In this paper, we investigate the stochastic evolution equations (SEEs) driven by a bounded log-Whittle-Matˊern (W-M) random diffusion coefficient field and Q-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.
Citation: Xiao Qi, Mejdi Azaiez, Can Huang, Chuanju Xu. An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise[J]. AIMS Mathematics, 2022, 7(12): 20684-20710. doi: 10.3934/math.20221134
[1] | Aliaa Burqan, Mohammed Shqair, Ahmad El-Ajou, Sherif M. E. Ismaeel, Zeyad AlZhour . Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method. AIMS Mathematics, 2023, 8(8): 19297-19312. doi: 10.3934/math.2023984 |
[2] | M. Sivakumar, M. Mallikarjuna, R. Senthamarai . A kinetic non-steady state analysis of immobilized enzyme systems with external mass transfer resistance. AIMS Mathematics, 2024, 9(7): 18083-18102. doi: 10.3934/math.2024882 |
[3] | Yudhveer Singh, Devendra Kumar, Kanak Modi, Vinod Gill . A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative. AIMS Mathematics, 2020, 5(2): 843-855. doi: 10.3934/math.2020057 |
[4] | Zihan Yue, Wei Jiang, Boying Wu, Biao Zhang . A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation. AIMS Mathematics, 2024, 9(3): 7040-7062. doi: 10.3934/math.2024343 |
[5] | Zeliha Korpinar, Mustafa Inc, Dumitru Baleanu . On the fractional model of Fokker-Planck equations with two different operator. AIMS Mathematics, 2020, 5(1): 236-248. doi: 10.3934/math.2020015 |
[6] | Ayyaz Ali, Zafar Ullah, Irfan Waheed, Moin-ud-Din Junjua, Muhammad Mohsen Saleem, Gulnaz Atta, Maimoona Karim, Ather Qayyum . New exact solitary wave solutions for fractional model. AIMS Mathematics, 2022, 7(10): 18587-18602. doi: 10.3934/math.20221022 |
[7] | Nisar Gul, Saima Noor, Abdulkafi Mohammed Saeed, Musaad S. Aldhabani, Roman Ullah . Analytical solution of the systems of nonlinear fractional partial differential equations using conformable Laplace transform iterative method. AIMS Mathematics, 2025, 10(2): 1945-1966. doi: 10.3934/math.2025091 |
[8] | Shabir Ahmad, Aman Ullah, Ali Akgül, Fahd Jarad . A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations. AIMS Mathematics, 2022, 7(5): 9389-9404. doi: 10.3934/math.2022521 |
[9] | Gulalai, Shabir Ahmad, Fathalla Ali Rihan, Aman Ullah, Qasem M. Al-Mdallal, Ali Akgül . Nonlinear analysis of a nonlinear modified KdV equation under Atangana Baleanu Caputo derivative. AIMS Mathematics, 2022, 7(5): 7847-7865. doi: 10.3934/math.2022439 |
[10] | Lynda Taleb, Rabah Gherdaoui . Approximation by the heat kernel of the solution to the transport-diffusion equation with the time-dependent diffusion coefficient. AIMS Mathematics, 2025, 10(2): 2392-2412. doi: 10.3934/math.2025111 |
In this paper, we investigate the stochastic evolution equations (SEEs) driven by a bounded log-Whittle-Matˊern (W-M) random diffusion coefficient field and Q-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.
In the present research, we prove existence results for a fourth-order differential equation system that takes the form:
{ϖ(4)(t)=f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t)),a.e. t∈J=[0,1],ϖ(0)=ϖ0,ϖ′(0)=ϖ1andϖ″∈(BC), | (1.1) |
where f:[0,1]×R4n→Rn represents an L1 -Carathéodory function, ϖ0,ϖ1∈Rn and (BC) can be the boundary conditions that are given by one of the following:
(SL) Strum-Liouville boundary conditions on J
A0ϖ(0)−β0ϖ′(0)=r0,A1ϖ(1)+β1ϖ′(1)=r1. | (1.2) |
(P) Periodic boundary conditions on J
ϖ(0)=ϖ(1),ϖ′(0)=ϖ′(1), | (1.3) |
where (Ai)i∈{0,1}∈Mn×n(R), such that
∀i∈{0,1},∃κi≥0:⟨ϖ,Aiϖ⟩≥κi‖ϖ‖2,∀ϖ∈Rn |
∀i∈{0,1},ri∈R:βi∈{0,1},κi+βi>0. |
We refer to [1,2,3] for further findings that were achieved in the specific instance of a boundary value issue for only one differential equation of the fourth-order (n=1), for more details, please see [4,5,6]. Existence results for higher-order differential equations can be found in [7,8], and the general case of Nth order systems is discussed in [9,10,11].
The concept of the solution-tube of problem (1.1) is presented in this work; see [12,13,14]. This idea is inspired by [15] and [16], where solution-tubes for second and third order differential equations systems are defined, respectively, as follows:
{ϖ″(t)=f(t,ϖ(t),ϖ′(t)),a.e. t∈J,ϖ∈(BC), | (1.4) |
and
{ϖ‴(t)=f(t,ϖ(t),ϖ′(t),ϖ″(t)),a.e. t∈J,ϖ(0)=ϖ0,ϖ′∈(BC). | (1.5) |
We prove that the system (1.1) has solutions. For this system, we employ the concept of a solution tube, which extends to systems the ideas of lower and upper solutions to the fourth-order differential equations presented in [17,18,19].
The structure of this paper is given as follows: This article will utilize the notations, definitions, and findings found in Section 2. In Section 3, we provide the idea of a solution-tube to get existence results for fourth-order differential equation systems. We then go on to demonstrate the practicality of our results through two examples.
In this section, we recall some notations, definitions, and results that we will use in this article. The scalar product and the Euclidian norm in Rn are denoted by ⟨,⟩ and ‖⋅‖, respectively. Also, let Ck(J,Rn) be the Banach space of the k-times continuously differentiable functions ϖ associated with the norm
‖ϖ‖k=max{‖ϖ‖0,‖ϖ′‖0,...,‖ϖ(k)‖0}, |
where
‖ϖ‖0=max{ϖ(t):t∈J}. |
The space of integral functions is denoted by L1(J,Rn), with the usual norm ‖⋅‖L1. The Sobolev space of functions in Ck−1(J,Rn), where k≥1 and the (k−1)th derivative is denoted by Wk,1(J,Rn).
For ϖ0,ϖ1∈Rn, we have the following:
Cϖ0(J,Rn):={ϖ∈C(J,Rn):ϖ(0)=ϖ0}, |
C1ϖ0,ϖ1(J,Rn):={ϖ∈C1(J,Rn):ϖ(0)=ϖ0,ϖ′(0)=ϖ1}, |
CkB(J,Rn)={ϖ∈Ck(J,Rn):ϖ∈(BC)}, |
Wk,1B(J,Rn))={ϖ∈Wk,1(J,Rn)):ϖ∈(BC)}, |
Ck+1ϖ0,B(J,Rn)={ϖ∈Ck+1(J,Rn):ϖ(0)=ϖ0,ϖ(k)∈(BC)}, |
Wk+1,1ϖ0,B(J,Rn))={ϖ∈Wk+1,1(J,Rn)):ϖ(0)=ϖ0,ϖ(k)∈(BC)}, |
Ck+2ϖ0,ϖ1,B(J,Rn)={ϖ∈Ck+2(J,Rn):ϖ(0)=ϖ0,ϖ′(0)=ϖ1,x(k)∈(BC)}, |
Wk+2,1ϖ0,ϖ1,B(J,Rn))={ϖ∈Wk+2,1(J,Rn)):ϖ(0)=ϖ0,ϖ′(0)=ϖ1,ϖ(k)∈(BC)}. |
Definition 2.1. A function f:J×R4n→Rn is called an L1-Carathéodory function if
(i) For every (ϖ,y,q,p)∈R4n, the function t↦f(t,ϖ,y,q,p) is measurable;
(ii) The function (ϖ,y,q,p)↦f(t,ϖ,y,q,p) is continuous for a.e. t∈J;
(iii) For every r>0, there exists a function hr∈L1(J,[0,∞)) such that ‖f(t,ϖ,y,q,p)‖≤hr(t) for a.e. t∈J and for all (ϖ,y,q,p)∈D, where
D={(ϖ,y,q,p)∈R4n:‖ϖ‖≤r, ‖y‖≤r, ‖q‖≤r, ‖p‖≤r}. |
Definition 2.2. A function F:C3(J,Rn)×J→L1(J,Rn) is integrally bounded, if for every bounded subset B⊂C3(J,Rn), there exists an integral function hB∈L1(J,[0,∞)) so that ‖F(ϖ,α)(t)‖≤hB(t), for ∀t∈J,(ϖ,α)∈B×J.
The operator NF:C3(J,Rn)×J→C0(J,Rn) will be associated with F and defined by
NF(ϖ)(t)=∫t0F(ϖ,α)(s)ds. |
We now state the following results:
Theorem 2.1. [20] Let F:C3(J,Rn)×J→L1(J,Rn) be continuous and integrally bounded, then NF is continuous and completely continuous.
Lemma 2.1. [21] Let E be a Banach space. Let v:J→E be an absolutely continuous function, then for
{t∈J:v(t)=0andv′(t)≠0}, |
the measure is zero.
Lemma 2.2. [22] For w∈W2,1(J;R) and ε≥0, assume that one of the next properties is satisfied:
(i) w″(t)−εw(t)≥0; for almost every t∈J,κ0w(0)−ν0w′(0)≤0,κ1w(1)+ν1w′(1)≤0; where κi,νi≥0,max{κi,νi}>0;i=0,1;andmax{κ0,κ1,ε}>0,
(ii) w″(t)−εw(t)≥0; for almost every t∈J,ε>0,w(0)=w(1), w′(1)−w′(0)≤0,
(iii) w″(t)−εw(t)≥0; for almost every t∈[0,t1]∪[t2,1],ε>0,w(0)=w(1), w′(1)−w′(0)≤0, w(t)≤0,t∈[t1,t2].
Then w(t)≤0, ∀t∈[0,1].
Lemma 2.3. [22] Let f∈C(J×R2n,Rn) be a L1-Carathéodory function (see definition in [22]). Consider the following problem:
{ϖ″(t)=f(t,ϖ(t),ϖ′(t)),a.e. t∈J,ϖ∈(BC). | (2.1) |
Let ε>0, and (z,N) a solution-tube of (2.1) given in Definition 2.3 of [22]. If ϖ∈W2,1B(J,Rn) satisfies
Π(t)=⟨ϖ(t)−z(t),ϖ″(t)−z″(t)⟩+‖ϖ′(t)−z′(t)‖2‖ϖ(t)−z(t)‖−⟨ϖ(t)−z(t),ϖ′(t)−z′(t)⟩2‖ϖ(t)−z(t)‖3−ε‖ϖ(t)−z(t)‖≥N″(t)−εN(t), |
a.e. on
{t∈J:‖ϖ(t)−z(t)‖>N(t)}. |
Then
‖ϖ(t)−z(t)‖≤N(t) for every t∈J. |
Now, we recall some properties of the Leray Schauder degree. The interested reader can see [23,24].
Theorem 2.2. Let E be a Banach space and U⊂E is an open bounded set. We define K∂U(¯U,E)={f:¯U→E, where f is compact and f(ϖ)≠ϖ, for every ϖ∈∂U}, the Leary-Schauder degree on U of (Id−f) is an integer deg(Id−f,U,0) satisfying the following properties:
(i) (Existence) If deg(Id−f,U,0)≠0, then ∃ϖ∈U, s.t.,
ϖ−f(ϖ)=0. |
(ii) (Normalization) If 0∈U, then deg(Id,U,0)=1.
(iii) (Homotopy invariance) If h:¯U×J→E is a compact such that ϖ−h(ϖ,α)≠0 for each (ϖ,α)∈∂U×J, then
deg(Id−h(.,α),U,0)=deg(Id−h(.,0),U,0), for every α∈J. |
(iv) (Excision) If V⊂U is open and ϖ−f(ϖ)≠0 for all ϖ∈¯U ∖V, then
deg(Id−f,U,0)=deg(Id−f,V,0). |
(v) (Additivity) If U1,U2⊂U are disjoint and open, such that ¯U=¯U1∪U2 and ϖ−f(ϖ)≠0 for all ϖ∈∂U1∪∂U2, then
deg(Id−f,U,0)=deg(Id−f,U1,0)+deg(Id−f,U2,0). |
In this section, we define the solution-tube to the problem (1.1). This definition is important for our discussion about the existence results. A solution to this problem is a function ϖ∈W4,1(J,Rn) satisfying (1.1). Now, we define the tube solution of problem (1.1), where the functions z∈W4,1(J,Rn) and N∈W4,1(J,[0,∞) are chosen before studying the existence of this problem.
Definition 3.1. Let (z,N)∈W4,1(J,Rn)×W4,1(J,[0,∞)). The couple (z,N) is solution-tube of (1.1), if
(i) N″(t)≥0,∀t∈J.
(ii) For almost every t∈J and for all (ϖ,y,q,p)∈F,
⟨q−z″(t),f(,t,ϖ,y,q)−z‴(t)⟩+‖p−z‴(t)‖2≥N″(t)N4(t)+(N‴(t))2, |
where
F={(ϖ,y,q,p)∈R4n:‖ϖ−z(t)‖≤N(t),‖y−z′(t)‖≤N′(t),‖q−z″(t)‖=N″(t),⟨q−z″(t),p−z‴(t)⟩=N″(t)N‴(t)}. |
(iii) z(4)(t)=f(t,ϖ,y,z″(t),z‴(t)),a.e.t∈[0,1] such that N″(t)=0 and (ϖ,y)∈R2n, such that ‖ϖ−z(t)‖≤N(t) and ‖y−z′(t)‖≤N′(t).
(iv) With (1.2), we have
‖r0−(A0z″(0)−β0z‴(0))‖≤κ0N″(0)−β0N‴(0), |
‖r1−(A1z″(1)+β1z‴(1))‖≤κ1N″(1)+β1N‴(1). |
If (BC) is given by (1.3), then
z″(0)=z″(1), N‴(0)=N″(1),‖z‴(1)−z‴(0)‖≤N‴(1)−N‴(0). |
(v) ‖ϖ0−z(0)‖≤N(0), ‖ϖ1−z′(0)‖≤N′(0).
The next notation will be used
T(z,N)={ϖ∈C2(J, Rn):‖ϖ″(t)−z″(t)‖≤N″(t),‖ϖ′(t)−z′(t)‖≤N′(t)and‖ϖ‴(t)−z‴(t)‖≤N″(t)forallt∈J}. |
The next hypotheses will be used:
(F1) f:J×R4n→Rn is a L1-Carathéodory function.
(H1) There exists (z,N)∈W4,1(J,Rn)×W4,1(J,[0,∞)) a solution-tube of the main system (1.1).
The next family of problems should be considered to prove the general existence theorem that will be presented:
{ϖ(4)(t)−εϖ″(t)=fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t)),a.e. t∈J,ϖ(0)=ϖ0, ϖ′(0)=ϖ1andϖ″∈(BC), | (3.1) |
where ε,α∈J and fεα:J×R4n→Rn is defined by
fεα(t,ϖ,y,q,p)={α(N″(t)‖q−z″(t)‖f1(t,ϖ,y,˜q,˘p)−ε˜q)−ε(1−α)z″(t)+(1−αN″(t)‖q−z″(t)‖)(z(4)(t)+N(4)(t)‖q−z″(t)‖(q−z″(t))),if‖q−z″(t)‖>N″(t),α(f1(t,ϖ,y,q,p)−εq)−ε(1−α)z″(t)+(1−α)(z(4)(t)+N(4)(t)N″(t)(q−z″(t))),otherwise, |
where (z,N) is the solution-tube of (1.1),
f1(t,ϖ,y,q,p)={f(t,ˉϖ,ˆy,q,p), if ‖ϖ−z(t)‖>N(t) and ‖y−z′(t)‖>N′(t),f(t,ϖ,y,q,p), otherwise, |
ˉϖ(t)=N(t)‖ϖ−z(t)‖(ϖ−z(t))+z(t), | (3.2) |
ˆy(t)=N′(t)‖y−z′(t)‖(y−z′(t))+z′(t), | (3.3) |
˜q(t)=N″(t)‖q−z″(t)‖(q−z″(t))+z″(t), | (3.4) |
˘p(t)=p+(N‴(t)−⟨q−z″(t),p−z‴(t)⟩‖q−z″(t)‖)(q−z″(t)‖q−z″(t)‖), | (3.5) |
and where we mean
N(4)(t)N″(t)(q−z″(t))=0on{t∈J:‖q(t)−z″(t)‖=N″(t)=0}. |
We associate with fεα the operator Fε:C3(J,Rn)×J→L1(J,Rn) defined by
Fε(ϖ,α)(t)=fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t)). |
Similarly to the Lemma 3.3 and Propositions 3.4 in [20] and results in [25], we need the following auxiliary results:
Lemma 3.1. Assume (H1). If a function ϖ∈W4,1ϖ0,ϖ1,B(J,Rn) satisfies
⟨ϖ″(t)−z″(4)(t)⟩+‖ϖ‴(t)−z‴(t)‖2‖ϖ″(t)−z′′(t)‖−⟨ϖ′′(t)−z′′(t),ϖ′′′(t)−z′′′(t)⟩2‖ϖ′′(t)−z′′(t)‖3−ε‖ϖ′′(t)−z′′(t)‖≥N(4)(t)−εN′′(t), |
for a.e. t∈{t∈J:‖ϖ″(t)−z″(t)‖>N″(t)}, then ϖ∈T(z,N).
Proof. By assumption
ϖ′∈W3,1ϖ1,B(J,Rn), ϖ″∈W2,1B(J,Rn), |
and thus, from applying Lemma 2.3 to ϖ″, we obtain
‖ϖ″(t)−z″(t)‖≤N″(t),∀t∈J. |
On
{t∈J:‖ϖ′(t)−z′(t)‖>N′(t), ‖ϖ′(t)−z′(t)‖′≤‖ϖ″(t)−z″(t)‖≤N″(t).} |
The function
t→‖ϖ′(t)−z′(t)‖−N′(t), |
is nonincreasing on J. Since
‖ϖ′0−z′(0)‖≤N′(0), |
we get
‖ϖ′(t)−z′(t)‖≤N′(t),∀t∈J, |
hence
‖ϖ(t)−z(t)‖′≤‖ϖ′(t)−z′(t)‖≤N′(t). |
The function
t→‖ϖ(t)−z(t)‖−N(t), |
is nonincreasing on J and since
‖ϖ(0)−z(0)‖≤N(0), |
we obtain
‖ϖ(t)−z(t)‖≤N(t),∀t∈J. |
Proposition 3.1. Assume (F1) and (H1) hold. Then the operator Fε that was defined earlier is continuous and integrally bounded.
Proof. First, we will prove that Fε is integrally bounded. If ϖ∈B, where B is a bounded set of C3(J,Rn), ∃K>0 that satisfies ‖ϖ(i)(t)‖≤K, ∀t∈J, where i=0,1,2,3. Then fεα(t,.,.,.,.) is bounded in E, it can be observed that
‖Fε(ϖ,α)(t)‖=‖fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))‖≤max{‖f(t,ϖ,y,q,p)‖,(ϖ,y,q,p)∈E}+|N″(t)|+‖z″(t)‖+‖z(4)(t)‖+|N(4)(t)|, |
for all α∈J and almost every t∈J, where
E={(u,y,q,p)∈R4n:‖u‖≤‖z‖0+‖N‖0,‖y‖≤‖z′‖0+‖N′‖0,‖q‖≤‖z″‖0+‖N″‖0,‖p‖≤2‖ϖ‴‖0+‖z‴‖0+‖N‴‖0}. |
As f is L1-Carathéodory, z∈W4,1(J,Rn) and N∈W4,1(J,[0,∞)), it is easy to see that Fε is integrally bounded.
In order prove the continuity, we should firstly prove that if (ϖp,αp)→(ϖ,α) in C3(J,Rn)×J, then
fεαp(t,ϖp(t),ϖ′p(t),ϖ″p(t),ϖ‴p(t))→fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))a.e. t∈J. | (3.6) |
Using the fact that f is L1-Carathéodory, and from the definition of fεα, it can be concluded that (3.6) is true a.e. on {t∈J:‖ϖ″(t)−z″(t)‖≠N″(t)}. Then, by Lemma 2.1 and Proposition 3.5 in [22], we easily show that ˘ϖ‴n(t)→ϖ‴(t) on
{t∈J:‖ϖ″(t)−z″(t)‖=N″(t)>0}, |
where ˘ϖ‴n(t), is defined as (3.5). Then, (3.6) is satisfied on
{t∈J:‖ϖ″(t)−z″(t)‖=N″(t)>0}. |
For
A={t∈J:‖ϖ″(t)−z″(t)‖=N″(t)=0}, |
where ϖ″(t)=z″(t), and by Lemma 2.1, it is not hard to see that ϖ‴(t)=z‴(t), N‴(t)=0 and N(4)(t)=0, ∀t∈A, which means,
fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))=α(f1(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))−εϖ″(t))+(1−α)(z(4)(t)−εz″(t))=αf1(t,ϖ(t),ϖ′(t),z″(t),z‴(t))−εz″(t)+(1−α)z(4)(t), |
a.e. on A. By the solution tube hypothesis (Definition 3.1 condition (iii)), we have
fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))=αz(4)(t)+(1−α)z(4)(t)−εz″(t)=z(4)(t)−εz″(t), |
a.e. on A. Consequently, (3.6) must be true a.e. on J. Using the Lebesgue-dominated convergence theorem, and since Fε is integrally bounded, the proof can be concluded.
Now, we can obtain our general existence result. We follow the method of proof given in [20].
Theorem 3.1. Assume (F1), (H1), and the following conditions are satisfied:
(Hk) For every solution ϖ of the related system (3.1), ∃K>0, so that
‖ϖ‴(t)‖<K,∀t∈J. |
Then, problem (1.1) has a solution ϖ∈W4,1(J,Rn)∩T(z,N).
Proof. We first show that if (ϖ,N)∈W4,1ϖ0,ϖ1,B(J,Rn)×W4,1(J,[0,∞)) is a solution of (3.1), then
‖ϖ″(t)−z″(t)‖≤N″(t),∀t∈J. |
For the set
{t∈J:‖ϖ″(t)−z″(t)‖>N″(t)}. |
By the definition of ~ϖ″ and ˘ϖ‴(t) (as (3.4) and (3.5)), we have
‖~ϖ″(t)−z″(t)‖=N″(t), | (3.7) |
<~ϖ″(t)−z″(t),˘ϖ‴(t)−z‴(t)>=N″(t)N‴(t). |
Also
‖˘ϖ‴(t)−z‴(t)‖2=‖ϖ‴(t)−z‴(t)‖2+(N‴(t))2−⟨ϖ″(t)−z″(t),ϖ‴(t)−z‴(t)⟩2‖ϖ″(t)−z″(t)‖2. |
Then, by (H1), we obtain
⟨ϖ″(t)−z″(t)−z(4)(t)⟩+‖ϖ‴(t)−z‴(t)‖2‖ϖ″(t)−z″(t)‖−⟨ϖ″(t)−z″(t),ϖ‴(t)−z‴(t)⟩2‖ϖ″(t)−z″(t)‖3−ε‖ϖ″(t)−z″(t)‖=⟨ϖ″(t)−z″(t),fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))+εϖ″(t)⟩‖ϖ″(t)−z″(t)‖+1‖ϖ″(t)−z″(t)‖(‖ϖ‴(t)−z‴(t)‖2−⟨ϖ″(t)−z″(t),ϖ‴(t)−z‴(t)⟩2‖ϖ″(t)−z″(t)‖2)−ε‖ϖ″(t)−z″(t)‖=⟨ϖ″(t)−z″(t),αN″(t)‖ϖ″(t)−z″(t)‖(f1(t,ϖ(t),ϖ′(t),~ϖ″(t),˘ϖ‴(t))−z(4)(t))⟩‖ϖ″(t)−z″(t)‖+⟨ϖ″(t)−z″(t),(1−αN″(t)‖ϖ″(t)−z″(t)‖)N(4)(t)(ϖ″(t)−z″(t))‖ϖ″(t)−z″(t)‖⟩‖ϖ″(t)−z″(t)‖−ε⟨ϖ″(t)−z″(t),α(~ϖ″(t)−z″(t))−(ϖ″(t)−z″(t))⟩‖ϖ″(t)−z″(t)‖+‖˘ϖ‴(t)−z‴(t)‖2−(N‴(t))2‖ϖ″(t)−z″(t)‖−ε‖ϖ″(t)−z″(t)‖=α‖ϖ″(t)−z″(t)‖⟨~ϖ″(t)−z″(t),f1(t,ϖ(t),ϖ′(t),~ϖ″(t),˘ϖ‴(t))−z(4)(t)⟩+N(4)(t)(1−αN″(t)‖ϖ″(t)−z″(t)‖)−ε‖ϖ″(t)−z″(t)‖+ε‖ϖ″(t)−z″(t)‖−ε⟨ϖ″(t)−z″(t),α(~ϖ″(t)−z″(t))⟩‖ϖ″(t)−z″(t)‖+‖˘ϖ‴(t)−z‴(t)‖2−(N‴(t))2‖ϖ″(t)−z″(t)‖≥α‖ϖ″(t)−z″(t)‖(N″(t)+(N‴(t))2−‖˘ϖ‴(t)−z‴(t)‖2)+N(4)(t)−αN″(t)‖ϖ″(t)−z″(t)‖−αεN″(t)+‖˘ϖ‴(t)−z″(t)‖2−(N‴(t))2‖ϖ″(t)−z″(t)‖=N(4)(t)−αεN″(t)+(1−α)(‖˘ϖ‴(t)−z‴(t)‖2−(N‴(t))2)‖ϖ″(t)−z″(t)‖≥N(4)(t)−εN″(t), |
on
{t∈J:‖ϖ″(t)−z″(t)‖>N″(t)}. |
Using Lemma 3.1, it can be observed that any solutions to system (3.1) are in T(z,N) and then, in U, where
U={ϖ∈C3(J,Rn):‖u(i)‖0≤‖z(i)‖0+‖N(i)‖0+1,i=1,0,2;‖ϖ‴‖0≤K}. |
Fix ε∈J such that the operator Lε:C1B(J,Rn)→C0(J,Rn) given by
Lε(ϖ)(t)=ϖ′(t)−ϖ′(0)−ε∫t0ϖ(s)ds |
is invertible.
Consider the linear operator D:C3ϖ0,ϖ1,B(J,Rn)→C1B(J,Rn) defined by
D(ϖ)=ϖ″. |
It can be easily confirmed that D is invertible.
A solution to (1.1) is a fixed point of the operator
K=D−1oL−1εoNFε:C3(J,Rn)×J→C3ϖ0,ϖ1,B(J,Rn)⊂C3(J,Rn). |
Using Proposition 3.1 and Theorem 2.1, and since the operators D and Lε are continuous, it can be concluded that K is completely continuous and fixed point free on ∂U. Let
K0:C3(J,Rn)×J→C3(J, Rn) |
by K0(ϖ,α)=αK(ϖ,0). Because Fε(.,0) is integrally bounded, there exists an open bounded set K⊂C3(J,Rn), where
U⊂K and K0(C3(J,Rn)×J)⊂K, |
it can be implied from the homotopic and the excision properties of the Leray-Schauder theorem that
1=deg(Id,K,0)=°(Id−K0(.,1),K,0)=deg(Id−K(.,0),K,0)=°(Id−K(.,0),U,0)=deg(Id−K(.,1),U,0). |
As a result, there exists a solution ϖ∈T(z,N) for α=1 to (3.1), which also can solve (1.1) by definition of fε1. The proof is complete.
Now, following from our general existence theorem (Theorem 3.1), other existence results will be presented. We will consider the following assumptions:
(H2) There exist a function γ∈L1(J,[0,∞)) and a Borel measurable function Ψ∈C([0,∞),[1,∞)) s.t.
(ⅰ) ‖f(t,ϖ,y,q,p)‖≤γ(t)Ψ(‖p‖),∀t∈J and ∀(ϖ,y,q,p)∈R4n, where ‖ϖ−z(t)‖≤N(t), ‖y−z′(t)‖≤N′(t) and ‖q−z″(t)‖≤N″(t),
(ⅱ) ∀c≥0, we have
∫∞cdτΨ(τ)=∞. |
(H3) There exist, a function γ∈L1(J,[0,∞)) and a Borel measurable function Ψ∈C([0,∞],]0,∞)) s.t.
(ⅰ) ‖⟨p,f(t,ϖ,y,q,p)⟩‖≤Ψ(‖p‖)(γ(t)+‖p‖),∀t∈J and ∀(ϖ,y,q,p)∈R4n, where ‖ϖ−z(t)‖≤N(t), ‖y−z′(t)‖≤N′(t) and ‖q−z″(t)‖≤N″(t),
(ⅱ) ∀c≥0, we have
∫∞cτdτΨ(τ)+τ=∞. |
(H4) ∃r,b>0, c≥0 and a function h∈L1(J,R) s.t. ∀t∈J, ∀(ϖ,y,q,p)∈R4n, where
‖ϖ−z(t)‖≤N(t),‖y−z′(t)‖≤N′(t),‖q−z″(t)‖≤N″(t), |
and ‖p‖≥r, then
(b+c‖q‖)σ(t,ϖ,y,q,p)≥‖p‖−h(t), |
where
σ(t,ϖ,y,q,p)=⟨q,f(t,ϖ,y,q,p)⟩+‖p‖2‖p‖−⟨p,f(t,ϖ,y,q,p)⟩⟨q,p⟩‖p‖3. |
(H5) ∃a≥0 and l∈L1(J,R) s.t.
‖f(t,ϖ,y,q,p)‖≤a(⟨q,f(t,ϖ,y,q,p)⟩+‖p‖2)+l(t), |
∀t∈J and ∀(ϖ,y,q,p)∈R4n, where
‖ϖ−z(t)‖≤N(t),‖y−z′(t)‖≤N′(t), |
and
‖q−z″(t)‖≤N″(t). |
Theorem 3.2. Assume (F1), (H1), and (\mathcal{H}2) aresatisfied.If (BC) isgivenby (1.2) with \max \left\{ \beta _{0}, \beta _{1}\right\} > $$ 0 $, then system $ (1.1) $ has at least one solution $ \varpi \in T(z, N)\cap W^{4,1}( \mathcal{J}, \mathbb{\ R}^{n}) $.
Proof. Theorem 3.1 will guarantee the existence of a solution if we can obtain a priori bound on the third derivative of any solution ϖ to (3.1). It is known that ϖ∈T(z,N) from the Theorem 3.1 proof. Therefore, since (BC) is given by (1.2) with max{β0,β1}>0, ∃k>0, s.t.
min{‖ϖ‴(0)‖,‖ϖ‴(1)‖}≤k. |
Now, let R>k such that
∫RkdsΨ(s)>L=‖γ‖L1+ε‖N″‖0+‖z(4)‖L1+‖N(4)‖L1. |
Suppose there exists t1∈[0,1] s.t. ‖ϖ‴(t1)‖≥R. Then, there exists t0≠t1∈[0,1] such that ‖ϖ‴(t0)‖=k and ‖ϖ‴(t)‖≥k, ∀t∈[t0,t1]. Let us assume that t0<t1. Thus, by (H2), almost everywhere on [t0,t1], we have
‖ϖ‴(t)‖′=⟨ϖ‴(t)⟩‖ϖ‴(t)‖≤‖ϖ(4)(t)‖≤‖f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))‖+ε‖ϖ″(t)−z″(t)‖+‖z(4)(t)‖+|N(4)(t)|≤‖γ(t)‖Ψ(‖ϖ‴(t)‖)+ε‖N″(t)‖0+‖z(4)(t)‖+‖N(4)‖L1. |
So,
∫t1t0‖ϖ‴(t)‖′tΨ(‖ϖ‴(t)‖)dt≤L. |
Then, we have
∫t1t0‖ϖ‴(t)‖′tΨ(‖ϖ‴(t)‖)dt=∫‖ϖ‴(t1)‖‖ϖ‴(t0)‖dsΨ(s)≥∫RkdsΨ(s)>L, |
which contradict the assumptions. So, for any solution ϖ of (3.1), ∃R>0 s.t. ‖ϖ‴(t)‖<R, ∀t∈J.
If (H2) is replaced by (H3), extra assumptions are needed.
Theorem 3.3. Assume (F1), (H1), (H3), and (\mathcal{H}4) or (\mathcal{H}5) aresatisfied.Then,thereexistsasolution \varpi \in T(z, N)\cap W^{4,1}(\mathcal{J}, \mathbb{\ R}^{n}) to (1.1) $.
For this end, we need the next three Lemmas.
Lemma 3.2. [20] Let r,k≥0, N∈L1([0,1],R) and Ψ∈C([0,∞[,]0,∞[) be a Borel measurable function s.t.
∫∞rτdτΨ(τ)>‖N‖L1([0,1],R)+k. |
Then ∃K>0, s.t. ‖ϖ′‖0<K, ∀ϖ∈W2,1([0,1],Rn) satisfy:
(i) mint∈[0,1]‖ϖ′(t)‖≤r;
(ii) ‖ϖ′‖L1([t0,t1],R)≤k for every interval [t0,t1]⊂{t∈[0,1]:‖ϖ′(t)‖≥r};
(iii) |⟨ϖ′(t),ϖ″(t)⟩|≤Ψ(‖ϖ′(t)‖)(N(t)+‖ϖ′(t)‖) a.e. on
{t∈[0,1]:‖ϖ′(t)‖≥r}. |
Lemma 3.3. [20] Let r,ν>0, γ≥0 and N∈L1([0,1],R). Then there exists a nondecreasing function ω∈C[0,∞[,[0,∞[) s.t.
‖ϖ′‖L1([t0,t1],R)≤ω(‖ϖ‖0), |
and
mint∈[0,1]‖ϖ′(t)‖≤max{r,;ω(‖ϖ‖0)}. |
∀u∈W2,1([0,1],Rn) and
{t∈[t0,t1]:‖ϖ′(t)‖≥r}, |
the following inequality
(ν+γ‖ϖ(t)‖)σ0(t,ϖ)+γ⟨ϖ(t),ϖ′(t)⟩2‖ϖ(t)‖ϖ′(t)‖≥‖ϖ′(t)‖−N(t) |
is satisfied, where
σ0(t,ϖ)=⟨ϖ(t),ϖ″(t)⟩+‖ϖ′2‖ϖ′(t)‖−⟨ϖ′(t),ϖ″(t)⟩⟨ϖ(t),ϖ′(t)⟩‖ϖ′(t)‖3. |
Lemma 3.4. [20] Let K>0, and N∈L1([0,1],R). Then there exists an increasing function ω∈C([0,∞[,]0,∞[) s.t. ‖ϖ′‖L1([0,1],R)≤ω(‖ϖ‖0) for all ϖ∈W2,1([0,1],Rn) that satisfies
‖ϖ″(t)‖≤k(⟨ϖ(t),ϖ″(t)⟩+‖ϖ′(t)‖2)+N(t), |
for almost every t∈[0,1].
Proof of Theorem 3.3. Similarly to the previous proof, we need Theorem 3.1 to prove that the third derivative of all solutions ϖ to (3.1) is bounded. Let ϖ be a solution to (3.1), where ϖ∈T(z,N) from Theorem 3.1 proof. We obtain from (H3),
|⟨ϖ‴(t),ϖ(4)(t)⟩|≤|⟨ϖ‴(t),f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))⟩|+(ε‖ϖ″(t)−z″(t)‖+‖z(4)(t)‖+|N(4)(t)|)‖ϖ‴(t)‖≤(γ(t)+‖ϖ‴(t)‖)Ψ(‖ϖ‴(t)‖)+(ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)|)‖ϖ‴(t)‖≤(Ψ(‖ϖ‴(t)‖)+‖ϖ‴(t)‖)+(γ(t)+‖ϖ‴(t)‖+ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)|), |
for almost every t∈[0,1]. Thus, condition (iii) of Lemma 3.2 is verified, where
ψ(τ)=Ψ(τ)+τandN(τ)=γ(τ)+ε|N″(τ)|+‖z(4)(τ)‖+|N(4)(t)|. |
Therefore, it is enough to prove that conditions (i) and (ii) are verified. (H4) guarantees that a.e. on
{t∈[0,1]:‖ϖ‴(t)‖≥r}, |
we have
σ0(t,ϖ″)=⟨ϖ″(t)⟩+‖ϖ‴(t)‖2‖ϖ‴(t)‖−⟨ϖ‴(t)⟩⟨ϖ″(t),ϖ‴(t)⟩‖ϖ‴(t)‖3=ασ(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))+(1−α)‖ϖ‴(t)‖ +(1−α)⟨ϖ″(t)+(ε+N(4)(t)N″(t))(ϖ″(t)−z″(t))⟩‖ϖ‴(t)‖ −(1−α)⟨ϖ‴(t)+(ε+N(4)(t)N″(t))(ϖ″(t)−z″(t))⟩⟨ϖ″(t),ϖ‴(t)⟩‖ϖ‴(t)‖3≥ασ(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))+(1−α)‖ϖ‴(t)‖ −2(‖z″(t)‖+|N″(t)|)(‖z(4)(t)‖+ε|N″(t)|+|N(4)(t)|)r. |
Thus, we have
(b+c‖ϖ″(t)‖)σ0(t,ϖ″)+c⟨ϖ″(t),ϖ‴(t)⟩2‖ϖ″(t)‖‖ϖ‴(t)‖≥α‖ϖ‴(t)‖+b(1−α)‖ϖ‴(t)‖−h(t)−δ0,(t), |
where
δ0(t)=2r(b+c‖z″(t)‖+c|N″(t)|)(‖z″(t)‖+|N″(t)|)(‖z(4)(t)‖+ε|N″(t)|+|N(4)(t)|). |
If we take
z=minα∈[0,1]{α+b(1−α)},ν=bzandθ=cη, |
we can apply Lemma 3.3 to ϖ″([0,1],Rn). Thus, conditions of Lemma 3.2 are verified. Moreover, if (H5) holds, we have
‖ϖ(4)(t)‖≤α‖f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))‖+ε‖ϖ″(t)−z″(t)‖+‖z(4)(t)‖+|N(4)(t)|≤αa(⟨ϖ″(t),f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))⟩+‖ϖ‴(t)‖2)+l(t) +ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)|≤a(⟨ϖ″(t),ϖ(4)(t)⟩+‖ϖ‴(t)‖2)+ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)| −a(1−α)⟨ϖ″(t),z(4)(t)+(N(4)(t)N″(t)+ε)(ϖ″(t)−z″(t))⟩≤a(⟨ϖ″(t),ϖ(4)(t)⟩+‖ϖ‴(t)‖2)+ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)| +a(‖z″(t)‖+|N″(t)|)(‖z(4)(t)‖+|N(4)(t)|+εN″(t)). |
Therefore, if Lemma 3.4 is applied to ϖ″([0,1],Rn), all conditions of Lemma 3.2 are satisfied. As a result, for all solutions ϖ of (3.1), ‖ϖ‴‖0<K for some constant K>0.
From the previous results, we obtain the following consequence:
Corollary 3.1. Assume (F1), (H1), (H2), and (H4) or (H5) are satisfied. Then, we have a solution ϖ∈T(z,N)∩W4,1([0,1],Rn) to the system (1.1).
Remark 3.1. Definition 3.1 is associated to the definitions of lower and upper solutions to the fourth-order differential equation. These definitions are used in [17], and introduce them for problems (1.1) and (1.2).
Definition 3.2. Let n=1 and ϖ0=ϖ1=0.
A function κ∈C4(]0,1[)∩C3(J) is called a lower solution to (1.1) and (1.2), if
(i) κ(4)(t)≥f(t,κ(t),κ′(t),κ″(t),κ‴(t)) for every t∈J;
(ii) κ(0)=κ′(0)=0;
(iii) A0κ″(0)−β0κ‴(0)≤r0 and A1κ″(1)+β1κ‴(1)≤r1.
On the other hand, an upper solution to (1.1) and (1.2) is a function ν∈C4(]0,1[)∩C3(J) that satisfies (i)–(iii) with reversed inequalities.
Similarly to Remark 3.2 in [20], we consider the following assumptions:
(A) There exist lower and upper solutions, κ and ν, respectively, to (1.1) and (1.2), where κ≤ν .
(B) There exists a solution-tube (z,N) to (1.1) and (1.2).
(C) There exist lower and upper solutions, κ≤ν, to (1.1) and (1.2) s.t.
(i) κ″(t)≤ν″(t)) for all t∈J;
(ii) f(t,ν(t),ν′(t),q,p)≤f(t,ϖ,y,q,p)≤f(t,κ(t),κ′(t),q,p); ∀t∈J and (ϖ,y,q,p)∈R4n such that κ(t)≤ϖ≤ν(t) and κ′(t)≤ϖ′(t)≤ν′(t).
It can be easily checked that
● If (B) holds with z and N of class C4, and z(0)=N(0)=0, then (A) holds.
Indeed, κ=z−N and ν=z+N are respectively lower and upper solutions of (1.1) and (1.2). However, (A) does not imply (B).
Noting that (B) is more general than (C), see [17]; i.e.,
● If (C) is verified, then (B) is verified.
Taking z=(κ+ν)/2 and N=(ν−κ)/2. But, (B) does not imply (C) (ii) and κ(0)=ν(0)=0.
Next, we present two examples to illustrate the applicability of Theorem 3.3.
Example 3.1. Consider the following system:
{ϖ(4)(t)=ϖ‴(t)+‖ϖ‴(t)‖(‖ϖ″(t)‖2ϖ′(t)−⟨ϖ′(t),ϖ″(t)⟩ϖ″(t))−ξ,a.e. t∈J,ϖ(0)=0,ϖ′(0)=0,A0ϖ″(0)=0,A1ϖ″(1)+βt=1ϖ‴(1)=0, | (3.8) |
here ξ∈Rn,‖ξ‖=1, and Aiandβi are given before for i=0,1. Show that when z≡0,N(t)=t36, (z,N) is a solution-tube of (3.8). We have (H3) and (H4) are verified for
Ψ(τ)=3τ+1,γ(t)=0,b=1,c=0,r>0,h(τ)=2τr+τ5. |
Owing to the Theorem 3.3, the problem (3.8) has at least one solution ϖ s.t.
‖ϖ(t)‖≤t36, ‖ϖ(t)′‖≤t22 and ‖ϖ(t)′′‖≤t for all t∈J. |
Example 3.2. Consider the following system:
{ϖ(4)(t)=ϖ″(t)(‖ϖ‴(t)‖2+1)+φ(t),a.e. t∈J,ϖ(0)=0,ϖ′(0)=0,ϖ″(0)=ϖ″(1),ϖ‴(0)=ϖ‴(1), | (3.9) |
where φ∈L∞(J,Rn) with ‖φ‖L∞≤1. Show that for z≡0,N(t)=t22, (z,N) is a solution-tube of (3.9). We have (H3) and (H5) are verified when
Ψ(τ)=τ2+2, γ(t)=0,a=1,l(t)=3. |
By Theorem 3.3, the problem (3.9) has at least one solution ϖ s.t.
‖ϖ(t)‖≤t22, ‖ϖ′(t)‖≤t, ‖ϖ′′(t)‖≤1,∀t∈J. |
Our paper discusses the existence of solutions for fourth-order differential equation systems, focusing particularly on cases involving L1-Carathéodory functions on the right-hand side of the equations. We first, introduced the concept of a solution-tube, which is an innovative approach that extends the concepts of upper and lower solutions applicable to fourth-order equations into the domain of systems. It outlines the mathematical framework necessary to demonstrate that solutions exist for these types of differential equation systems under specified boundary conditions (such as Sturm-Liouville and periodic conditions). The paper stands on prior results regarding higher-order differential equations, providing a fresh perspective and methodology that can be used to explore further developments in the field. In addition to presenting the theoretical underpinnings, we also illustrated the practicality of our results with examples, contributing to the mathematical discourse on differential equations and our solutions, which ultimately serves as a scholarly contribution to understanding the dynamics of fourth-order systems and the existence of their solutions; please see [26,27].
Bouharket Bendouma: Conceptualization, formal analysis, Writing-original draft preparation; Fatima Zohra Ladrani and Keltoum Bouhali: investigation, Methodology; Ahmed Hammoudi and Loay Alkhalifa: Writing-review and editing. All authors have read and approved the final version of the manuscript for publication.
The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).
The authors declare that there is no conflict of interest.
[1] |
E. J. Allen, S. J. Novosel, Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics, 64 (1998), 117–142. https://doi.org/10.1080/17442509808834159 doi: 10.1080/17442509808834159
![]() |
[2] |
A. Andersson, S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation, Math. Comput., 85 (2016), 1335–1358. https://doi.org/10.1090/mcom/3016 doi: 10.1090/mcom/3016
![]() |
[3] |
I. Babuˇska, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), 1005–1034. https://doi.org/10.1137/050645142 doi: 10.1137/050645142
![]() |
[4] |
I. Babuˇska, R. Tempone, G. E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42 (2004), 800–825. https://doi.org/10.1137/S0036142902418680 doi: 10.1137/S0036142902418680
![]() |
[5] | M. Beccari, M. Hutzenthaler, A. Jentzen, R. Kurniawan, F. Lindner, D. Salimova, Strong and weak divergence of exponential and linear-implicit euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities, arXiv preprint arXiv: 1903.06066, 2019. |
[6] | C. E. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Anal., 40 (2014), 1–40. |
[7] |
C. E. Bréhier, J. Cui, J. Hong, Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation, IMA J. Numer. Anal., 39 (2019), 2096–2134. https://doi.org/10.1093/imanum/dry052 doi: 10.1093/imanum/dry052
![]() |
[8] | M. Cai, S. Gan, X. Wang, Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise, J. Sci. Comput., 86 (2021), 1–30. |
[9] |
Y. Cao, J. Hong, Z. Liu, Approximating stochastic evolution equations with additive white and rough noises, SIAM J. Numer. Anal., 55 (2017), 1958–1981. https://doi.org/10.1137/16M1056122 doi: 10.1137/16M1056122
![]() |
[10] |
N. Chopin, Fast simulation of truncated Gaussian distributions, Stat. Comput., 21 (2011), 275–288. https://doi.org/10.1007/s11222-009-9168-1 doi: 10.1007/s11222-009-9168-1
![]() |
[11] |
J. Cui, J. Hong, Analysis of a splitting scheme for damped stochastic nonlinear Schrödinger equation with multiplicative noise, SIAM J. Numer. Anal., 56 (2018), 2045–2069. https://doi.org/10.1137/17M1154904 doi: 10.1137/17M1154904
![]() |
[12] |
J. Cui, J. Hong, Strong and weak convergence rates of a spatial approximation for stochastic partial differential equation with one-sided Lipschitz coefficient, SIAM J. Numer. Anal., 57 (2019), 1815–1841. https://doi.org/10.1137/18M1215554 doi: 10.1137/18M1215554
![]() |
[13] |
J. Cui, J. Hong, Z. Liu, Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations, J. Differ. Equations, 263 (2017), 3687–3713. https://doi.org/10.1016/j.jde.2017.05.002 doi: 10.1016/j.jde.2017.05.002
![]() |
[14] |
J. Cui, J. Hong, Z. Liu, W. Zhou, Strong convergence rate of splitting schemes for stochastic nonlinear Schrödinger equations, J. Differ. Equations, 266 (2019), 5625–5663. https://doi.org/10.1016/j.jde.2018.10.034 doi: 10.1016/j.jde.2018.10.034
![]() |
[15] |
J. Cui, J. Hong, L. Sun, Strong Convergence of Full Discretization for Stochastic Cahn-Hilliard Equation Driven by Additive Noise, SIAM J. Numer. Anal., 59 (2021), 2866–2899. https://doi.org/10.1137/20M1382131 doi: 10.1137/20M1382131
![]() |
[16] | J. Cui, J. Hong, L. Sun, Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficients, Stoch. Proc. their Appl., 134 (2021), 55–93. |
[17] | G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge university press, 2014. |
[18] |
A. Davie, J. Gaines, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math. Comput., 70 (2001), 121–134. https://doi.org/10.1090/S0025-5718-00-01224-2 doi: 10.1090/S0025-5718-00-01224-2
![]() |
[19] |
A. De Bouard, A. Debussche, Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation, Appl. Math. Optim., 54 (2006), 369–399. https://doi.org/10.1007/s00245-006-0875-0 doi: 10.1007/s00245-006-0875-0
![]() |
[20] |
A. Debussche, Weak approximation of stochastic partial differential equations: The nonlinear case, Math. Comput., 80 (2011), 89–117. https://doi.org/10.1090/S0025-5718-2010-02395-6 doi: 10.1090/S0025-5718-2010-02395-6
![]() |
[21] |
A. Debussche, J. Printems, Weak order for the discretization of the stochastic heat equation, Math. Comput., 78 (2009), 845–863. https://doi.org/10.1090/S0025-5718-08-02184-4 doi: 10.1090/S0025-5718-08-02184-4
![]() |
[22] |
C. R. Dietrich, A simple and efficient space domain implementation of the turning bands method, Water Resour. Res., 31 (1995), 147–156. https://doi.org/10.1029/94WR01457 doi: 10.1029/94WR01457
![]() |
[23] |
C. R. Dietrich, G. N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput., 18 (1997), 1088–1107. https://doi.org/10.1137/S1064827592240555 doi: 10.1137/S1064827592240555
![]() |
[24] | Q. Du, T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises, SIAM J. Numer. Anal., 140 (2002), 1421–1445. |
[25] | K. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, Semigroup Forum, 63 (1999), 278–280. |
[26] |
X. Feng, Y. Li, Y. Zhang, Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise, SIAM J. Numer. Anal., 55 (2017), 194–216. https://doi.org/10.1137/15M1022124 doi: 10.1137/15M1022124
![]() |
[27] | M. Geissert, M. Kovács, S. Larsson, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise, BIT Numer. Math., 549 (2009), 343–356. |
[28] | I. Gohberg, S. Goldberg, M. A. Kaashoek, Hilbert-schmidt operators, In: Classes of Linear Operators Vol. I, 138–147. Springer, 1990. |
[29] | I. Gy¨ongy, S. Sabanis, D. ˇSiˇska, Convergence of tamed Euler schemes for a class of stochastic evolution equations, Stochast. Partial Differ. Equations: Anal. Comput., 4 (2016), 225–245. |
[30] |
I. Gyöngy, A. Millet, Rate of convergence of space time approximations for stochastic evolution equations, Potential Anal., 30 (2009), 29–64. https://doi.org/10.1111/j.1755-3768.1986.tb06988.x doi: 10.1111/j.1755-3768.1986.tb06988.x
![]() |
[31] |
E. Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal., 18 (2003), 141–186. https://doi.org/10.1023/A:1020552804087 doi: 10.1023/A:1020552804087
![]() |
[32] | E. Hausenblas, Weak approximation for semilinear stochastic evolution equations, In: Stochastic analysis and related topics VIII, 111–128. Springer, 2003. |
[33] | M. Hutzenthaler, A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, volume 236. American Mathematical Society, 2015. |
[34] |
M. Hutzenthaler, A. Jentzen, P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, P. Royal Soc. A-Math. Phy., 467 (2011), 1563–1576. https://doi.org/10.1098/rspa.2010.0348 doi: 10.1098/rspa.2010.0348
![]() |
[35] |
A. Jentzen, P. E. Kloeden, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise, P. Royal Soc. A-Math. Phy., 465 (2008), 649–667. https://doi.org/10.1098/rspa.2008.0325 doi: 10.1098/rspa.2008.0325
![]() |
[36] | A. Jentzen, P. Puˇsnik, Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities, IMA J. Numer. Anal., 40 (2020), 1005–1050. |
[37] | N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions, volume 289, John wiley & sons, 1995. |
[38] |
Y. Kazashi, Quasi-monte carlo integration with product weights for elliptic PDEs with log-normal coeffcients, IMA J. Numer. Anal., 39 (2019), 1563–1593. https://doi.org/10.1093/imanum/dry028 doi: 10.1093/imanum/dry028
![]() |
[39] |
P. E. Kloeden, G. J. Lord, A. et al Neuenkirch, The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds, J. Comput. Appl. Math., 235 (2011), 1245–1260. https://doi.org/10.1016/j.cam.2010.08.011 doi: 10.1016/j.cam.2010.08.011
![]() |
[40] | P. E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, Springer Science and Business Media, 2013. |
[41] |
M. Kovács, S. Larsson, F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise, BIT Numer. Math., 52 (2012), 85–108. https://doi.org/10.1007/s10543-011-0344-2 doi: 10.1007/s10543-011-0344-2
![]() |
[42] | M. Kovács, S. Larsson, F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II, Fully discrete schemes, BIT Numer. Math., 53 (2013), 497–525. |
[43] |
M. Kovˊacs, S. Larsson, F. Lindgren, On the discretisation in time of the stochastic Allen–Cahn equation, Math. Nachr., 291 (2018), 966–995. https://doi.org/10.1002/mana.201600283 doi: 10.1002/mana.201600283
![]() |
[44] |
R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal., 34 (2014), 217–251. https://doi.org/10.1093/imanum/drs055 doi: 10.1093/imanum/drs055
![]() |
[45] | R. Kruse, Strong and weak approximation of semilinear stochastic evolution equations, Springer, 2014. |
[46] | C. Li, Y. Huang, N. Yi, An unconditionally energy stable second order finite element method for solving the Allen–Cahn equation, J. Comput. Appl. Math., 353 (2019), 38–48. |
[47] | N. Li, B. Meng, X. Feng, D. Gui, The spectral collocation method for the stochastic Allen-Cahn equation via generalized polynomial chaos, Numer. Heat Tr. B-Fund., 68 (2015), 11–29. |
[48] |
N. Li, J. Zhao, X. Feng, D. Gui, Generalized polynomial chaos for the convection diffusion equation with uncertainty, Int. J. Heat Mass Transfer, 97 (2016), 289–300. https://doi.org/10.1016/j.ijheatmasstransfer.2016.02.006 doi: 10.1016/j.ijheatmasstransfer.2016.02.006
![]() |
[49] |
Y. Li, H. G. Lee, D. Jeong, J. Kim, An unconditionally stable hybrid numerical method for solving the Allen–Cahn equation, Comput. Math. Appl., 60 (2010), 1591–1606. https://doi.org/10.1016/j.camwa.2010.06.041 doi: 10.1016/j.camwa.2010.06.041
![]() |
[50] |
F. Lindner, R. Schilling, Weak order for the discretization of the stochastic heat equation driven by impulsive noise, Potential Anal., 38 (2013), 345–379. https://doi.org/10.1007/s11118-012-9276-y doi: 10.1007/s11118-012-9276-y
![]() |
[51] |
Z. Liu, Z. Qiao, Strong approximation of monotone stochastic partial differential equations driven by white noise, IMA J. Numer. Anal., 40 (2020), 1074–1093. https://doi.org/10.1093/imanum/dry088 doi: 10.1093/imanum/dry088
![]() |
[52] |
Z. Liu, Z. Qiao, Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise, Stoch. Partial Differ., 9 (2021), 559–602. https://doi.org/10.1007/s40072-020-00179-2 doi: 10.1007/s40072-020-00179-2
![]() |
[53] | G. J. Lord, C. E. Powell, T. Shardlow, An introduction to computational stochastic PDEs, Cambridge University Press, 2014. |
[54] |
A. Mantoglou, J. L. Wilson, The turning bands method for simulation of random fields using line generation by a spectral method, Water Resour. Res., 18 (1982), 1379–1394. https://doi.org/10.1029/WR018i005p01379 doi: 10.1029/WR018i005p01379
![]() |
[55] |
G. Milstein, M. V. Tretyakov, Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients, SIAM J. Numer. Anal., 43 (2005), 1139–1154. https://doi.org/10.1137/040612026 doi: 10.1137/040612026
![]() |
[56] | G. N. Milstein, M. V. Tretyakov, Stochastic numerics for mathematical physics, Springer Science and Business Media, 2013. |
[57] |
G. N. Newsam, C. R. Dietrich, Bounds on the size of nonnegative definite circulant embeddings of positive definite Toeplitz matrices, IEEE T. Inform. Theory, 40 (1994), 1218–1220. https://doi.org/10.1109/18.335952 doi: 10.1109/18.335952
![]() |
[58] |
J. Printems, On the discretization in time of parabolic stochastic partial differential equations, ESAIM: Math. Model. Numer. Anal., 35 (2001), 1055–1078. https://doi.org/10.1051/m2an:2001148 doi: 10.1051/m2an:2001148
![]() |
[59] |
M. Sauer, W. Stannat, Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition, Math. Comput., 84 (2015), 743–766. https://doi.org/10.1090/S0025-5718-2014-02873-1 doi: 10.1090/S0025-5718-2014-02873-1
![]() |
[60] |
M. Shinozuka, Simulation of multivariate and multidimensional random processes, J. Acoust. Soc. Am., 49 (1971), 357–368. https://doi.org/10.1121/1.1912338 doi: 10.1121/1.1912338
![]() |
[61] |
M. Shinozuka, C. M. Jan, Digital simulation of random processes and its applications, J. Sound Vib., 25 (1972), 111–128. https://doi.org/10.1016/0022-460X(72)90600-1 doi: 10.1016/0022-460X(72)90600-1
![]() |
[62] | J. L. Wadsworth, J. A. Tawn, Efficient inference for spatial extreme value processes associated to log-Gaussian random functions, Biometrika, 101 (2014), 1–15. |
[63] | J. B. Walsh, Finite element methods for parabolic stochastic PDEs, Potential Anal., 23 (2005), 1–43. |
[64] | X. Wang, Strong convergence rates of the linear implicit Euler method for the finite element discretization of SPDEs with additive noise IMA J. Numer. Anal., 37 (2017), 965–984. |
[65] |
X. Wang, An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation, Stoch. Proc. Appl., 130 (2020), 6271–6299. https://doi.org/10.1016/j.spa.2020.05.011 doi: 10.1016/j.spa.2020.05.011
![]() |
[66] |
X. Wang, S. Gan, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise, J. Math. Anal. Appl., 398 (2013), 151–169. https://doi.org/10.1016/j.jmaa.2012.08.038 doi: 10.1016/j.jmaa.2012.08.038
![]() |
[67] | A. TA. Wood, G. Chan, Simulation of stationary Gaussian processes in [0,1]d. J. Comput. Graph. Stat., 3 (1994), 409–432. https://doi.org/10.1086/174579 |
[68] |
D. Xiu, J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Comput. Phys., 228 (2009), 266–281. https://doi.org/10.1016/j.jcp.2008.09.008 doi: 10.1016/j.jcp.2008.09.008
![]() |
[69] |
Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363–1384. https://doi.org/10.1137/040605278 doi: 10.1137/040605278
![]() |
[70] | Z. Zhang, G. Karniadakis, Numerical methods for stochastic partial differential equations with white noise, Springer, 2017. |
1. | Peng E, Tingting Xu, Linhua Deng, Yulin Shan, Miao Wan, Weihong Zhou, Solutions of a class of higher order variable coefficient homogeneous differential equations, 2025, 20, 1556-1801, 213, 10.3934/nhm.2025011 |