The restoration mechanism (RM) and subgrain characteristics of 0.05C-1.52Cu-1.51Mn steel in single-hit plane strain compression (PSC) were investigated using a thermomechanical simulator (Gleeble). It was observed that at diminished deformation temperature (DT) and larger strain rate, the austenitic phase (during deformation) showed some thermal/dynamic softening (TH/DRS), but it did not reach the condition where the "work hardening rate" (WH rate)became constant with the stress, i.e., dynamic recovery (DRV) softening balances work hardening (WH). However, it was observed that at higher DT and lower strain rate, the "WH rate" for samples deformed at 850 ℃ (at a strain rate of 0.01 s−1), 950 ℃ (at strain rates of 0.1 and 0.01 s−1) and 1000 ℃ (at strain rates of 0.1 and 0.01 s−1) increased to negative peak, and then decreased to almost zero (for samples deformed at 950 and 1000 ℃ at a strain rate of 0.01 s−1), which is the onset of steady-state flow. When the sample deformed at 750 ℃ followed by quenching, the microstructure was indicative of a deformed microstructure rather than a transformed microstructure. It was observed that there was an increase in the extent of substructure formation and a decrease in mean subgrain size with increasing strain rate. When samples deformed at 850,950 and 1000 ℃, these temperature ranges were above Ar3 temperature. Hence quenching would lead to a phase transformation and hence the deformed microstructure would be eliminated. The room temperature microstructures when the sample deformed at a strain rate of 1 s−1, were nicely equiaxed and clean with no dislocations present. However, at lower strain rates of 0.1 and 0.01 s−1, microstructure showed substructures.
Citation: Pawan Kumar. Subgrain characterization of low carbon copper bearing steel under plane strain compression using electron backscattered diffraction[J]. AIMS Materials Science, 2023, 10(6): 1121-1143. doi: 10.3934/matersci.2023060
[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 |
The restoration mechanism (RM) and subgrain characteristics of 0.05C-1.52Cu-1.51Mn steel in single-hit plane strain compression (PSC) were investigated using a thermomechanical simulator (Gleeble). It was observed that at diminished deformation temperature (DT) and larger strain rate, the austenitic phase (during deformation) showed some thermal/dynamic softening (TH/DRS), but it did not reach the condition where the "work hardening rate" (WH rate)became constant with the stress, i.e., dynamic recovery (DRV) softening balances work hardening (WH). However, it was observed that at higher DT and lower strain rate, the "WH rate" for samples deformed at 850 ℃ (at a strain rate of 0.01 s−1), 950 ℃ (at strain rates of 0.1 and 0.01 s−1) and 1000 ℃ (at strain rates of 0.1 and 0.01 s−1) increased to negative peak, and then decreased to almost zero (for samples deformed at 950 and 1000 ℃ at a strain rate of 0.01 s−1), which is the onset of steady-state flow. When the sample deformed at 750 ℃ followed by quenching, the microstructure was indicative of a deformed microstructure rather than a transformed microstructure. It was observed that there was an increase in the extent of substructure formation and a decrease in mean subgrain size with increasing strain rate. When samples deformed at 850,950 and 1000 ℃, these temperature ranges were above Ar3 temperature. Hence quenching would lead to a phase transformation and hence the deformed microstructure would be eliminated. The room temperature microstructures when the sample deformed at a strain rate of 1 s−1, were nicely equiaxed and clean with no dislocations present. However, at lower strain rates of 0.1 and 0.01 s−1, microstructure showed substructures.
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] |
Makhatha ME (2022) Effect of titanium addition on sub-structural characteristics of low carbon copper bearing steel in hot rolling. AIMS Mater Sci 9: 604–616. https://www.aimspress.com/article/doi/10.3934/matersci.2022036 doi: 10.3934/matersci.2022036
![]() |
[2] |
Sakai T, Belyakov A, Kaibyshev R, et al. (2014) Dynamic and post-dynamic recrystallization under hot, cold and severe plastic deformation conditions. Prog Mater Sci 60: 130–207. https://doi.org/10.1016/j.pmatsci.2013.09.002 doi: 10.1016/j.pmatsci.2013.09.002
![]() |
[3] | Link TM, Hance BM (2003) Effects of strain rate and temperature on the work hardening behavior of high strength sheet steels. SAE Trans 112: 211–219. http://www.jstor.org/stable/44699575 |
[4] |
Sakai T, Jonas JJ (1984) Overview no. 35 Dynamic recrystallization: Mechanical and microstructural considerations. Acta Metall 32: 189–209. https://doi.org/10.1016/0001-6160(84)90049-X doi: 10.1016/0001-6160(84)90049-X
![]() |
[5] |
Gourdet S, Montheillet F (2000) An experimental study of the recrystallization mechanism during hot deformation of aluminium. Mater Sci Eng A 283: 274–288. https://doi.org/10.1016/S0921-5093(00)00733-4 doi: 10.1016/S0921-5093(00)00733-4
![]() |
[6] |
Belyakov A, Tsuzaki K, Miura H, et al. (2003) Effect of initial microstructures on grain refinement in a stainless steel by large strain deformation. Acta Mater 51: 847–861. https://doi.org/10.1016/S1359-6454(02)00476-7 doi: 10.1016/S1359-6454(02)00476-7
![]() |
[7] |
Solberg JK, McQueen HJ, Ryum N, et al. (1989) Influence of ultra-high strains at elevated temperatures on the microstructure of aluminium. Philos Mag A 60: 447–471. https://doi.org/10.1080/01418618908213872 doi: 10.1080/01418618908213872
![]() |
[8] |
Hales SJ, McNelley TR, McQueen HJ (1991) Recrystallization and superplasticity at 300 ℃ in an aluminum-magnesium alloy. Metall Trans A 22: 1037–1047. https://doi.org/10.1007/BF02661097 doi: 10.1007/BF02661097
![]() |
[9] |
Tsuji N, Matsubara Y, Saito Y (1997) Dynamic recrystallization of ferrite in interstitial free steel. Scripta Mater 37: 477–484. https://doi.org/10.1016/S1359-6462(97)00123-1 doi: 10.1016/S1359-6462(97)00123-1
![]() |
[10] |
McNelley TR, McMahon ME (1997) Microtexture and grain boundary evolution during microstructural refinement processes in SUPRAL 2004. Metall Mater Trans A 28: 1879–1887. https://doi.org/10.1007/s11661-997-0118-2 doi: 10.1007/s11661-997-0118-2
![]() |
[11] |
Jorge AM, Balancin O (2005) Prediction of steel flow stresses under hot working conditions. Mat Res 8: 309–315. https://doi.org/10.1590/S1516-14392005000300015 doi: 10.1590/S1516-14392005000300015
![]() |
[12] |
Kingkam W, Li N, Zhang HX, et al. (2017) Hot deformation behavior of high strength low alloy steel by thermo mechanical simulator and finite element method. IOP Conf Ser Mater Sci Eng 205: 012001. https://dx.doi.org/10.1088/1757-899X/205/1/012001 doi: 10.1088/1757-899X/205/1/012001
![]() |
[13] |
Doherty RD, Hughes DA, Humphreys FJ, et al. (1997) Current issues in recrystallization: A review. Mater Sci Eng A 238: 219–274. https://doi.org/10.1016/S0921-5093(97)00424-3 doi: 10.1016/S0921-5093(97)00424-3
![]() |
[14] |
Quan GZ, Wang Y, Liu YY, et al. (2013) Effect of temperatures and strain rates on the average size of grains refined by dynamic recrystallization for as-extruded 42CrMo steel. Mat Res 16: 1092–1105. https://doi.org/10.1590/S1516-14392013005000091 doi: 10.1590/S1516-14392013005000091
![]() |
[15] |
Kumar P, Hodgson P, Beladi H, et al. (2020) EBSD investigation to study the restoration mechanism and substructural characteristics of 23Cr-6Ni-3Mo duplex stainless steel during post-deformation annealing. Trans Indian Inst Met 73: 1421–1431. https://doi.org/10.1007/s12666-020-01884-1 doi: 10.1007/s12666-020-01884-1
![]() |
[16] |
Montheillet F, Jonas JJ (1996) Temperature dependence of the rate sensitivity and its effect on the activation energy for high-temperature flow. Metall Mater Trans A 27: 3346–3348. https://doi.org/10.1007/BF02663887 doi: 10.1007/BF02663887
![]() |
[17] |
Davenport SB, Silk NJ, Sparks CN, et al. (2000) Development of constitutive equations for modelling of hot rolling. Mater Sci Technol 16: 539–546. https://doi.org/10.1179/026708300101508045 doi: 10.1179/026708300101508045
![]() |
[18] |
Sarkar A, Chakravartty JK (2013) Investigation of progress in dynamic recrystallization in two austenitic stainless steels exhibiting flow softening. J Metall Eng 2: 130–136. https://doi.org/10.5923/j.ijmee.20130202.03 doi: 10.5923/j.ijmee.20130202.03
![]() |
[19] |
Sarkar A, Kapoor R, Verma A, et al. (2012) Hot deformation behavior of Nb-1Zr-0.1C alloy in the temperature range 700–1700 ℃. J Nucl Mater 422: 1–7. https://doi.org/10.1016/j.jnucmat.2011.11.064 doi: 10.1016/j.jnucmat.2011.11.064
![]() |
[20] |
De Oliveira TS, Silva ES, Rodrigues SF, et al. (2017) Softening mechanisms of the AISI 410 martensitic stainless steel under hot torsion simulation. Mat Res 20: 395–406. https://doi.org/10.1590/1980-5373-MR-2016-0795 doi: 10.1590/1980-5373-mr-2016-0795
![]() |
[21] |
Prasad YVRK, Ravichandran N (1991) Effect of stacking fault energy on the dynamic recrystallization during hot working of FCC metals: A study using processing maps. Bull Mater Sci 14: 1241–1248. https://doi.org/10.1007/BF02744618 doi: 10.1007/BF02744618
![]() |
[22] |
Kumar P, Hodgson P, Beladi H, et al. (2021) Restoration mechanism and sub-structural characteristics of duplex stainless steel with an initial equiaxed austenite morphology during post-deformation annealing. Key Eng Mater 882: 64–73. https://doi.org/10.4028/www.scientific.net/KEM.882.64 doi: 10.4028/www.scientific.net/KEM.882.64
![]() |