Citation: Daniel Borchert, Diego A. Suarez-Zuluaga, Yvonne E. Thomassen, Christoph Herwig. Risk assessment and integrated process modeling–an improved QbD approach for the development of the bioprocess control strategy[J]. AIMS Bioengineering, 2020, 7(4): 254-271. doi: 10.3934/bioeng.2020022
[1] | Ricardo Almeida . Variational problems of variable fractional order involving arbitrary kernels. AIMS Mathematics, 2022, 7(10): 18690-18707. doi: 10.3934/math.20221028 |
[2] | Xiaojing Du, Xiaotong Liang, Yonghong Xie . Integral expressions of solutions to higher order $ \lambda $-weighted Dirac equations valued in the parameter dependent Clifford algebra. AIMS Mathematics, 2025, 10(1): 1043-1060. doi: 10.3934/math.2025050 |
[3] | Yongjian Hu, Huifeng Hao, Xuzhou Zhan . On the solvability of the indefinite Hamburger moment problem. AIMS Mathematics, 2023, 8(12): 30023-30037. doi: 10.3934/math.20231535 |
[4] | Hong Li, Keyu Zhang, Hongyan Xu . Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables. AIMS Mathematics, 2021, 6(11): 11796-11814. doi: 10.3934/math.2021685 |
[5] | Kun Li, Peng Wang . Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions. AIMS Mathematics, 2022, 7(6): 11487-11508. doi: 10.3934/math.2022640 |
[6] | Tuba Gulsen, Emrah Yilmaz, Ayse Çiğdem Yar . Proportional fractional Dirac dynamic system. AIMS Mathematics, 2024, 9(4): 9951-9968. doi: 10.3934/math.2024487 |
[7] | Valérie Gauthier-Umaña, Henryk Gzyl, Enrique ter Horst . Decoding as a linear ill-posed problem: The entropy minimization approach. AIMS Mathematics, 2025, 10(2): 4139-4152. doi: 10.3934/math.2025192 |
[8] | Yong Liu, Chaofeng Gao, Shuai Jiang . On meromorphic solutions of certain differential-difference equations. AIMS Mathematics, 2021, 6(9): 10343-10354. doi: 10.3934/math.2021599 |
[9] | Clara Burgos, Juan Carlos Cortés, Elena López-Navarro, Rafael Jacinto Villanueva . Probabilistic analysis of linear-quadratic logistic-type models with hybrid uncertainties via probability density functions. AIMS Mathematics, 2021, 6(5): 4938-4957. doi: 10.3934/math.2021290 |
[10] | Noureddine Bahri, Abderrahmane Beniani, Abdelkader Braik, Svetlin G. Georgiev, Zayd Hajjej, Khaled Zennir . Global existence and energy decay for a transmission problem under a boundary fractional derivative type. AIMS Mathematics, 2023, 8(11): 27605-27625. doi: 10.3934/math.20231412 |
We consider the system of Dirac equations
ℓy(x):=By′(x)+Q(x)y(x)=λy(x), x∈[a,b], | (1) |
where B=(01−10), Q(x)=(p(x)q(x)q(x)−p(x)), y(x)=(y1(x)y2(x)), p(x), q(x) are real valued functions in L2(a,b) and λ is a spectral parameter, with boundary conditions
U(y):=y2(a)+f1(λ)y1(a)=0 | (2) |
V(y):=y2(b)+f2(λ)y1(b)=0 | (3) |
and with transmission conditions
{y1(wi+0)=αiy1(wi−0)y2(wi+0)=α−1iy2(wi−0)+hi(λ)y1(wi−0)(i=1,2) | (4) |
where fi(λ), hi(λ)(i=1,2) are rational functions of Herglotz-Nevanlinna type such that
fi(λ)=aiλ+bi−Ni∑k=1fikλ−gik | (5) |
hi(λ)=miλ+ni−Pi∑k=1uikλ−tik (i=1,2) | (6) |
ai, bi, fik, gik,mi,ni,uik and tik are real numbers, a1<0, f1k<0, a2>0, f2k>0,mi>0, uik>0 and gi1<gi2<...<giNi, ti1<ti2<...<tiPi, αi>0 and a<w1<w2<b. In special case, when fi(λ)=∞, conditions (2) and (3) turn to Dirichlet conditions y1(a)=y1(b)=0 respectively. Moreover, when hi(λ)=∞, conditions (4) turn to y1(w2+0)=α2y1(w2−0), y2(w2+0)=α−12y2(w2−0)+h2(λ)y1(w2−0) and y1(w1+0)=α1y1(w1−0), y2(w1+0)=α−11y2(w1−0)+h1(λ)y1(w1−0) according to order i=1,2.
Inverse problems of spectral analysis compose of recovering operators from their spectral data. Such problems arise in mathematics, physics, geophysics, mechanics, electronics, meteorology and other branches of natural sciences. Inverse problems also play important role in solving many equations in mathematical physics.
R1(λ)y1(a)+R2(λ)y2(a)=0 is a boundary condition depending spectral parameter where R1(λ) and R2(λ) are polynomials. When degR1(λ)=degR2(λ)=1, this equality depends on spectral parameter as linearly. On the other hand, it is more difficult to search for higher orders of polynomials R1(λ) and R2(λ). When R1(λ)R2(λ) is rational function of Herglotz-Nevanlinna type such that f(λ)=aλ+b−N∑k=1fkλ−gk in boundary conditions, direct and inverse problems for Sturm-Liouville operator have been studied [1,2,3,4,5,6,7,8,9,10,11]. In this paper, direct and inverse spectral problem is studied for the system of Dirac equations with rational function of Herglotz-Nevanlinna in boundary and transmission conditions.
On the other hand, inverse problem firstly was studied by Ambarzumian in 1929 [12]. After that, G. Borg was proved the most important uniqueness theorem in 1946 [13]. In the light of these studies, we note that for the classical Sturm-Liouville operator and Dirac operator, the inverse problem has been studied fairly (see [14,15,16,17,18,19,20], where further references and links to applications can be found). Then, results in these studies have been extended to other inverse problems with boundary conditions depending spectral parameter and with transmission conditions. Therefore, spectral problems for differential operator with transmission conditions inside an interval and with eigenvalue dependent boundary and transmission conditions as linearly and non-linearly have been studied in so many problems of mathematics as well as in applications (see [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43] and other works, and see [44,45,46,47,48,49,50,51,52,53,54] and other works cited therein respectively).
The aim of this article is to get some uniqueness theorems for mentioned above Dirac problem with eigenvalue dependent as rational function of Herglotz-Nevanlinna type in both of the boundary conditions and also transmission conditions at two different points. We take into account inverse problem for reconstruction of considered boundary value problem by Weyl function and by spectral data {λn,ρn}n∈Z and {λn,μn}n∈Z. Although the boundary and transmission conditions of the problem are not linearly dependent on the spectral parameter, this allows the eigenvalues to be real and to define normalizing numbers.
Consider the space H:=L2(a,b)⊕L2(a,b)⊕CN1+1⊕CN2+1⊕CP1+1⊕ CP2+1 and element Y in H is in the form of Y=(y1(x),y2(x),τ,η,β,γ), such that τ=(Y1,Y2,…,YN1,YN1+1), η=(L1,L2,…,LN2,LN2+1), β=(R1,R2,…,RP1,RP1+1), γ=(V1,V2,…,VP2,VP2+1). H is a Hilbert space with the inner product defined by
<Y,Z>:=b∫a(y1(x)¯z1(x)+y2(x)¯z2(x))dx−YN1+1¯Y′N1+1a1+LN2+1¯L′N2+1a2+α1m1RP1+1¯R′P1+1+α2m2VP2+1¯V′P2+1+N1∑k=1Yk¯Y′k(−1f1k)+N2∑k=1Lk¯L′kf2k+P1∑k=1α1Rr¯R′ru1k+P2∑k=1α2Vr¯V′ru2k | (7) |
for Y=(y1(x),y2(x),τ,η,β,γ) ve Z=(z1(x),z2(x),τ′,η′,β′,γ′) in H. Define the operator T on the domain
D(T)={Y∈H:y1(x),y2(x)∈AC(a,b),
ly∈L2(a,b), y1(w+i)=αiy1(w−i),i=1,2
YN1+1:=−a1y1(a), LN2+1:=−a2y1(b),
RP1+1:=−m1y1(w−1),VP2+1:=−m2y1(w−2)}
such that
TY:=(ly,Tτ,Tη,Tβ,Tγ) | (8) |
where
Tτ=TYi={g1iYi−f1iy1(a), i=¯1,N1y2(a)+b1y1(a)+N1∑k=1Yk, i=N1+1 | (9) |
Tη=TLi={g2iLi−f2iy1(b), i=¯1,N2y2(b)+b2y1(b)+N2∑k=1Lk, i=N2+1 | (10) |
Tβ=TRi={t1iRi−u1iy1(w−1), i=¯1,P1−y2(w+1)+α−11y2(w−1)+n1y1(w−1)+P1∑k=1Rk, i=P1+1 | (11) |
Tγ=TVi={t2iVi−u2iy1(w−2), i=¯1,P2−y2(w+2)+α−12y2(w−2)+n2y1(w−2)+∑k=1P2Vk, i=P2+1. | (12) |
Accordingly, equality TY=λY corresponds to problem (1)-(4) under the domain D(T)⊂H.
Theorem 1. The eigenvalues of the operator T and the problem (1)-(4) coincide.
Proof. Assume that λ is an eigenvalue of T and Y(x)=(y1(x),y2(x),τ,η,β,γ)∈H is the eigenvector corresponding to λ. Since Y∈D(T), it is obvious that the condition y1(wi+0)−αiy1(wi−0)=0 and Eq (1) hold. On the other hand, boundary conditions (2)-(3) and the second condition of (4) are satisfied by the following equalities;
Tτ=TYi=g1i−Yi−f1iy1(a)=λYi, i=¯1,N1
TYN1+1=y2(a)+b1y1(a)+N1∑k=1Yk=−a1y1(a)λ
Tη=TLi=g2iLi−f2iy1(b)=λLi, i=¯1,N2
TLN2+1=y2(b)+b2y1(b)+N2∑k=1Lk=−a2y1(b)λ
Tβ=TRi=t1iRi−u1iy1(w−1), i=¯1,P1
TRP1+1=−y2(w+1)+α−11y2(w−1)+n1y1(w−1)+P1∑k=1Rk=−m1y1(w−1)λ
Tγ=TVi=t2iVi−u2iy1(w−2), i=¯1,P2
TVP2+1=−y2(w+2)+α−12y2(w−2)+n2y1(w−2)+∑k=1P2Vk=−m2y1(w−2)λ.
If λ=gik(i=1,2 and k={1,2,…Ni}) are eigenvalues of operator T, then, from above equalities and the domain of T, equalities (1), y1(a,g1k)=0, y1(b,g2k)=0 and (4) are satisfied.
Moreover, If λ=tik(i=1,2 and k={1,2,…Pi}) are eigenvalues of operator T, from above equalities and the domain of T, Eqs (1)-(3) and y1(w−i,tik)=0=y1(w+i,tik) are valid. In that case, λ is also an eigenvalue of L.
Conversely, let λ be an eigenvalue of L and (y1(x)y2(x)) be an eigenfunction corresponding to λ. If λ≠gik(i=1,2, k={1,2,…Ni}) and λ≠tik(i=1,2, k={1,2,…Pi}) then, it is clear that λ is an eigenvalue of T and the vector
Y=(y1(x),y2(x),f11g11−λy1(a),f12g12−λy1(a),…,f1N1g1N1−λy1(a),−a1y1(a),
f21g21−λy1(b),f22g22−λy1(b),…,f2N2g2N2−λy1(b),−a2y1(b),
u11t11−λy1(w−1),u12t12−λy1(w−1),…,u1P1t1P1−λy1(w−1),−m1y1(w−1),
u21t21−λy1(w−2),u22t22−λy1(w−2),…,u2P2t2P2−λy1(w−2),−m2y1(w−2)) is the eigenvector corresponding to λ.
If λ=g1k(k={1,2,…N1}), then,
Y=(y1(x),y2(x),Y1,Y2,…,YN1,0,L1,L2,…,LN2,LN2+1,R1,R2,…,RP1,RP1+1, V1,V2,…,VP2,VP2+1),
Yi={0, i≠k−y2(a),i=k,i=1,2,…,N1 is the eigenvector of T corresponding to g1k.
If λ=g2k(k={1,2,…N2}), then,
Y=(y1(x),y2(x),Y1,Y2,…,YN1,YN1+1,L1,L2,…,LN2,0,R1,R2,…,RP1,RP1+1,V1,V2,…,VP2,VP2+1), Li={0, i≠k−y2(b),i=k,i=1,2,…,N2 is the eigenvector of T corresponding to g2k.
Furthermore, if λ=t1k(k={1,2,…P1}), then,
Y=(y1(x),y2(x),Y1,Y2,…,YN1,YN1,L1,L2,…,LN2,LN2+1,R1,R2,…,RP1,0,V1,V2,…,VP2,VP2+1), Ri={0, i≠ky2(w+1)−α−11y2(w−1),i=k,i=1,2,…,P1 is the eigenvector corresponding to t1k.
If λ=t2k(k={1,2,…P2}), then, Y=(y1(x),y2(x),Y1,Y2,…,YN1,YN1,L1,L2,…,LN2,LN2+1,R1,R2,…,RP1,RP1+1, V1,V2,…,VP2,0), Vi={0, i≠ky2(w+2)−α−12y2(w−2),i=k,i=1,2,…,P2 is the eigenvector corresponding to t2k.
It is possible to write fi(λ) as follows:
fi(λ)=ai(λ)bi(λ), i=1,2 where
ai(λ)=(aiλ+bi)Ni∏k=1(λ−gik)−Ni∑k=1Ni∏j=1(j≠k)fik(λ−gij)
bi(λ)=Ni∏k=1(λ−gik).
Assume that a2(λ) and b2(λ) do not have common zeros.
Let functions φ(x,λ) and ψ(x,λ) be the solutions of (1) under the initial conditions
φ(a,λ)=(−b1(λ)a1(λ)),ψ(b,λ)=(−b2(λ)a2(λ)) | (13) |
as well as the transmission conditions (4) respectively such that
φ(x,λ)={φ1(x,λ), x<w1φ2(x,λ), w1<x<w2φ3(x,λ), w2<x<b and ψ(x,λ)={ψ3(x,λ), x<w1ψ2(x,λ), w1<x<w2ψ1(x,λ), w2<x<b.
Then it can be easily proven that φi(x,λ) and ψi(x,λ), i=¯1,3 are the solutions of the following integral equations;
φi+1,1(x,λ)=αiφi1(wi,λ)cosλ(x−wi)
−[α−1iφi2(wi,λ)+hi(λ)φi1(wi,λ)]sinλ(x−wi)
+x∫wi[p(t)sinλ(x−t)+q(t)cosλ(x−t)]φi+1,1(t,λ)dt
+x∫wi[q(t)sinλ(x−t)−p(t)cosλ(x−t)]φi+1,2(t,λ)dt,
φi+1,2(x,λ)=αiφi1(wi,λ)sinλ(x−wi)+[α−1iφi2(wi,λ)+hi(λ)φi1(wi,λ)]cosλ(x−wi)+x∫wi[−p(t)cosλ(x−t)+q(t)sinλ(x−t)]φi+1,1(t,λ)dt+x∫wi[−q(t)cosλ(x−t)−p(t)sinλ(x−t)]φi+1,2(t,λ)dt,for i=1,2 |
and
ψi1(x,λ)=α−1iψi+1,1(wi,λ)cosλ(x−wi)+(−αiψi+1,2(wi,λ)+hi(λ)ψi+1,1(wi,λ))sinλ(x−wi)−wi∫x[p(t)sinλ(x−t)+q(t)cosλ(x−t)]ψi1(t,λ)dt+wi∫x[−q(t)sinλ(x−t)+p(t)cosλ(x−t)]ψi2(t,λ)dtψi2(x,λ)=α−1iψi+1,1(wi,λ)sinλ(x−wi)+(αiψi+1,2(wi,λ)−hi(λ)ψi+1,1(wi,λ))cosλ(x−wi)+wi∫x[p(t)cosλ(x−t)−q(t)sinλ(x−t)]ψi1(t,λ)dt+w2∫x[q(t)cosλ(x−t)+p(t)sinλ(x−t)]ψi2(t,λ)dt,for i=2,1 |
Lemma 1. For the solutions φi(x,λ) and ψi(x,λ), i=¯1,3 as |λ|→∞, the following asymptotic estimates hold;
φ11(x,λ)={a1λN1+1sinλ(x−a)+o(|λ|N1+1exp|Imλ|[(x−a)]),
φ12(x,λ)={a1λN1+1cosλ(x−a)+o(|λ|N1+1exp|Imλ|[(x−a)]),
φ21(x,λ)={a1m1λL1+N1+2sinλ(w1−a)sinλ(x−w1)+o(|λ|L1+N1+2exp|Imλ|[(w1−a)+(x−w1)])
φ22(x,λ)={a1m1λL1+N1+2sinλ(w1−a)cosλ(x−w1)+o(|λ|L1+N1+2exp|Imλ|[(w1−a)+(x−w1)])
φ31(x,λ)={−m2m1a1λL1+L2+N1+3sinλ(w1−a)sinλ(w2−w1)sinλ(x−w2)+o(|λ|L1+L2+N1+3exp|Imλ|[(w1−a)+(w2−w1)+(x−w2)])
φ32(x,λ)={m2m1a1λL1+L2+N1+3sinλ(w1−a)sinλ(w2−w1)cosλ(x−w2)+o(|λ|L1+L2+N1+3exp|Imλ|[(w1−a)+(w2−w1)+(x−w2)])
ψ11(x,λ)={−a2λN2+1sinλ(x−b)+o(|λ|N2+1exp|Imλ|[(x−b)])
ψ12(x,λ)={a2λN2+1cosλ(x−b)+o(|λ|N2+1exp|Imλ|[(x−b)])
ψ21(x,λ)={−m2a2λN2+L2+2sinλ(w2−b)sinλ(x−w2)+o(|λ|N2+L2+2exp|Imλ|[(w2−b)+(x−w2)])
ψ22(x,λ)={m2a2λN2+L2+2sinλ(w2−b)cosλ(x−w2)+o(|λ|N2+L2+2exp|Imλ|[(w2−b)+(x−w2)])
ψ31(x,λ)={−m1m2a2λN2+L1+L2+3sinλ(w2−b)sinλ(w1−w2)sinλ(x−w1)+o(|λ|N2+L1+L2+3exp|Imλ|[(w2−b)+(w1−w2)+(x−w2)])
ψ32(x,λ)={m1m2a2λN2+L1+L2+3sinλ(w2−b)sinλ(w1−w2)cosλ(x−w1)+o(|λ|N2+L1+L2+3exp|Imλ|[(w2−b)+(w1−w2)+(x−w1)])
Theorem 2. The eigenvalues {λn}n∈Z of problem L are real numbers.
Proof. It is enough to prove that eigenvalues of operator T are real. By using inner product (7), for Y in D(T), we compute that
⟨TY,Y⟩=b∫alyˉydx−1a1TYN1+1¯YN1+1+1a2TLN2+1¯LN2+1+α1m1TRP1+1¯RP1+1+α2m2TVP2+1¯VP2+1N1−∑k=1TYk¯Yk(1f1k)+N2∑k=1TLk¯Lk(1f2k)+P1∑k=1α1TRk¯Rk(1u1k)+P2∑k=1α2TVk¯Vk(1u2k). |
If necessary arrangements are made, we get
⟨TY,Y⟩=b∫ap(x)(|y1|2−|y2|2)dx+b∫aq(x)2Re(y2¯y1)dx+b1|y1(a)|+N1∑k=12Re(Yk¯y1(a))−b2|y1(b)|2−N2∑k=12Re(Lk¯y1(b))−a1n1|y1(w−1)|2−P1∑k=1a12Re(Rky1(w−1))−a2n2|y1(w−2)|2−P2∑k=1a22Re(Vky1(w−2))−N1∑k=1g1k|Yk|21f1k+N2∑k=1g2kf2k|Lk|2+P1∑k=1a1t1ku1k|Rk|2+P2∑k=1a2t2ku2k|Vk|2−b∫a2Re(y2¯y1′)dx. |
Accordingly, since ⟨TY,Y⟩ is real for each Y in D(T), λ∈R is obtained.
Lemma 2. The equality ‖ is valid such that is eigenvector corresponding to eigenvalue of .
Proof. Let . When , following proof is done with minor changes. By using the structure of and the Eqs (8)-(12), we get
(14) |
On the other hand, the expression
is called characteristic function of problem (1)-(4). Moreover, since solutions and satisfy the problem ,
for
is obtained. Furthermore, since solutions and also satisfy transmission conditions (4), we get
Therefore, since characteristic function is independent from ,
can be written.
It is clear that is an entire function and its zeros namely coincide with the eigenvalues of the problem .
Accordingly, for each eigenvalue equality is valid where .
On the other hand, since ve for and , is an eigenvalue if and only if , i.e., .
At the same time, is an eigenvalue if and only if i.e., such that and .
Theorem 3. Eigenvalues of problem are simple.
Proof. Let and be eigenfunction corresponds to the eigenvalue . In that case, the Eq (1) can be written for and as follows;
If we multiply these equations by and respectively and add side by side, we get the following equality;
Then if last equality is integrated over the interval and the initial conditions (13) and transmission conditions (4) are used to get
Then, considering that
if the limit is passed when is obtained.
If and are non-simple eigenvalues then , and so is obtained. Since , and for all , , are positive, we have a contradiction. Therefore, eigenvalues are also simple.
Using expressions , and asymptotic behaviour of solution , we obtain the following asymptotic of characteristic function as ; .
Let be the solution of Eq (1) under the conditions , as well as the transmission conditions (4).
Since , it can be supposed that where is a constant.
By the relation , we get . Since , we obtain for .
Let and be solutions of (1) satify the conditions and transmission conditions (4).
Accordingly, the following equalities are obtained:
(15) |
(16) |
The function is called Weyl solution and the function is called Weyl function of problem . Therefore, since , we set .
Consider the boundary value problem in the same form with but different coefficients. Here, the expressions related to the problem are shown with and the ones related to are shown with . According to this statement, we set the problem as follows:
where .
Theorem 4. If , , then almost everywhere in , , and .
Proof. Introduce a matrix by the equality as follows;
.
According to this, we get
(17) |
or by using the relation ,
we obtain
(18) |
Taking into account the Eqs (15) and (16) and , we can easily get
Hence, the functions are entire in . Denote
and
where is sufficiently small and fixed.
Clearly, for , .
Therefore, \ for sufficiently large and from (18) we see that are bounded with respect to where and sufficiently large. From Liouville's theorem, it is obtained that these functions do not depend on .
On the other hand, from (18)
.
If it is considered that do not depend on and asymptotic formulas of solutions and , we obtain
for all in . Hence, .
Thus, and similarly, and .
Substitute these relations in (17), to obtain
,
, for all and .
Taking into account these results and Eq (1), we have
Therefore, i.e., . Moreover, it is considered that
and
we get . As we have said above, , as well as , do not have common zeros. Hence, , i.e., .
On the other hand, substituting and into transmission conditions (4), we get
,
,
, .
Therefore, since , these yield that ,
and , .
Theorem 5. If , then almost everywhere in , , and .
Proof. Since , . On the other hand, also since and , we get that . Therefore, is obtained.
Denote which is an entire function in . Since , and so . Hence, . As a result, the proof of theorem is finished by Theorem 4.
We examine the boundary value problem with the condition instead of (2) in problem . Let be eigenvalues of the problem . It is clear that are zeros of .
Theorem 6. If , and such that , then almost everywhere in , , and .
Proof. Since for all , and , and are entire functions in and in respectively. On the other hand, taking into account the asymptotic behaviours of , and , we obtain and . Therefore, since and , we get and . If we consider the case , then is obtained. Furthermore, since , . Hence, the proof is completed by Theorem 4.
The purpose of this paper is to state and prove some uniqueness theorems for Dirac equations with boundary and transmission conditions depending rational function of Herglotz-Nevanlinna. Accordingly, it has been proved that while in condition (2) is known, the coefficients of the boundary value problem (1)-(4) can be determined uniquely by each of the following;
i) The Weyl function
ii) Spectral data forming eigenvalues and normalizing constants respectively
iii) Two given spectra
These results are the application of the classical uniqueness theorems of Marchenko, Gelfand, Levitan and Borg to such Dirac equations. Considering this study, similar studies can be made for classical Sturm-Liouville operators, the system of Dirac equations and diffusion operators with finite number of transmission conditions depending spectral parameter as Herglotz-Nevanlinna function.
There is no conflict of interest.
[1] | FDA, Guidance for Industry, 2011. Available from: https://www.fda.gov/downloads/drugs/guidances/ucm070336.pdf. |
[2] | Katz P, Campbell C (2012) FDA 2011 process validation guidance: process validation revisited. J GXP Compliance 16: 18-29. |
[3] | Guideline IHT (2005) Quality risk management Q9. |
[4] |
Politis NS, Colombo P, Colombo G, et al. (2017) Design of experiments (DoE) in pharmaceutical development. Drug Dev Ind Pharm 43: 889-901. doi: 10.1080/03639045.2017.1291672
![]() |
[5] |
Burdick RK, LeBlond DJ, Pfahler LB, et al. (2017) Process Design: Stage 1 of the FDA process validation guidance. Statistical Applications for Chemistry, Manufacturing and Controls (CMC) in the Pharmaceutical Industry Cham: Springer International Publishing, 115-154. doi: 10.1007/978-3-319-50186-4_3
![]() |
[6] | Doran PM (2013) Bioprocess Engineering Principles Boston: Elsevier. |
[7] |
Bunnak P, Allmendinger R, Ramasamy SV, et al. (2016) Life-cycle and cost of goods assessment of fed-batch and perfusion-based manufacturing processes for mAbs. Biotechnol Prog 32: 1324-1335. doi: 10.1002/btpr.2323
![]() |
[8] |
Diab S, Gerogiorgis DI (2018) Process modelling, simulation and technoeconomic evaluation of crystallisation antisolvents for the continuous pharmaceutical manufacturing of rufinamide. Comput Chem Eng 111: 102-114. doi: 10.1016/j.compchemeng.2017.12.014
![]() |
[9] |
Langer ES, Rader RA (2014) Continuous bioprocessing and perfusion: wider adoption coming as bioprocessing matures. BioProcessing J 13: 43-49. doi: 10.12665/J131.Langer
![]() |
[10] |
Fisher AC, Kamga MH, Agarabi C, et al. (2019) The current scientific and regulatory landscape in advancing integrated continuous biopharmaceutical manufacturing. Trends Biotechnol 37: 253-267. doi: 10.1016/j.tibtech.2018.08.008
![]() |
[11] | Herwig C, Glassey J, Kockmann N, et al. (2017) Better by Design: Qualtiy by design must be viewed as an opportunity not as a regulatory burden. The Chemical Engineer 915: 41-43. |
[12] |
Herwig C, Garcia-Aponte OF, Golabgir A, et al. (2015) Knowledge management in the QbD paradigm: manufacturing of biotech therapeutics. Trends Biotechnol 33: 381-387. doi: 10.1016/j.tibtech.2015.04.004
![]() |
[13] | CMC Biotech Working Group (2009) A-Mab: a Case Study in Bioprocess Development. |
[14] | Welcome to Python.org. Available from: https://www.python.org/. |
[15] |
Steinwandter V, Borchert D, Herwig C (2019) Data science tools and applications on the way to Pharma 4.0. Drug discov today 24: 1795-1805. doi: 10.1016/j.drudis.2019.06.005
![]() |
[16] | Guideline IHT (2009) Pharmaceutical Development Q8 (R2). |
[17] |
Thomassen YE, Van Sprang ENM, Van der Pol LA, et al. (2010) Multivariate data analysis on historical IPV production data for better process understanding and future improvements. Biotechnol Bioeng 107: 96-104. doi: 10.1002/bit.22788
![]() |
[18] |
Borchert D, Suarez-Zuluaga DA, Sagmeister P, et al. (2019) Comparison of data science workflows for root cause analysis of bioprocesses. Bioproc Biosyst Eng 42: 245-256. doi: 10.1007/s00449-018-2029-6
![]() |
[19] |
Suarez-Zuluaga DA, Borchert D, Driessen NN, et al. (2019) Accelerating bioprocess development by analysis of all available data: A USP case study. Vaccine 37: 7081-7089. doi: 10.1016/j.vaccine.2019.07.026
![]() |
[20] |
Kirdar AO, Green KD, Rathore AS (2008) Application of multivariate data analysis for identification and successful resolution of a root cause for a bioprocessing application. Biotechnol Prog 24: 720-726. doi: 10.1021/bp0704384
![]() |
[21] |
Sagmeister P, Wechselberger P, Herwig C (2012) Information processing: rate-based investigation of cell physiological changes along design space development. PDA J Pharm Sci Technol 66: 526-541. doi: 10.5731/pdajpst.2012.00889
![]() |
[22] |
Golabgir A, Gutierrez JM, Hefzi H, et al. (2016) Quantitative feature extraction from the Chinese hamster ovary bioprocess bibliome using a novel meta-analysis workflow. Biotechnol Adv 34: 621-633. doi: 10.1016/j.biotechadv.2016.02.011
![]() |
[23] |
Bowles JB (2003) An assessment of RPN prioritization in a failure modes effects and criticality analysis. The 2003 Proceedings Annual Reliability and Maintainability Symposium IEEE, 380-386. doi: 10.1109/RAMS.2003.1182019
![]() |
[24] | Sharma KD, Srivastava S (2018) Failure mode and effect analysis (FMEA) implementation: a literature review. J Adv Res Aeronaut Space Sci 5: 1-17. |
[25] |
Sharma RK, Kumar D, Kumar P (2007) Modeling system behavior for risk and reliability analysis using KBARM. Qual Reliab Eng Int 23: 973-998. doi: 10.1002/qre.849
![]() |
[26] |
Braglia M, Frosolini M, Montanari R (2003) Fuzzy TOPSIS approach for failure mode, effects and criticality analysis. Qual Reliab Eng Int 19: 425-443. doi: 10.1002/qre.528
![]() |
[27] |
Borchert D, Zahel T, Thomassen YE, et al. (2019) Quantitative CPP evaluation from risk assessment using integrated process modeling. Bioengineering 6: 114. doi: 10.3390/bioengineering6040114
![]() |
[28] |
Zahel T, Hauer S, Mueller EM, et al. (2017) Integrated process modeling—a process validation life cycle companion. Bioengineering 4: 86. doi: 10.3390/bioengineering4040086
![]() |
[29] |
Bonate PL (2001) A brief introduction to Monte Carlo simulation. Clin Pharmacokinet 40: 15-22. doi: 10.2165/00003088-200140010-00002
![]() |
[30] |
Wechselberger P, Sagmeister P, Engelking H, et al. (2012) Efficient feeding profile optimization for recombinant protein production using physiological information. Bioprocess Biosyst Eng 35: 1637-1649. doi: 10.1007/s00449-012-0754-9
![]() |
[31] |
Abu-Absi SF, Yang LY, Thompson P, et al. (2010) Defining process design space for monoclonal antibody cell culture. Biotechnol Bioeng 106: 894-905. doi: 10.1002/bit.22764
![]() |
[32] |
Liu H, Ricart B, Stanton C, et al. (2019) Design space determination and process optimization in at-scale continuous twin screw wet granulation. Comput Chem Eng 125: 271-286. doi: 10.1016/j.compchemeng.2019.03.026
![]() |
[33] | Guideline IHT (2011) Development and manufacture of drug substances (chemical entities and biotechnological/biological entities) Q11. |
1. | Mehmet Kayalar, A uniqueness theorem for singular Sturm-liouville operator, 2023, 34, 1012-9405, 10.1007/s13370-023-01097-x |