Citation: Antonio Barrera, Patricia Román-Roán, Francisco Torres-Ruiz. Hyperbolastic type-III diffusion process: Obtaining from the generalized Weibull diffusion process[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 814-833. doi: 10.3934/mbe.2020043
[1] | Ridha Dida, Hamid Boulares, Bahaaeldin Abdalla, Manar A. Alqudah, Thabet Abdeljawad . On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives. AIMS Mathematics, 2023, 8(10): 23032-23045. doi: 10.3934/math.20231172 |
[2] | Hamid Boulares, Manar A. Alqudah, Thabet Abdeljawad . Existence of solutions for a semipositone fractional boundary value pantograph problem. AIMS Mathematics, 2022, 7(10): 19510-19519. doi: 10.3934/math.20221070 |
[3] | Reny George, Fahad Al-shammari, Mehran Ghaderi, Shahram Rezapour . On the boundedness of the solution set for the ψ-Caputo fractional pantograph equation with a measure of non-compactness via simulation analysis. AIMS Mathematics, 2023, 8(9): 20125-20142. doi: 10.3934/math.20231025 |
[4] | Saeed M. Ali, Mohammed S. Abdo, Bhausaheb Sontakke, Kamal Shah, Thabet Abdeljawad . New results on a coupled system for second-order pantograph equations with ABC fractional derivatives. AIMS Mathematics, 2022, 7(10): 19520-19538. doi: 10.3934/math.20221071 |
[5] | Weerawat Sudsutad, Chatthai Thaiprayoon, Aphirak Aphithana, Jutarat Kongson, Weerapan Sae-dan . Qualitative results and numerical approximations of the (k,ψ)-Caputo proportional fractional differential equations and applications to blood alcohol levels model. AIMS Mathematics, 2024, 9(12): 34013-34041. doi: 10.3934/math.20241622 |
[6] | Choukri Derbazi, Zidane Baitiche, Mohammed S. Abdo, Thabet Abdeljawad . Qualitative analysis of fractional relaxation equation and coupled system with Ψ-Caputo fractional derivative in Banach spaces. AIMS Mathematics, 2021, 6(3): 2486-2509. doi: 10.3934/math.2021151 |
[7] | Abdelkader Moumen, Ramsha Shafqat, Zakia Hammouch, Azmat Ullah Khan Niazi, Mdi Begum Jeelani . Stability results for fractional integral pantograph differential equations involving two Caputo operators. AIMS Mathematics, 2023, 8(3): 6009-6025. doi: 10.3934/math.2023303 |
[8] | Iyad Suwan, Mohammed S. Abdo, Thabet Abdeljawad, Mohammed M. Matar, Abdellatif Boutiara, Mohammed A. Almalahi . Existence theorems for Ψ-fractional hybrid systems with periodic boundary conditions. AIMS Mathematics, 2022, 7(1): 171-186. doi: 10.3934/math.2022010 |
[9] | Mohamed Houas, Kirti Kaushik, Anoop Kumar, Aziz Khan, Thabet Abdeljawad . Existence and stability results of pantograph equation with three sequential fractional derivatives. AIMS Mathematics, 2023, 8(3): 5216-5232. doi: 10.3934/math.2023262 |
[10] | Karim Guida, Lahcen Ibnelazyz, Khalid Hilal, Said Melliani . Existence and uniqueness results for sequential ψ-Hilfer fractional pantograph differential equations with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(8): 8239-8255. doi: 10.3934/math.2021477 |
Recently, fractional calculus methods became of great interest, because it is a powerful tool for calculating the derivation of multiples systems. These methods study real world phenomena in many areas of natural sciences including biomedical, radiography, biology, chemistry, and physics [1,2,3,4,5,6,7]. Abundant publications focus on the Caputo fractional derivative (CFD) and the Caputo-Hadamard derivative. Additionally, other generalization of the previous derivatives, such as Ψ-Caputo, study the existence of solutions to some FDEs (see [8,9,10,11,12,13,14]).
In general, an m-point fractional boundary problem involves a fractional differential equation with fractional boundary conditions that are specified at m different points on the boundary of a domain. The fractional derivative is defined using the Riemann-Liouville fractional derivative or the Caputo fractional derivative. Solving these types of problems can be challenging due to the non-local nature of fractional derivatives. However, there are various numerical and analytical methods available for solving such problems, including the spectral method, the finite difference method, the finite element method, and the homotopy analysis method. The applications of m-point fractional boundary problems can be found in various fields, including physics, engineering, finance, and biology. These problems are useful in modeling and analyzing phenomena that exhibit non-local behavior or involve memory effects (see [15,16,17,18]).
Pantograph equations are a set of differential equations that describe the motion of a pantograph, which is a mechanism used for copying and scaling drawings or diagrams. The equations are based on the assumption that the pantograph arms are rigid and do not deform during operation, we can simply say that see [19]. One important application of the pantograph equations is in the field of drafting and technical drawing. Before the advent of computer-aided design (CAD) software, pantographs were commonly used to produce scaled copies of drawings and diagrams. By adjusting the lengths of the arms and the position of the stylus, a pantograph can produce copies that are larger or smaller than the original [20], electrodynamics [21] and electrical pantograph of locomotive [22].
Many authors studied a huge number of positive solutions for nonlinear fractional BVP using fixed point theorems (FPTs) such as SFPT, Leggett-Williams and Guo-Krasnosel'skii (see [23,24]). Some studies addressed the sign-changing of solution of BVPs [25,26,27,28,29].
In this work, we use Schauder's fixed point theorem (SFPT) to solve the semipostone multipoint Ψ-Caputo fractional pantograph problem
Dν;ψrϰ(ς)+F(ς,ϰ(ς),ϰ(r+λς))=0, ς in (r,ℑ) | (1.1) |
ϰ(r)=ϑ1, ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2, ϑi∈R, i∈{1,2}, | (1.2) |
where λ∈(0,ℑ−rℑ),Dν;ψr is Ψ-Caputo fractional derivative (Ψ-CFD) of order ν, 1<ν≤2, ζi∈R+(1≤i≤m−2) such that 0<Σm−2i=1ζi<1, ηi∈(r,ℑ), and F:[r,ℑ]×R×R→R.
The most important aspect of this research is to prove the existence of a positive solution of the above m-point FBVP. Note that in [30], the author considered a two-point BVP using Liouville-Caputo derivative.
The article is organized as follows. In the next section, we provide some basic definitions and arguments pertinent to fractional calculus (FC). Section 3 is devoted to proving the the main result and an illustrative example is given in Section 4.
In the sequel, Ψ denotes an increasing map Ψ:[r1,r2]→R via Ψ′(ς)≠0, ∀ ς, and [α] indicates the integer part of the real number α.
Definition 2.1. [4,5] Suppose the continuous function ϰ:(0,∞)→R. We define (RLFD) the Riemann-Liouville fractional derivative of order α>0,n=[α]+1 by
RLDα0+ϰ(ς)=1Γ(n−α)(ddς)n∫ς0(ς−τ)n−α−1ϰ(τ)dτ, |
where n−1<α<n.
Definition 2.2. [4,5] The Ψ-Riemann-Liouville fractional integral (Ψ-RLFI) of order α>0 of a continuous function ϰ:[r,ℑ]→R is defined by
Iα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))α−1Γ(α)Ψ′(τ)ϰ(τ)dτ. |
Definition 2.3. [4,5] The CFD of order α>0 of a function ϰ:[0,+∞)→R is defined by
Dαϰ(ς)=1Γ(n−α)∫ς0(ς−τ)n−α−1ϰ(n)(τ)dτ, α∈(n−1,n),n∈N. |
Definition 2.4. [4,5] We define the Ψ-CFD of order α>0 of a continuous function ϰ:[r,ℑ]→R by
Dα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))n−α−1Γ(n−α)Ψ′(τ)∂nΨϰ(τ)dτ, ς>r, α∈(n−1,n), |
where ∂nΨ=(1Ψ′(ς)ddς)n,n∈N.
Lemma 2.1. [4,5] Suppose q,ℓ>0, and ϰinC([r,ℑ],R). Then ∀ς∈[r,ℑ] and by assuming Fr(ς)=Ψ(ς)−Ψ(r), we have
1) Iq;ΨrIℓ;Ψrϰ(ς)=Iq+ℓ;Ψrϰ(ς),
2) Dq;ΨrIq;Ψrϰ(ς)=ϰ(ς),
3) Iq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ+q)(Fr(ς))ℓ+q−1,
4) Dq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ−q)(Fr(ς))ℓ−q−1,
5) Dq;Ψr(Fr(ς))k=0, k=0,…,n−1, n∈N, qin(n−1,n].
Lemma 2.2. [4,5] Let n−1<α1≤n,α2>0, r>0, ϰ∈L(r,ℑ), Dα1;Ψrϰ∈L(r,ℑ). Then the differential equation
Dα1;Ψrϰ=0 |
has the unique solution
ϰ(ς)=W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
and
Iα1;ΨrDα1;Ψrϰ(ς)=ϰ(ς)+W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
with Wℓ∈R, ℓ∈{0,1,…,n−1}.
Furthermore,
Dα1;ΨrIα1;Ψrϰ(ς)=ϰ(ς), |
and
Iα1;ΨrIα2;Ψrϰ(ς)=Iα2;ΨrIα1;Ψrϰ(ς)=Iα1+α2;Ψrϰ(ς). |
Here we will deal with the FDE solution of (1.1) and (1.2), by considering the solution of
−Dν;ψrϰ(ς)=h(ς), | (2.1) |
bounded by the condition (1.2). We set
Δ:=Ψ(ℑ)−Ψ(r)−Σm−2i=1ζi(Ψ(ηi)−Ψ(r)). |
Lemma 2.3. Let ν∈(1,2] and ς∈[r,ℑ]. Then, the FBVP (2.1) and (1.2) have a solution ϰ of the form
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where
ϖ(ς,τ)=1Γ(ν){[(Ψ(ℑ)−Ψ(r))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ς)−Ψ(r)Δ−(Ψ(ς)−Ψ(τ))ν−1,τ≤ς,ηi−1<τ≤ηi,[(Ψ(ℑ)−Ψ(τ))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ℑ)−Ψ(r)Δ,ς≤τ,ηi−1<τ≤ηi, | (2.2) |
i=1,2,...,m−2.
Proof. According to the Lemma 2.2 the solution of Dν;ψrϰ(ς)=−h(ς) is given by
ϰ(ς)=−1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+c0+c1(Ψ(ς)−Ψ(r)), | (2.3) |
where c0,c1∈R. Since ϰ(r)=ϑ1 and ϰ(ℑ)=∑m−2i=1ζiϰ(ηi)+ϑ2, we get c0=ϑ1 and
c1=1Δ(−1Γ(ν)m−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+1Γ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+ϑ1[m−2∑i=1ζi−1]+ϑ2). |
By substituting c0,c1 into Eq (2.3) we find,
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2−1Γ(ν)(∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+(Ψ(ς)−Ψ(r))Δm−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ−Ψ(ς)−Ψ(r)Δ∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where ϖ(ς,τ) is given by (2.2). Hence the required result.
Lemma 2.4. If 0<∑m−2i=1ζi<1, then
i) Δ>0,
ii) (Ψ(ℑ)−Ψ(τ))ν−1−∑m−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0.
Proof. i) Since ηi<ℑ, we have
ζi(Ψ(ηi)−Ψ(r))<ζi(Ψ(ℑ)−Ψ(r)), |
−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r)), |
Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r))=(Ψ(ℑ)−Ψ(r))[1−m−2∑i=1ζi]. |
If 1−Σm−2i=1ζi>0, then (Ψ(ℑ)−Ψ(r))−Σm−2i=1ζi(Ψ(ηi)−Ψ(r))>0. So we have Δ>0.
ii) Since 0<ν−1≤1, we have (Ψ(ηi)−Ψ(τ))ν−1<(Ψ(ℑ)−Ψ(τ))ν−1. Then we obtain
m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1<m−2∑j=iζj(Ψ(ℑ)−Ψ(τ))ν−1≤(Ψ(ℑ)−Ψ(τ))ν−1m−2∑i=1ζi<(Ψ(ℑ)−Ψ(τ))ν−1, |
and so
(Ψ(ℑ)−Ψ(τ))ν−1−m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0. |
Remark 2.1. Note that ∫ℑrϖ(ς,τ)Ψ′(τ)dτ is bounded ∀ς∈[r,ℑ]. Indeed
∫ℑr|ϖ(ς,τ)|Ψ′(τ)dτ≤1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)Γ(ν)Δm−2∑i=1ζi∫ηir(Ψ(ηj)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)ΔΓ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1Ψ′(τ)dτ=(Ψ(ς)−Ψ(r))νΓ(ν+1)+Ψ(ς)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+Ψ(ς)−Ψ(r)ΔΓ(ν+1)(Ψ(ℑ)−Ψ(r))ν≤(Ψ(ℑ)−Ψ(r))νΓ(ν+1)+Ψ(ℑ)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+(Ψ(ℑ)−Ψ(r))ν+1ΔΓ(ν+1)=M. | (2.4) |
Remark 2.2. Suppose Υ(ς)∈L1[r,ℑ], and w(ς) verify
{Dν;ψrw(ς)+Υ(ς)=0,w(r)=0, w(ℑ)=Σm−2i=1ζiw(ηi), | (2.5) |
then w(ς)=∫ℑrϖ(ς,τ)Υ(τ)Ψ′(τ)dτ.
Next we recall the Schauder fixed point theorem.
Theorem 2.1. [23] [SFPT] Consider the Banach space Ω. Assume ℵ bounded, convex, closed subset in Ω. If ϝ:ℵ→ℵ is compact, then it has a fixed point in ℵ.
We start this section by listing two conditions which will be used in the sequel.
● (Σ1) There exists a nonnegative function Υ∈L1[r,ℑ] such that ∫ℑrΥ(ς)dς>0 and F(ς,ϰ,v)≥−Υ(ς) for all (ς,ϰ,v)∈[r,ℑ]×R×R.
● (Σ2) G(ς,ϰ,v)≠0, for (ς,ϰ,v)∈[r,ℑ]×R×R.
Let ℵ=C([r,ℑ],R) the Banach space of CFs (continuous functions) with the following norm
‖ϰ‖=sup{|ϰ(ς)|:ς∈[r,ℑ]}. |
First of all, it seems that the FDE below is valid
Dν;ψrϰ(ς)+G(ς,ϰ∗(ς),ϰ∗(r+λς))=0, ς∈[r,ℑ]. | (3.1) |
Here the existence of solution satisfying the condition (1.2), such that G:[r,ℑ]×R×R→R
G(ς,z1,z2)={F(ς,z1,z2)+Υ(ς), z1,z2≥0,F(ς,0,0)+Υ(ς), z1≤0 or z2≤0, | (3.2) |
and ϰ∗(ς)=max{(ϰ−w)(ς),0}, hence the problem (2.5) has w as unique solution. The mapping Q:ℵ→ℵ accompanied with the (3.1) and (1.2) defined as
(Qϰ)(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(ς,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ, | (3.3) |
where the relation (2.2) define ϖ(ς,τ). The existence of solution of the problems (3.1) and (1.2) give the existence of a fixed point for Q.
Theorem 3.1. Suppose the conditions (Σ1) and (Σ2) hold. If there exists ρ>0 such that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
where L≥max{|G(ς,ϰ,v)|:ς∈[r,ℑ], |ϰ|,|v|≤ρ} and M is defined in (2.4), then, the problems (3.1) and (3.2) have a solution ϰ(ς).
Proof. Since P:={ϰ∈ℵ:‖ϰ‖≤ρ} is a convex, closed and bounded subset of B described in the Eq (3.3), the SFPT is applicable to P. Define Q:P→ℵ by (3.3). Clearly Q is continuous mapping. We claim that range of Q is subset of P. Suppose ϰ∈P and let ϰ∗(ς)≤ϰ(ς)≤ρ, ∀ς∈[r,ℑ]. So
|Qϰ(ς)|=|[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ|≤[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
for all ς∈[r,ℑ]. This indicates that ‖Qϰ‖≤ρ, which proves our claim. Thus, by using the Arzela-Ascoli theorem, Q:ℵ→ℵ is compact. As a result of SFPT, Q has a fixed point ϰ in P. Hence, the problems (3.1) and (1.2) has ϰ as solution.
Lemma 3.1. ϰ∗(ς) is a solution of the FBVP (1.1), (1.2) and ϰ(ς)>w(ς) for every ς∈[r,ℑ] iff the positive solution of FBVP (3.1) and (1.2) is ϰ=ϰ∗+w.
Proof. Let ϰ(ς) be a solution of FBVP (3.1) and (1.2). Then
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)(F(τ,ϰ∗(τ),ϰ∗(r+λτ))+p(τ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+1Γ(ν)∫ℑrϖ(ς,τ)p(τ)Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+w(ς). |
So,
ϰ(ς)−w(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ. |
Then we get the existence of the solution with the condition
ϰ∗(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ. |
For the converse, if ϰ∗ is a solution of the FBVP (1.1) and (1.2), we get
Dν;ψr(ϰ∗(ς)+w(ς))=Dν;ψrϰ∗(ς)+Dν;ψrw(ς)=−F(ς,ϰ∗(ς),ϰ∗(r+λς))−p(ς)=−[F(ς,ϰ∗(ς),ϰ∗(r+λς))+p(ς)]=−G(ς,ϰ∗(ς),ϰ∗(r+λς)), |
which leads to
Dν;ψrϰ(ς)=−G(ς,ϰ∗(ς),ϰ∗(r+λς)). |
We easily see that
ϰ∗(r)=ϰ(r)−w(r)=ϰ(r)−0=ϑ1, |
i.e., ϰ(r)=ϑ1 and
ϰ∗(ℑ)=m−2∑i=1ζiϰ∗(ηi)+ϑ2, |
ϰ(ℑ)−w(ℑ)=m−2∑i=1ζiϰ(ηi)−m−2∑i=1ζjw(ηi)+ϑ2=m−2∑i=1ζi(ϰ(ηi)−w(ηi))+ϑ2. |
So,
ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2. |
Thus ϰ(ς) is solution of the problem FBVP (3.1) and (3.2).
We propose the given FBVP as follows
D75ϰ(ς)+F(ς,ϰ(ς),ϰ(1+0.5ς))=0, ς∈(1,e), | (4.1) |
ϰ(1)=1, ϰ(e)=17ϰ(52)+15ϰ(74)+19ϰ(115)−1. | (4.2) |
Let Ψ(ς)=logς, where F(ς,ϰ(ς),ϰ(1+12ς))=ς1+ςarctan(ϰ(ς)+ϰ(1+12ς)).
Taking Υ(ς)=ς we get ∫e1ςdς=e2−12>0, then the hypotheses (Σ1) and (Σ2) hold. Evaluate Δ≅0.366, M≅3.25 we also get |G(ς,ϰ,v)|<π+e=L such that |ϰ|≤ρ, ρ=17, we could just confirm that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≅16.35≤17. | (4.3) |
By applying the Theorem 3.1 there exit a solution ϰ(ς) of the problem (4.1) and (4.2).
In this paper, we have provided the proof of BVP solutions to a nonlinear Ψ-Caputo fractional pantograph problem or for a semi-positone multi-point of (1.1) and(1.2). What's new here is that even using the generalized Ψ-Caputo fractional derivative, we were able to explicitly prove that there is one solution to this problem, and that in our findings, we utilize the SFPT. The results obtained in our work are significantly generalized and the exclusive result concern the semi-positone multi-point Ψ-Caputo fractional differential pantograph problem (1.1) and (1.2).
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Small Groups (RGP.1/350/43).
The authors declare no conflict of interest.
[1] | A. Tsoularis and J. Wallace, Analysis of logistic growth models, Math. Biosci., 179 (2002), 21-55, DOI: 10.1016/S0025-5564(02)00096-2 |
[2] | J. Xia and S. Wan, Independent effects of warming and nitrogen addition on plant phenology in the Inner Mongolian steppe, Ann. Bot., 111 (2013), 1207-1217, DOI: 10.1093/aob/mct079 |
[3] | S. A. Pardo, A. B.Cooper and N. K. Dulvy, Avoiding fishy growth curves, Methods Ecol. Evol., 4 (2013), 353-360, DOI: 10.1111/2041-210x.12020 |
[4] | Y. H. Hsieh, D. N. Fisman and J. Wu, On epidemic modeling in real time: an application to the 2009 Novel A (H1N1) influenza outbreak in Canada, BMC Res. Notes, 3 (2010), 283, DOI: 10.1186/1756-0500-3-283 |
[5] | S. P. Cairns, D. M. Robinson and D. S. Loiselle, Double-sigmoid model for fitting fatigue profiles in mouse fast- and slow-twitch muscle, Exp. Physiol., 93 (2008), 851-862, DOI: 10.1113/expphysiol.2007.041285 |
[6] | L. E. Bergues-Cabrales, A. Ramírez-Aguilera, R. Placeres-Jiménez, et al., Mathematical modeling of tumor growth in mice following low-level direct electric current, Math. Comput. Simul., 78 (2008), 112-120, DOI: 10.1016/j.matcom.2007.06.004 |
[7] | D. Dingli, M. D. Cascino, K. Josic, et al., Mathematical modeling of cancer radiovirotherapy, Math. Biosci., 199 (2006), 55-78, DOI: 10.1016/j.mbs.2005.11.001 |
[8] | B. Gallagher, Peak oil analyzed with a logistic function and idealized Hubbert curve, Energy Policy, 39 (2011), 709-802, DOI: 10.1016/j.enpol.2010.10.053 |
[9] | M. Höök, L. Junchen, N. Oba, et al., Descriptive and predictive growth curves in energy system analysis, Nat. Resour. Res., 20 (2011), 103-116, DOI:10.1007/s11053-011-9139-z |
[10] | P. R. Koya and A. T. Goshu, Solutions of rate-state equation describing biological growths, Am. J. Math. Stat., 3 (2013), 305-331, DOI: 10.5923/j.ajms.20130306.02 |
[11] | P. R. Koya and A. T. Goshu, Generalized mathematical model for biological growths, Open J. Model. Simul., 1 (2013), 42-53, DOI: 10.4236/ojmsi.2013.14008 |
[12] | R. Bürger, G. Chowell and L. Y. Lara-Díaz, Comparative analysis of phenomenological growth models applied to epidemic outbreaks, Math. Biosci. Eng., 16 (2019), 4250-4273, DOI: 10.3934/mbe.2019212 |
[13] | M. Tabatatai, D. K. Williams and Z. Bursac, Hyperbolastic growth models: theory and application, Theor. Biol. Med. Model., 2 (2005), 1-13, DOI: 10.1186/1742-4682-2-14 |
[14] | W. Eby, M. Tabatabai and Z. Bursac, Hyperbolastic modeling of tumor growth with a combined treatment of iodoacetate and dimethylsulphoxide, BMC Cancer, 10 (2010), 509, DOI: 10.1186/1471-2407-10-509 |
[15] | M. Tabatabai, Z. Bursac, W. Eby, et al., Mathematical modeling of stem cell proliferation, Med. Biol. Eng. Comput., 49 (2011), 253-262, DOI: 10.1007/s11517-010-0686-y |
[16] | M. Tabatabai, W. Eby and Z. Bursac, Oscillabolastic model, a new model for oscillatory dynamics, applied to the analysis of Hes1 gene expression and Ehrlich ascites tumor growth, J. Biomed. Inform., 45 (2012), 401-407, DOI: 10.1016/j.jbi.2011.11.016 |
[17] | M. Tabatabai, W. Eby, K. Singh, et al., T model of growth and its application in systems of tumorimmune dynamics, Math. Biosci. Eng., 10 (2013), 925-938, DOI: 10.3934/mbe.2013.10.925 |
[18] | W. Weibull, A statistical distribution function of wide applicability, J. Appl. Mech. Trans. ASME, 18 (1951), 293-297. |
[19] | P. Rosin and E. Rammler, The laws governing the fineness of powdered coal, J. Inst. Fuel., 7(1933), 29-36. |
[20] | G. A. Seber and C. J. Wild, Nonlinear Regression, John Wiley & Sons Inc, Hoboken, New Jersey, 2003. |
[21] | D. J. Mahanta and M. Borah, Parameter Estimation of Weibull Growth Models in Forestry, Int. J. Math. Trends Techn., 8 (2014), 157-163, DOI: 10.14445/22315373/IJMTT-V8P521 |
[22] | J. Swintek, M. Etterson, K. Flynn, et al., Optimized temporal sampling designs of the Weibull growth curve with extensions to the von Bertalanffy model, Environmetrics, e2552 (2019), DOI:doi.org/10.1002/env.2552 |
[23] |
A. K. Pradhan, M. Li, Y. Li, et al., A modified Weibull model for growth and survival of Listeria innocua and Salmonella Typhimurium in chicken breasts during refrigerated and frozen storage, Poultry Sci., 91 (2012), 1482-1488, DOI: 10.3382/ps.2011-01851 doi: 10.3382/ps.2011-01851
![]() |
[24] | J. L. Rocha and S. M. Aleixo, An extension of Gompertzian growth dynamics: Weibull and Fréchet models, Math. Biosci. Eng., 10 (2013), 379-398, DOI: 10.3934/mbe.2013.10.379 |
[25] | D. Rodríguez-Pérez, Package growth models, R package version 1.2 (2013). Available from http://cran.r-project.org/package=growthmodels |
[26] | H. C. Tuckwell and J. A. Koziol, Logistic population growth under random dispersal, Bull. Math. Biol., 49 (1987), 495-506, DOI: 10.1016/S0092-8240(87)80010-1 |
[27] | R. M. Capocelli and L. M. Ricciardi, Growth with regulation in random environment, Kybernetik, 15 (1974), 147-157. DOI:10.1007/BF00274586 |
[28] | L. Ferrante, S. Bompadre, L. Possati, et al., Parameter estimation in a Gompertzian stochastic model for tumor growth, Biometrics, 56 (2000), 1076-1081, DOI: 10.1111/j.0006-341X.2000.01076.x |
[29] | C. F. Lo, A modified stochastic Gompertz model for tumor cell growth, Comput. Math. Method M., 11 (2010), 3-11, DOI: 10.1080/17486700802545543 |
[30] | G. Albano, V. Giorno, P. Román-Román, et al., Inference on a stochastic two-compartment model in tumor growth, Comput. Stat. Data An., 56 (2012), 1723-1736, DOI: 10.1016/j.csda.2011.10.016 |
[31] | P. Román-Román, S. Román-Román, J. J. Serrano-Pérez, et al., Modeling tumor growth in the presence of a therapy with an effect on rate growth and variability by means of a modified Gompertz diffusion process, J. Theor. Biol., 407 (2016), 1-17, DOI: 10.1016/j.jtbi.2016.07.023 |
[32] | V. Giorno, P. Román-Román, S. Spina, et al., Estimating a non-homogeneous Gompertz process with jumps as model of tumor dynamics, Comput. Stat. Data An., 107 (2017), 18-31, DOI: 10.1016/j.csda.2016.10.005 |
[33] | Q. Lv and J. W. Pitchford, Stochastic von Bertalanffy models, with applications to fish recruitment, J. Theor. Biol., 244 (2007), 640-655, DOI:10.1016/j.jtbi.2006.09.009 |
[34] | P. Román-Román, D. Romero and F. Torres-Ruiz, A diffusion process to model generalized von Bertalanffy growth patterns: fitting to real data, J. Theor. Biol., 263 (2010), 59-69, DOI: 10.1016/j.jtbi.2009.12.009 |
[35] | P. Román-Román and F. Torres-Ruiz, Modeling logistic growth by a new diffusion process: application to biological systems, BioSystems, 110 (2012), 9-21, DOI: 10.1016/j.biosystems.2012.06.004 |
[36] | P. Román-Román and F. Torres-Ruiz, A stochastic model related to the Richards-type growth curve. Estimation by means of simulated annealing and variable neighborhood search, App. Math. Comput., 266 (2015), 579-598, DOI:10.1016/j.amc.2015.05.096 |
[37] | A. Barrera, P. Román-Román and F. Torres-Ruiz, A hyperbolastic type-I diffusion process: Parameter estimation by means of the firefly algorithm, BioSystems, 163 (2018), 11-22, DOI: 10.1016/j.biosystems.2017.11.001 |
[38] | P. Román-Román, J. J. Serrano-Pérez and F. Torres-Ruiz, A note on estimation of multi-sigmoidal Gompertz functions with random noise, Mathematics, 7 (2019), 541, DOI:10.3390/math7060541 |
[39] | P. Román-Román, J. J. Serrano-Pérez and F. Torres-Ruiz, Some notes about inference for the lognormal diffusion process with exogenous factors, Mathematics, 6 (2018), 85, DOI:10.3390/math6050085 |
[40] | R. G. Rutledge and D. Stewart, A kinetic-based sigmoidal model for the polymerase chain reaction and its application to high-capacity absolute quantitative real-time PCR, BMC Biotechnol., 8 (2008), 48, DOI:10.1186/1472-6750-8-47 |
[41] | R. G. Rutledge and C. Côt, Mathematics of quantitative kinetic PCR and the application of standard curves, Nucleic Acids Res., 31 (2003), e93, DOI:10.1093/nar/gng093 |
[42] | G. J. Boggy and P. J. Woolf, A mechanistic model of PCR for accurate quantification of quantitative PCR data, PLoS One, 5 (2010), e12355, DOI:10.1371/journal.pone.0012355 |
[43] | A. N. Spiess, C. Feig and C. Ritz, Highly accurate sigmoidal fitting of real-time PCR data by introducing a parameter for asymmetry, BMC Bioinforma., 9 (2008), 221, DOI:10.1186/14712105-9-221 |
[44] | A. N. Spiess, S. Rdiger, M. Burdukiewicz, et al., System-specific periodicity in quantitative realtime polymerase chain reaction data questions threshold-based quantitation, Sci. Rep., 6 (2016), 38951, DOI:10.1038/srep38951 |