
by |
|||||
6 | 0.2427E-02 | 1.97 | 0.1027E+00 | 1.02 | 3072 |
7 | 0.6100E-03 | 1.99 | 0.5105E-01 | 1.01 | 12288 |
8 | 0.1527E-03 | 2.00 | 0.2546E-01 | 1.00 | 49152 |
by |
|||||
5 | 0.1533E-03 | 3.00 | 0.2042E-01 | 2.03 | 1536 |
6 | 0.1915E-04 | 3.00 | 0.5061E-02 | 2.01 | 6144 |
7 | 0.2394E-05 | 3.00 | 0.1260E-02 | 2.01 | 24576 |
by |
|||||
5 | 0.7959E-05 | 4.00 | 0.1965E-02 | 3.00 | 2560 |
6 | 0.4971E-06 | 4.00 | 0.2451E-03 | 3.00 | 10240 |
7 | 0.3140E-07 | 3.98 | 0.3059E-04 | 3.00 | 40960 |
by |
|||||
4 | 0.1055E-04 | 4.97 | 0.1421E-02 | 4.05 | 960 |
5 | 0.3314E-06 | 4.99 | 0.8735E-04 | 4.02 | 3840 |
6 | 0.1057E-07 | 4.97 | 0.5417E-05 | 4.01 | 15360 |
by |
|||||
2 | 0.2835E-02 | 6.24 | 0.1450E+00 | 5.49 | 84 |
3 | 0.4532E-04 | 5.97 | 0.4718E-02 | 4.94 | 336 |
4 | 0.7115E-06 | 5.99 | 0.1478E-03 | 5.00 | 1344 |
level | rate | rate | |||
by |
|||||
6 | 0.4006E-03 | 1.99 | 0.2389E-02 | 1.99 | 4096 |
7 | 0.1003E-03 | 2.00 | 0.5982E-03 | 2.00 | 16384 |
8 | 0.2510E-04 | 2.00 | 0.1496E-03 | 2.00 | 65536 |
by |
|||||
6 | 0.2360E-04 | 2.99 | 0.3186E-02 | 1.99 | 9216 |
7 | 0.2953E-05 | 3.00 | 0.7976E-03 | 2.00 | 36864 |
8 | 0.3692E-06 | 3.00 | 0.1995E-03 | 2.00 | 147456 |
by |
|||||
5 | 0.1413E-04 | 4.08 | 0.1650E-02 | 2.97 | 4096 |
6 | 0.8676E-06 | 4.03 | 0.2072E-03 | 2.99 | 16384 |
7 | 0.5398E-07 | 4.01 | 0.2593E-04 | 3.00 | 65536 |
by |
|||||
3 | 0.2226E-02 | 4.59 | 0.5414E-01 | 3.52 | 400 |
4 | 0.9610E-04 | 4.53 | 0.3723E-02 | 3.86 | 1600 |
5 | 0.3279E-05 | 4.87 | 0.2392E-03 | 3.96 | 6400 |
level | rate | rate | |||
by |
|||||
3 | 0.8265E-02 | 1.06 | 0.4577E-01 | 1.14 | 16 |
4 | 0.2772E-02 | 1.58 | 0.1732E-01 | 1.40 | 64 |
5 | 0.7965E-03 | 1.80 | 0.6331E-02 | 1.45 | 256 |
6 | 0.2142E-03 | 1.90 | 0.2290E-02 | 1.47 | 1024 |
7 | 0.5564E-04 | 1.94 | 0.8213E-03 | 1.48 | 4096 |
8 | 0.1419E-04 | 1.97 | 0.2928E-03 | 1.49 | 16384 |
level | rate | rate | |||
by |
|||||
3 | 0.4929E-02 | 0.97 | 0.5371E-01 | 0.80 | 16 |
4 | 0.1917E-02 | 1.36 | 0.2401E-01 | 1.16 | 64 |
5 | 0.6004E-03 | 1.67 | 0.9407E-02 | 1.35 | 256 |
6 | 0.1682E-03 | 1.84 | 0.3507E-02 | 1.42 | 1024 |
7 | 0.4457E-04 | 1.92 | 0.1275E-02 | 1.46 | 4096 |
8 | 0.1148E-04 | 1.96 | 0.4576E-03 | 1.48 | 16384 |