Mathematical Methods in the Applied Sciences

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Online ISSN: 1099-1476
Print ISSN: 0170-4214
Discipline: Mathematical Modeling
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Aims and scope

Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome.

Recent publications
In this paper, a discrete-time seasonally forced SIR epidemic model with a nonstandard discretization scheme is investigated for different types of bifurcations. Although many researchers have already suggested numerically that this model can exhibit chaotic dynamics, not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Second, the flip, Neimark–Sacker, and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. The model is discretized by a novel technique, namely a nonstandard finite difference discretization scheme (NSFD). Some graphical representations of the model are presented to verify the obtained results.
Schematic illustrating the notation and jump process, from t$$ t $$ to t+τ$$ t+\tau $$, for the proposed generalized Einstein random walk paradigm for x=(x1,x2)∈ℝ2$$ x=\left({x}_1,{x}_2\right)\in {\mathrm{\mathbb{R}}}^2 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Example numerical simulations showing finite speed of propagation of initially compact solutions of the PDE (60) from the initial condition (61) (shown as dash‐dotted curves in the left panels), for different choices of nonlinearity exponents γ0$$ {\gamma}_0 $$ and m$$ m $$, up to t=T=10$$ t=T=10 $$. The speed of propagation, as predicted by the analysis in Section 6.2, is represented by the slope of the dashed trendlines in the right panels. Note that (B) is the special case corresponding to the “Barenblatt solution” of the porous medium equation. The time evolution in the left panels is in the direction of the arrow from dark to light colored curves. The plots in the right panels are on a log‐log scale. [Colour figure can be viewed at wileyonlinelibrary.com]
Schematic illustrating of the location of the points on the x$$ x $$‐axis.
We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in non-divergence form. Specifically, in these equations, the diffusion coefficient can depend on both the dependent variable and its gradient, and it vanishes when either one of the latter does. It is known that solution of such degenerate equations can exhibit finite speed of propagation (so-called localization property of solutions). We give a proof of this property using a De Giorgi--Ladyzhenskaya iteration procedure for non-divergence-from equations. A mapping theorem is then established to a divergence-form version of the governing equation for the case of one spatial dimension. Numerical results via a finite-difference scheme are used to illustrate the main mathematical results for this special case. For completeness, we also provide an explicit construction of the one-dimensional self-similar solution with finite speed of propagation function, in the sense of Kompaneets--Zel'dovich--Barenblatt. We thus show how the finite speed of propagation quantitatively depends on the model's parameters.
  • Kayyunnapara Divya Joseph
    Kayyunnapara Divya Joseph
This paper is concerned with the study of interaction of waves originating from the Riemann problem centered at two different points for a system of equations modeling propagation of elastic waves. The system consists of two equations for (u,σ)$$ \left(u,\sigma \right) $$, where u$$ u $$ is the velocity and σ$$ \sigma $$ is the stress and is strictly hyperbolic and nonconservative. The study of interaction of waves is one of the most important steps in the construction of a global solution with initial data in the space of functions of bounded variation using an approximation procedure like Glimm's scheme. This amounts to constructing a solution with initial data consisting of three states (uL,σL)$$ \left({u}_L,{\sigma}_L\right) $$, (um,σm)$$ \left({u}_m,{\sigma}_m\right) $$, and (uR,σR)$$ \left({u}_R,{\sigma}_R\right) $$. Usually, this analysis is done for the states which are in a small neighborhood of a fixed state. Here, we get an explicit formula for the solution of the system when the data lie in the level sets of Riemann invariants. The speciality of the work is that we do not assume smallness conditions on the initial data.
The solutions to fractional differential equations are a developing area of current research, given that these equations arise in various domains. In this article, we provide some necessary criteria for the existence, uniqueness, and different types of Ulam stability for a coupled implicit system requiring the conditions for nonlocal Riemann–Liouville and Erdélyi–Kober q‐fractional integral conditions. The uniqueness and existence results for the suggested coupled system are demonstrated using Banach fixed point theorem and Leray–Schauder of cone type. We also explore the various types of stability using classical methods of nonlinear functional analysis. To verify the effectiveness of our theoretical outcomes, we study an interesting example.
  • Liya Jiang
    Liya Jiang
  • Youhua Wei
    Youhua Wei
  • Kaige Yang
    Kaige Yang
The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations but also are physically relevant and mathematically important. Unfortunately, it is not often well‐understood for many ranges of fractional powers. This paper focuses on a system of the 3D Boussinesq equations with fractional horizontal (−Δh)αu$$ {\left(-{\Delta}_h\right)}^{\alpha }u $$ and (−Δh)βθ$$ {\left(-{\Delta}_h\right)}^{\beta}\theta $$ dissipation and proves that if the initial data ( u0,θ0$$ {u}_0,{\theta}_0 $$) in the Sobolev space H3(ℝ3)$$ {H}^3\left({\mathrm{\mathbb{R}}}^3\right) $$ are close enough to the hydrostatic balance state, respectively, the equations with α,β∈12,1$$ \alpha, \beta \in \left(\frac{1}{2},1\right] $$ then always lead to a steady solution.
Graphs of the exact solution and the fourth approximate solution obtained by the iterative method. [Colour figure can be viewed at wileyonlinelibrary.com]
Illustration of the artificial neural network corresponding to functions of the set NN$$ \mathcal{NN} $$ such that N0=4,N1=7,N2=8,N3=8,N4=5,N5=3$$ {N}_0=4,{N}_1=7,{N}_2=8,{N}_3=8,{N}_4=5,{N}_5=3 $$, an activation function ρ$$ \rho $$. [Colour figure can be viewed at wileyonlinelibrary.com]
The corresponding architecture of an artificial neural networks of functions of the set NN1,1,4ρ$$ \mathcal{N}{\mathcal{N}}_{1,1,4}^{\rho } $$. [Colour figure can be viewed at wileyonlinelibrary.com]
The graph of approximate function of problem (30) and its exact solution via artificial neuron networks. h=0.01$$ h=0.01 $$, loss function = mean square error (MSE), optimizer = Adam, learning rate = 0.01, number of epochs = 1000. [Colour figure can be viewed at wileyonlinelibrary.com]
In this work, we discussed the existence and uniqueness of the solution for a fractional boundary value problem involving the Ψ-Riemann-Liouville operators where we used Banach's fixed point theorem; after that, we applied an iterative method and artificial neural networks technique to solve our main problem. Finally, the approximate solution was plotted. KEYWORDS approximate solution, boundary value problem, iterative method, Ψ-Riemann-Liouville operators MSC CLASSIFICATION 26A33, 34A08, 65R20
The graphs of state variables. [Colour figure can be viewed at wileyonlinelibrary.com]
The graphs of the control input variables. [Colour figure can be viewed at wileyonlinelibrary.com]
  • Wenhua Xia
    Wenhua Xia
  • Yiping Luo
    Yiping Luo
  • Bifeng Zhou
    Bifeng Zhou
  • Zhujun Wang
    Zhujun Wang
A class of time‐varying delay distributed parameter systems with input saturation is investigated in this paper. The periodic intermittent control method is adopted to make the system stable in finite time, improve the control performance of the system, and save on control cost. A periodic intermittent controller combined saturated input is designed to ensure the stability of the proposed system in finite time. Lyapunov–Krasoviskii stability theory and matrix inequality techniques are used to analyze the finite‐time stability of the system, and sufficient conditions for the system to be stable in finite time are obtained. Finally, the correctness of the theorems is verified by simulation experiments.
  • Ling Li
    Ling Li
  • Xiaoqian Liu
    Xiaoqian Liu
In this paper, we are concerned with an integral system u(x)=Wβ,γup−1v(x),u>0inRn,v(x)=Iαup(x),v>0inRn,$$ \left\{\begin{array}{l}u(x)={W}_{\beta, \gamma}\left({u}^{p-1}v\right)(x),u>0\kern0.5em \mathrm{in}\kern0.5em {R}^n,\\ {}v(x)={I}_{\alpha}\left({u}^p\right)(x),v>0\kern0.5em \mathrm{in}\kern0.5em {R}^n,\end{array}\right. $$ where p≥1,0<α,βγ<n,γ>1$$ p\ge 1,0<\alpha, \beta \gamma <n,\gamma >1 $$. Base on the integrability of positive solutions, we obtain some Liouville theorems and the decay rates of positive solutions at infinity. In addition, we use the properties of the contraction map and the shrinking map to prove that u$$ u $$ is Lipschitz continuous. In particular, the Serrin type condition is established, which plays an important role to classify the positive solutions.
  • Jacek Turski
    Jacek Turski
Two‐dimensional (2D) images of light beams reflected off the objects in space impinge on the retinal photoreceptors of our two laterally separated eyes. Nevertheless, we experience our visual percept as a single 3D entity—our visual world that we tend to identify with the physical world. However, experiments point to different geometries in these two worlds. Using the binocular system with asymmetric eyes (AEs), this article studies the global geometric aspects of visual space in the Riemannian geometry framework. The constant‐depth curves in the horizontal field of binocular fixations consist of families of arcs of ellipses or hyperbolas depending on the AE parameters and the eyes' fixation point. For a single set of AE parameters, there is a unique symmetric fixation at the abathic distance such that the constant‐depth conics are straight frontal lines. Critically, the distribution of these constant‐depth lines is independent of such fixations. In these cases, a two‐parameter family of the Riemann metrics is proposed to account for the retinotopy of the brain's visual pathways and simulated constant‐depth lines. The obtained geodesics for a subset of metric parameters include incomplete geodesics that give finite distances to the horizon. The Gaussian curvature of the phenomenal horizontal field is analyzed for all the metric parameters. The sign of the curvature can be inferred from the global behavior of the constant‐depth ellipses and hyperbolas only when for the metric parameters for which the constant‐depth frontal lines at the abathic distance fixations are geodesics.
  • Hazal Yüksekkaya
    Hazal Yüksekkaya
  • Erhan Piskin
    Erhan Piskin
  • Mohammad M. Kafini
    Mohammad M. Kafini
  • Adel M. Al‐Mahdi
    Adel M. Al‐Mahdi
This paper is concerned with the viscoelastic damped swelling porous elastic soils with time delay. Under more general assumption on the relaxation function and some specific conditions on the weight of the delay, we establish general decay results by using multiplier method and some properties of convex functions. These results generalize and improve some earlier related results in the literature.
This contribution aims at studying a general class of random differential equations with Dirac-delta impulse terms at a finite number of time instants. Our approach directly addresses calculating the so-called first probability density function, from which all the relevant statistical information about the solution, a stochastic process, can be extracted. We combine the Liouville partial differential equation and the random variable transformation method to conduct our study. Finally, all our theoretical findings are illustrated on two stochastic models, widely used in mathematical modeling, for which numerical simulations are carried out.
  • Munish Kansal
    Munish Kansal
  • Litika Rani
    Litika Rani
In this article, we have proposed a novel matrix iteration for computing the matrix sign function numerically. An extensive convergence analysis is carried out in matrix case to achieve fourth‐order convergence. The basins of attraction are drawn to illustrate the global convergence behavior of the proposed method. Analytically, it is shown that the presented scheme is asymptotically stable. Further, the theoretical developments are verified numerically by discussing eigenvalues clustering around ±1$$ \pm 1 $$. Moreover, a class of numerical examples for matrices of various dimensions is assessed to exhibit the efficacy of proposed method.
  • Khumoyun Jabbarkhanov
    Khumoyun Jabbarkhanov
  • Durvudkhan Suragan
    Durvudkhan Suragan
In the present note, we show that if the initial date is between zero and one, then the global solution of a Dirichlet Fisher‐KPP equation with the fractional p$$ p $$‐Laplacian is also between zero and one. As a consequence, a large‐time behavior of the global solution is discussed. In addition, some blow‐up results including a general Lipschitz nonlinearity are given. The proofs mainly rely on special symmetry and invariance properties of the fractional p$$ p $$‐Laplacian.
  • Vijay K. Yadav
    Vijay K. Yadav
  • Nilam
Precision in the measurement of glucose levels in the artificial pancreas is a challenging task and a mandatory requirement for the proper functioning of an artificial pancreas. A suitable machine learning (ML) technique for the measurement of glucose levels in an artificial pancreas may play a crucial role in the management of diabetes. Therefore in the present work, a comparison has been made among a few ML techniques for the measurement of glucose levels in the artificial pancreas because ML is an astounding technology of artificial intelligence and widely applicable in various fields such as medical science, robotics, and environmental science. The models, namely, decision tree (DT), random forest (RF), support vector machine (SVM), and K‐nearest neighbor (KNN), based on supervised learning, are proposed for the dataset of Pima Indian to predict and classify diabetes mellitus. Ensuring the predictions and accuracy up to the level of diabetes mellitus type 2 (DMT2), the comparative behavior of all four models has been discussed. The ML models developed here stratify and predict whether an individual is diabetic or not based on the features available in the dataset. The dataset passes through pre‐processing, and ML algorithms are fitted to train the dataset, and then the performance of the test results is discussed. An error matrix (EM) has been generated to measure the accuracy score of the models. The accuracies in the prediction and classification of DMT2 models are 71%, 77%, 78%, and 80% for DT, SVM, RF, and KNN algorithms, respectively. The KNN model has shown a more precise result in comparison to other models. The proposed methods have shown astounding behavior in terms of accuracy in the prediction of diabetes mellitus as compared to previously developed methods.
This paper is concerned with the well‐posedness of a diffusion–reaction system for a susceptible‐exposed‐infected‐recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio‐temporal spread of the COVID‐19 pandemic and is a variation of the one recently introduced, discussed, and tested in a paper by Viguerie et al (2020). Here, we deal with the mathematical analysis of the resulting Cauchy–Neumann problem: The existence of solutions is proved in a rather general setting, and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions.
In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering.
In this paper, we study the semidiscrete mixed finite element scheme and construct a two‐grid algorithm for the two‐dimensional time‐dependent Schrödinger equation. We analyze error results of the mixed finite element solution in L2$$ {L}^2 $$‐norm by some projection operators. Then, we propose a two‐grid method of the semidiscrete mixed finite element. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two elliptic equations on the fine grid. We also obtain the error estimate of two‐grid solution with exact solution in L2$$ {L}^2 $$‐norm. Finally, a numerical experiment indicates that our two‐grid algorithm is more efficient than the standard mixed finite element method.
This figure shows the approximation of f(x,y)$$ f\left(x,y\right) $$ (blue) by the series (Iwχf)w>0$$ {\left({I}_w^{\chi }f\right)}_{w>0} $$ for w=10$$ w=10 $$ and 40 (yellow and pink, respectively). [Colour figure can be viewed at wileyonlinelibrary.com]
This figure shows the convergence of g(x,y)$$ g\left(x,y\right) $$ (green) by the series (Iwχg)w>0$$ {\left({I}_w^{\chi }g\right)}_{w>0} $$ for w=8$$ w=8 $$ and 35 (blue and pink, respectively). [Colour figure can be viewed at wileyonlinelibrary.com]
In this paper, we introduce and analyze the approximation properties of bivariate generalization for the family of Kantorovich-type exponential sampling series. We derive the basic convergence result and Voronovskaya-type theorem for the proposed sampling series. Using the logarithmic modulus of smoothness, we establish the quantitative estimate of the order of convergence for the Kantorovich-type exponential sampling series. Furthermore, we study the convergence results for the generalized Boolean sum (GBS) operator associated with the bivariate Kantorovich exponential sampling series. In the end, we provide a few examples of kernels to which the presented theory can be applied along with the graphical representation and error estimates.
In this work, an efficient numerical approximation for the solution of the time fractional nonlinear diffuse interface model is studied. The solution to this problem has a weak singularity near the initial time t=0$$ t=0 $$. The fractional order nonlinear diffusion model is transformed into a system of nonlinear functional equations. The Daftardar–Gejji and Jafari method is employed to solve the corresponding nonlinear system. The L1 scheme is used to discretize the Caputo fractional derivative on a graded mesh in the time direction. In contrast, the spatial derivative is approximated by applying a classical central finite difference scheme to a uniform mesh. The convergence analysis and the error bounds are carried out. The analysis and the computational findings exhibit the effectiveness of the proposed method.
We introduce a new version of ψ$$ \psi $$‐Hilfer fractional derivative, on an arbitrary time scale. The fundamental properties of the new operator are investigated, and in particular, we prove an integration by parts formula. Using the Laplace transform and the obtained integration by parts formula, we then propose a ψ$$ \psi $$‐Riemann–Liouville fractional integral on times scales. The applicability of the new operators is illustrated by considering a fractional initial value problem on an arbitrary time scale, for which we prove existence, uniqueness, and controllability of solutions in a suitable Banach space. The obtained results are interesting and nontrivial even for the following particular choices: (i) of the time scale, (ii) of the order of differentiation, and/or (iii) function ψ$$ \psi $$, opening new directions of investigation. Finally, we end the article with comments and future work.
In this paper, we consider the following quasilinear Schrödinger equations with critical growth −Δu+καΔ|u|2α|u|2α−2u+V(x)u=|u|q−2u+|u|2∗−2u,x∈ℝN,$$ -\Delta u+\kappa \alpha \left(\Delta \left({\left|u\right|}^{2\alpha}\right)\right){\left|u\right|}^{2\alpha -2}u+V(x)u={\left|u\right|}^{q-2}u+{\left|u\right|}^{2^{\ast }-2}u,x\in {\mathrm{\mathbb{R}}}^N, $$ where κ>0,34<α≤43,V:ℝN↦ℝ+,N≥3$$ \kappa >0,\frac{3}{4}<\alpha \le \frac{4}{3},V:{\mathrm{\mathbb{R}}}^N\mapsto {\mathrm{\mathbb{R}}}^{+},N\ge 3 $$, and 2<q<2∗=2NN−2$$ 2<q<{2}^{\ast }=\frac{2N}{N-2} $$. The existence of positive solutions will be established through using the variational approach.
Plots of the sum Sv(n;λ,p)$$ {S}_v\left(n;\lambda, p\right) $$ for v=3,n∈{0,1,2,3,4},λ∈−12,12,p=10$$ v=3,n\in \left\{0,1,2,3,4\right\},\lambda \in \left[-\frac{1}{2},\frac{1}{2}\right],p=10 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Plots of the polynomials Qv(x,n;λ,p)$$ {\mathcal{Q}}_v\left(x,n;\lambda, p\right) $$ for v∈{1,2,3,4},x∈−15,15,n=2,λ=12,p=10$$ v\in \left\{1,2,3,4\right\},x\in \left[-\frac{1}{5},\frac{1}{5}\right],n=2,\lambda =\frac{1}{2},p=10 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Plots of the polynomials Qv(x,n;λ,p)$$ {\mathcal{Q}}_v\left(x,n;\lambda, p\right) $$ for v∈{1,2,3,4},x=12,n=2,λ∈−5,5,p=10$$ v\in \left\{1,2,3,4\right\},x=\frac{1}{2},n=2,\lambda \in \left[-5,5\right],p=10 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Surface plots of the polynomials Qv(x,n;λ,p)$$ {\mathcal{Q}}_v\left(x,n;\lambda, p\right) $$ for x∈−1,1,n=2,λ∈−12,12,p=10$$ x\in \left[-1,1\right],n=2,\lambda \in \left[-\frac{1}{2},\frac{1}{2}\right],p=10 $$ with cases: (A) v=1$$ v=1 $$, (B) v=2$$ v=2 $$, (C) v=3$$ v=3 $$, (D) v=4$$ v=4 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler–Frobenius polynomial and B‐spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived. Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.
The phase‐field model for the description of the solidification processes with the glass‐crystal competition is suggested. The model combines the first‐order phase transition model in the phase‐field formalism and gauge‐field theory of glass transition. We present a self‐consistent system of stochastic motion equations for unconserved order parameters describing the crystal‐like short‐range ordering and vitrification. It is shown that the model qualitatively describes the glass‐crystal competition during quenching with finite cooling speed. The nucleation of the crystalline phase at slow cooling speeds and low undercoolings proceeds by a fluctuation mechanism. The model demonstrates the tendency to amorphization with the increase of its cooling rate.
Surface plots for Example 4.1. [Colour figure can be viewed at wileyonlinelibrary.com]
Surface plots for Example 4.2. [Colour figure can be viewed at wileyonlinelibrary.com]
Contour plots for Example 4.2. [Colour figure can be viewed at wileyonlinelibrary.com]
Contour plots of numerical electron density eh$$ {e}_h $$ for Example 4.3. [Colour figure can be viewed at wileyonlinelibrary.com]
Contour plots of numerical hole density ph$$ {p}_h $$ for Example 4.3. [Colour figure can be viewed at wileyonlinelibrary.com]
  • Xindong Li
    Xindong Li
  • Wenwen Xu
    Wenwen Xu
  • Qing Guo
    Qing Guo
We consider a mass conservative type method for semiconductor device problem by employing mixed finite element method (FEM) for electric potential equation and mass conservative characteristic FEM for both electron and hole density equations. The boundedness of numerical solution without certain time‐step restriction and optimal L2$$ {L}^2 $$ error estimates of full discrete scheme are proved. Numerical experiment is presented to verify the effectiveness and unconditional stability of the proposed method.
In this paper, we study an elliptic variational problem regarding the p$$ p $$‐fractional Laplacian in ℝN$$ {\mathrm{\mathbb{R}}}^N $$ on the basis of recent result which generalizes some nice published work, and then give some sufficient conditions under which some weak solutions to our studied elliptic variational problem are continuous in ℝN$$ {\mathrm{\mathbb{R}}}^N $$. In the final appendix, we correct the proofs of two published lemmas for 1<p<2$$ 1<p<2 $$.
This paper investigates the complete regularity of the weak solutions, the existence of the strong (X,X2α)$$ \left(X,{X}_{2\alpha}\right) $$‐global and exponential attractors, and their stability on dissipative index α$$ \alpha $$ for the structurally damped Kirchhoff wave equation: utt−M(‖∇u‖2)Δu+(−Δ)αut+f(u)=g(x)$$ {u}_{tt}-M\left({\left\Vert \nabla u\right\Vert}^2\right)\Delta u+{\left(-\Delta \right)}^{\alpha }{u}_t+f(u)=g(x) $$, together with the Dirichlet boundary condition, where the perturbed parameter α∈(1/2,1)$$ \alpha \in \left(1/2,1\right) $$ is called a dissipative index, X$$ X $$ is energy space, and X2α$$ {X}_{2\alpha } $$ is strong solution space. We show that when the nonlinearity f(u)$$ f(u) $$ is of supercritical growth: p∗=N+2N−2≤p<p¯α$$ {p}^{\ast}\left(=\frac{N+2}{N-2}\right)\le p<{\overline{p}}_{\alpha } $$, (i) the weak solutions of the model are just the strong ones; (ii) the global and exponential attractors of the related dynamical system (Sα(t),X)$$ \left({S}^{\alpha }(t),X\right) $$ obtained in literature before are exactly the strong (X,X2α)$$ \left(X,{X}_{2\alpha}\right) $$‐ones, and the family of strong (X,X2α)$$ \left(X,{X}_{2\alpha}\right) $$‐global attractors {Aα}α∈(1/2,1)$$ {\left\{{\mathcal{A}}_{\alpha}\right\}}_{\alpha \in \left(1/2,1\right)} $$ is upper semicontinuous on α$$ \alpha $$ in X2α$$ {X}_{2\alpha } $$‐topology; (iii) for each α0∈(1/2,1)$$ {\alpha}_0\in \left(1/2,1\right) $$, Sα(t)$$ {S}^{\alpha }(t) $$ has a family of strong (X,X2α)$$ \left(X,{X}_{2\alpha}\right) $$‐exponential attractors {Aexpα}$$ \left\{{\mathfrak{A}}_{exp}^{\alpha}\right\} $$, which is Hölder continuous at α0$$ {\alpha}_0 $$ in X2α0$$ {X}_{2{\alpha}_0} $$‐topology. The method developed here allows establishing the above‐mentioned results, which breakthrough the restriction 1≤p<p∗$$ 1\le p<{p}^{\ast } $$ on this topic in literature before.
In this paper, the fourth‐order parabolic equation with the first Neumann boundary value conditions is concerned, where the values of the first and second spatial derivatives of the unknown function at the boundary are given. A compact difference scheme is established for this kind of problem by using the weighted average and order reduction methods. The difficulty lies in the challenges of handling the boundary conditions with high accuracy. The unique solvability, convergence and stability of the proposed compact difference scheme are proved by the energy method. Some novel techniques are introduced for the analysis. Then, the extension to a more general case with the reaction term is briefly explored. Finally, two numerical examples are numerically calculated to show the efficiency of the proposed numerical schemes.
Paths of the susceptible, infected, vaccinated, and recovered populations, under the first condition of the disease extinction, together with a comparison between different types of noise with the deterministic case. [Colour figure can be viewed at wileyonlinelibrary.com]
Paths of the susceptible, infected, vaccinated, and recovered populations with Lévy noise, under the second disease extinction condition and in the presence and absence of time delays in the stages of vaccination. [Colour figure can be viewed at wileyonlinelibrary.com]
Paths of the susceptible, infected, vaccinated, and recovered populations with Lévy noise, under the disease persistence condition and in the presence and absence of time delays in the stages of vaccination. [Colour figure can be viewed at wileyonlinelibrary.com]
The focal point of this paper is to further enhance the existing stochastic epidemic models by incorporating several new disease characteristics, such as the validation time of the vaccination procedure, the stages of vaccine required to gain a long-period immunity together with the time separating each stage, the deaths linked to the vaccine, and finally, the sudden environmental noise which is exhibited by sociocultural changes, such as antivaccination movements. To incorporate all the aforementioned characteristics, we extend the standard Susceptible-Vaccinated-Infected-Recovered (SVIR) epidemic model to a new mathematical model, which is governed by a system of coupled stochastic delay differential equations, in which the disease transmission rates are driven by Gaussian noise and Lévy-type jump stochastic process. First, under suitable conditions on the jump intensities, we address the mathematical well-posedness and biological feasibility of the model, by virtue of the Lyapunov method and the stopping-time technique. Then, by choosing an adequate positively invariant set for the considered model, we establish sufficient conditions guaranteeing the disease extinction and persistence. Lastly, to support the theoretical results, we provide the outcome of several numerical simulations which, together with our conducted analysis, indicate that the spread of the disease can be majorly altered by all the new considered characteristics.
The emergence of COVID-19 pandemic has been a major social as well as economic challenges around the globe. Infections from the infected surfaces have also been identified as drivers of COVID-19 transmission, but most of the epidemic models do not include the effect of environmental contamination to account for the indirect transmission of the disease. The present study is devoted to the investigation of the effect of environmental contamination on the spread of the coronavirus pandemic by means of a mathematical model. We also consider the impact of vaccination coverage as an effective control measure against COVID-19. The proposed model is analyzed to discuss the feasibility as well as stability of the disease-free and endemic equilibria; an epidemic threshold in the form of basic reproduction number is obtained. Further, we incorporate the effect of seasonal periodic changes by letting the rate of direct transmission of disease as time dependent, and find sufficient conditions for the global attractivity of the positive periodic solution. We employ sophisticated techniques of sensitivity analysis to identify model parameters which significantly alter the epidemic threshold and the disease prevalence. We find that by enhancing the vaccination of the susceptible population and hospitalization of the symptomatic/asymptomatic individuals, the basic reproduction number can be lowered to a value less than unity. The findings show that the prevalence of disease can be potentially suppressed by increasing the vaccination of susceptible population, hospitalization of infected people and depletion of environmental contamination. Moreover, we observe that seasonal pattern in the disease transmission causes persistence of the pandemic in the population for a longer period.
Original envelop of 3D image.
The flow‐process diagram of using analytic signal's amplitude to envelop detection.
Wang's analytic signal associated with CFT [27].
This paper's method: analytic signal associated with CLCT, where A1=(1,10;−0.05,0.5)$$ {A}_1=\left(1,10;-0.05,0.5\right) $$, A2=(1,10;−0.05,0.5)$$ {A}_2=\left(1,10;-0.05,0.5\right) $$, and A3=(1000,1;−0.1,0.0009)$$ {A}_3=\left(1000,1;-0.1,0.0009\right) $$.
Monogenic signal approach of [12].
Analytic signal is a useful mathematical tool. It separates qualitative and quantitative information of a signal in form of the local phase and local amplitude. Clifford Fourier transform (CFT) plays a vital role in the representation of multidimensional signals. By generalizing the CFT to Clifford linear canonical transform (CLCT), we present a new type of Clifford biquaternionic analytic signal. Due to the advantages of more freedom, the envelop detection problems of 3D images, with the help of this new analytic signal, can get a better visual appearance. Synthesis examples are presented to demonstrate these advantages.
In this paper, we study the null controllability for the problems associated to the operators yt−Ay−λb(x)y+∫01K(t,x,τ)y(t,τ)dτ,(t,x)∈(0,T)×(0,1),$$ {y}_t- Ay-\frac{\lambda }{b(x)}y+\int_0^1K\left(t,x,\tau \right)y\left(t,\tau \right)\kern0.1em d\tau, \kern0.30em \left(t,x\right)\in \left(0,T\right)\times \left(0,1\right), $$ where Ay:=ayxx$$ Ay:= a{y}_{xx} $$ or Ay:=(ayx)x$$ Ay:= {\left(a{y}_x\right)}_x $$ and the functions a$$ a $$ and b$$ b $$ degenerate at an interior point x0∈(0,1)$$ {x}_0\in \left(0,1\right) $$. To this aim, as a first step, we study the well‐posedness, the Carleman estimates, and the null controllability for the associated nonhomogeneous degenerate and singular heat equations. Then, using Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.
In this paper, we propose a new iterative scheme with memory for solving nonlinear equations numerically in order to achieve higher order of convergence in comparison to the cubically convergent Chebyshev–Halley‐type method. Several modifications on Chebyshev–Halley‐type methods without memory have been considered in order to extend it to the scheme with memory. We have used self accelerating parameter in order to attain acceleration of convergence speed which is estimated from the current and previous iterations using divided differences. Therefore, the order of convergence increases from 3 to 3.30 without any further functional evaluation. We study the complex and real dynamics of the proposed family. The parameter spaces and dynamical planes are presented. From the parameter spaces, we can detect different members of the proposed family that have good and bad convergence properties. From the dynamical planes, we can analyze the stability of the proposed family in terms of different values of the parameter involved. This study aids in determining the family members with stable behavior which in turn are suitable for practical problems. Numerical examples and comparisons with some of the existing methods are included to confirm the theoretical results. Furthermore, basins of attraction are included to describe a clear picture of the convergence of the proposed as well as some of the existing methods.
Forensic linguistics and stylometry have in the exploration of linguistic patterns one of their fundamental tools. Mathematical structures such as complex multilayer networks and hypergraphs provide remarkable resources to represent and analyze texts. In this paper, we present a model that includes some specific mesoscopic relations between the different types of words in a corpus (lexical words, verbs, linking words, other words) according to the sentences or paragraphs in which they appear. This model is supported by various mathematical structures such as partial multiline graphs, multilayer hypergraphs, and their derivative graphs. The methodology proposed from this new point of view is of singular help to find meaningful sentences from any text to set up an automatic summary of the text and, eventually, to determine its linguistic level.
The paper is a new approach to the Duhamel's convolution integral. It contains a brief description of the Duhamel's convolution integral and an overview of applications of this integral. The scientific novelty presented in the work is the original algorithm for determining the frequency characteristic, together with its mathematical justification as well as with hardware and software implementation. The proposed algorithm was used in computer measurement systems enabling the determination of Bode and Nyquist characteristics of objects used in automation. The measurement system was built by the authors on the basis of the proposed algorithm. Measurements were carried out using it. During the measurements of the selected automation object, actual results were obtained and referenced to theoretical results. Then their correctness was assessed using the theory of measurement errors. In this way, the correctness of the mathematical considerations concerning the application of the Duhamel's convolution integral in computer measurement systems enabling the determination of the frequency characteristics of objects used in automation was confirmed.
This paper is devoted to the robust approximation with a variational phase field approach of multiphase mean curvature flows with possibly highly contrasted mobilities. The case of harmonically additive mobilities has been addressed recently using a suitable metric to define the gradient flow of the phase field approximate energy. We generalize this approach to arbitrary nonnegative mobilities using a decomposition as sums of harmonically additive mobilities. We establish the consistency of the resulting method by analyzing the sharp interface limit of the flow: A formal expansion of the phase field shows that the method is of second order. We propose a simple numerical scheme to approximate the solutions to our new model. Finally, we present some numerical experiments in dimensions 2 and 3 that illustrate the interest and effectiveness of our approach, in particular for approximating flows in which the mobility of some phases is zero.
A kind of nonlinear noninstantaneous impulsive equation with state‐dependent delay is studied here. By utilizing suitable fixed point theorem and the theory of semigroup in Banach space, the uniqueness and existence results of S$$ \mathcal{S} $$‐asymptotically w$$ w $$‐periodic mild solutions are obtained, respectively. In the end, two examples are presented to demonstrate the validity of the obtained results.
The solution curve of S(t),E(t),I1(t),I2(t)$$ S(t),E(t),{I}_1(t),{I}_2(t) $$ for system (2) with different intensity of white noises. [Colour figure can be viewed at wileyonlinelibrary.com]
In this paper, by using Itô's formula, we study the dynamic behaviors of a stochastic human immune deficiency virus (HIV) model, which describes Acquired Immune Deficiency Syndrome (AIDS) with the effect of treatment. First, we discuss the global positivity of the solution to such a model with positive initial value. Then, we consider the limiting behavior of the solution. Moreover, we obtain the sufficient condition to guarantee the extinction of AIDS. Meanwhile, by constructing an appropriate Lyapunov function, we prove that the solution has a unique ergodic stationary distribution under a certain assumptions. Finally, we illustrate the validity of our results by numerical simulations.
We study the global dynamics of a susceptible‐vaccinated‐infected‐recovered model that incorporates nonlocal diffusion. By identifying the basic reproduction number ℛ0$$ {\mathrm{\mathcal{R}}}_0 $$ of the model, we obtain the following threshold‐type results: (i) If ℛ0<1$$ {\mathrm{\mathcal{R}}}_0<1 $$, then the epidemic becomes extinct in the sense that the infection‐free equilibrium is globally attractive; (ii) if ℛ0>1$$ {\mathrm{\mathcal{R}}}_0>1 $$ and the diffusion coefficients are the same for all classes, then the epidemic persists in the sense that the system is uniformly persistent; and (iii) if ℛ0>1$$ {\mathrm{\mathcal{R}}}_0>1 $$, the diffusion coefficients for susceptible, and the vaccinated classes are zero, then the system admits a unique endemic equilibrium, and the omega‐limit set is included in the singleton of the endemic equilibrium. Our results show that ℛ0$$ {\mathrm{\mathcal{R}}}_0 $$ is an essential value for determining global epidemic dynamics in our model.
This paper investigates the theoretical modeling and coupled free vibration behaviors of a rotating double‐bladed shaft assembly resting on elastic supports in a spacecraft system. According to the Kirchhoff plate theory and the Euler–Bernoulli beam theory, the theoretical model is established. The presented rotor is considered to be made of porous foam metal matrix and graphene nanoplatelet (GPL) reinforcement. Nonuniform distributions of porosity and GPLs are taken into account, which leads to the functionally graded (FG) structure. The effective material properties of the double‐bladed shaft assembly are varying along the radius and thickness directions of the shaft and blade, respectively. Moreover, the rule of mixture, the Halpin–Tsai model, and the open‐cell scheme are employed to determine its material properties. Considering the gyroscopic effect, the Lagrange equation is utilized to derive the coupled equations of motion. Then the traveling wave frequencies of the double‐bladed shaft assembly are obtained by applying the assumed modes method and substructure modal synthesis method. A detailed parametric analysis is conducted to examine the effects of the rotating speed, GPL weight fraction, GPL distribution pattern, GPL length‐to‐thickness ratio, GPL length‐to‐width ratio, porosity coefficient, porosity distribution pattern, shaft length‐to‐radius ratio, blade length‐to‐thickness ratio, support stiffness and support location on the free vibration behaviors of the double‐bladed shaft assembly.
First, the solution uniqueness, existence and regularity for stationary anisotropic (linear) Stokes and generalised Oseen systems with constant viscosity coefficients in a compressible framework are analysed in a range of periodic Sobolev (Bessel‐potential) spaces in ℝn$$ {\mathrm{\mathbb{R}}}^n $$. By the Galerkin algorithm and the Brower fixed point theorem, the existence of solution to the stationary anisotropic (nonlinear) Navier–Stokes incompressible system is shown in a periodic Sobolev space for any n≥2$$ n\ge 2 $$. Then the solution uniqueness and regularity results for stationary anisotropic periodic Navier–Stokes system are established for n∈{2,3,4}$$ n\in \left\{2,3,4\right\} $$.
In this work, we propose the construction of discrete‐time systems with two time scales in which infectious diseases dynamics are involved. We deal with two general situations. In the first, we consider that individuals affected by the disease move between generalized sites on a faster time scale than the dynamics of the disease itself. The second situation includes the dynamics of the disease acting faster together with another slower general process. Once the models have been built, conditions are established so that the analysis of the asymptotic behavior of their solutions can be carried out through reduced models. This is done using known reduction results for discrete‐time systems with two time scales. These results are applied in the analysis of two new models. The first of them illustrates the first proposed situation, being the local dynamics of the SIS‐type disease. Conditions are found for the eradication or global endemicity of the disease. In the second model, a case of co‐infection with a primary disease and an opportunistic disease is treated, the latter acting faster than the former. Conditions for eradication and endemicity of co‐infection are proposed.
The directions of the components of the velocity vector.
Distributions of the flow variables in the region behind a cylindrical shock front for different values of γ$$ \gamma $$ with m1=−0.1,d∗=0.1,S∗=5$$ {m}_1=-0.1,{d}^{\ast }=0.1,{S}_{\ast }=5 $$, and MA−2=0.01$$ {M}_A^{-2}=0.01 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Distributions of the flow variables in the region behind a cylindrical shock front for different values of m1$$ {m}_1 $$ with γ=7/5,d∗=0.1,S∗=5$$ \gamma =7/5,{d}^{\ast }=0.1,{S}_{\ast }=5 $$, and MA−2=0.02$$ {M}_A^{-2}=0.02 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Distributions of flow variables in the region behind a cylindrical shock front for different values of MA−2$$ {M}_A^{-2} $$ with γ=7/5,d∗=0.1,S∗=5$$ \gamma =7/5,{d}^{\ast }=0.1,{S}_{\ast }=5 $$, and m1=−0.1$$ {m}_1=-0.1 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Distributions of the flow variables in the region behind a cylindrical shock front for different values of d∗$$ {d}^{\ast } $$ with MA−2=0.01,γ=7/5,S∗=5$$ {M}_A^{-2}=0.01,\gamma =7/5,{S}_{\ast }=5 $$, and m1=−0.1$$ {m}_1=-0.1 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
In this manuscript, we studied a quasi‐linear hyperbolic system of partial differential equations which describes the one‐dimensional adiabatic unsteady flow behind a strong cylindrical shock wave propagating in a rotating non‐ideal gas, which has a varying azimuthal fluid velocity together with a varying axial fluid velocity, with radiation heat flux, and similarity solutions are obtained by using the Lie group‐theoretic method. The basic idea of this method is that it changes the system of PDEs representing the one‐dimensional flow through the similarity variable to the system of ODEs. The flow profiles are drawn behind the shock followed by a brief discussion on the behavior of the solutions via graphs. The effects of variation in non‐ideal parameter, adiabatic index of the gas, Alfven–Mach number and ambient azimuthal velocity exponent on the flow variables are described. All numerical calculations have been performed using the “Mathematica” software.
In this article, families of solitary waves solutions of a general third-order nonlinear non-ohmic cable equation in cardio-electro-physiology are obtained using the $\exp(-\varphi(\xi))$-expansion method. In this equation, the unknown function represents the transmembrane current, and the exact soliton-like solutions are thoroughly derived using this analytic technique, and illustrated through surface and contour plots. To support the analytical findings presented in this work, the existence of solution and their optimal regularity is proved rigorously using Shauder's fixed-point theorem.
Dynamical planes of M4,γ$$ {M}_{4,\gamma } $$ and their methods with memory. [Colour figure can be viewed at wileyonlinelibrary.com]
Dynamical planes of M7,γ$$ {M}_{7,\gamma } $$ and their methods with memory. [Colour figure can be viewed at wileyonlinelibrary.com]
In this paper, we design two parametric classes of iterative methods without memory to solve nonlinear systems, whose convergence order is 4 and 7, respectively. From their error equations and to increase the convergence order without performing new functional evaluations, memory is introduced in these families of different forms. That allows us to increase from 4 to 6 the convergence order in the first family and from 7 to 11 in the second one. We perform some numerical experiments with big size systems for confirming the theoretical results and comparing the proposed methods along other known schemes.
Graph of B2,2(x1,x2)$$ {B}_{2,2}\left({x}_1,{x}_2\right) $$ with parameter n=m=2$$ n=m=2 $$ in the domain ]0,3]×]0,3]$$ \left]0,3\right]\times \left]0,3\right] $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Graph of J(x1,x2)$$ J\left({x}_1,{x}_2\right) $$ with parameter γ1=γ2=1$$ {\gamma}_1={\gamma}_2=1 $$ and β1=β2=2$$ {\beta}_1={\beta}_2=2 $$ in the domain (0,1]×(0,1]$$ \left(0,1\right]\times \left(0,1\right] $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Exponential model function described by Equation (18). On the left, the 2D plot; on the right one, 1D profile of the same function along a generic direction θ$$ \theta $$. [Colour figure can be viewed at wileyonlinelibrary.com]
Examples of noisy signals for different values of the SNR$$ SNR $$: in blue, the noise‐free function; in green, the function with the Gaussian noise. From left to right: SNR=1,10,50$$ SNR=1,10,50 $$. In the second row, the corresponding 1D profile for a generic direction θ$$ \theta $$. It is evident how having a good SNR$$ SNR $$ makes the noise quite negligible (see, e.g, SNR=50$$ SNR=50 $$). [Colour figure can be viewed at wileyonlinelibrary.com]
Two examples of reconstruction and filtering of a noisy function: the profile of the original function in blue, the profile of the noisy one in green (Gaussian noise, zero mean), and the profile of the reconstruction by means of the Durrmeyer operator in red. The results have been achieved with N=30$$ N=30 $$ and w=10$$ w=10 $$, SNR=10$$ SNR=10 $$ (left), SNR=1$$ SNR=1 $$ (right). [Colour figure can be viewed at wileyonlinelibrary.com]
In this paper, we study the convergence properties of certain semi‐discrete exponential‐type sampling series in a multidimensional frame. In particular, we obtain an asymptotic formula of Voronovskaya type, which gives a precise order of approximation in the space of continuous functions, and we give some particular example illustrating the theory. Applications to the study of the seismic waves are illustrated.
(A) Graph of the function t↦𝔼|Xξ(t)|2; (B) the corresponding graph of the function t↦1tln𝔼|Xξ(t)|2 versus z=−0.5$$ z=-0.5 $$; (C) 100 trajectories of Xξ(t)$$ {X}^{\xi }(t) $$; (D) the corresponding graph of the function t↦1tln|Xξ(t)|$$ t\mapsto \frac{1}{t}\ln \mid {X}^{\xi }(t)\mid $$ versus z=−0.25$$ z=-0.25 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
(A) Graph of the function t↦𝔼|Xζ(t)|2; (B) the corresponding graph of the function t↦1tln𝔼|Xζ(t)|2 versus z=−0.5$$ z=-0.5 $$. [Colour figure can be viewed at wileyonlinelibrary.com]
This paper focuses on exponential stability of numerical solutions of neutral stochastic delay differential equations. A novel approach is introduced to study numerical approximation methods of neutral stochastic differential equations with time‐dependent delay. In contrast to the advances in the literature, this work provides a new method and new criteria for which the Euler‐Maruyama approximation method and the backward Euler‐Maruyama approximation method can reproduce exponential stability in mean square and almost sure exponential stability for sufficiently small step sizes. Two examples are provided to demonstrate the effectiveness of our criteria.
This paper develops a Müntz–Legendre wavelet method for solving a fractional optimal control problem with dynamic constraint as a fractional Sturm–Liouville problem. We first derive the necessary optimality conditions as a two‐point boundary value problem by using the calculus of variation and integration by part formula. The operational matrices for Müntz–Legendre scaling functions have been obtained, and by utilizing it, the two‐point boundary value problem has been converted into a system of algebraic equations. The L2$$ {L}_2 $$‐error estimates in the approximation of operations matrices and in the approximation of unknown variable by the Müntz–Legendre wavelet has been derived. In the last, illustrative examples have been taken to show the applicability of the proposed method.
In this work, we show how it is possible to test the nullity of covariances, in a set of variables, using a simple univariate procedure. The methodology proposed enables us to perform the multivariate test of independence of several variables, under specific conditions for the covariance structure. The methodology proposed may be used in the high‐dimensional setting and, given its simplicity, allows to overcome the difficulties in using the exact distribution of the statistic used in the likelihood ratio testing procedure. A simulation study is provided to assess the power and significance level, in different scenarios, of the testing procedure proposed when compared with different likelihood ratio tests and a methodology available in the literature.
Concentration versus temperature (phase equilibrium diagram with liquidus and solidus curves). [Colour figure can be viewed at wileyonlinelibrary.com]
Similarity solutions for the two‐phase Rubinstein binary‐alloy solidification problem in a semi‐infinite material are developed. These new explicit solutions are obtained by considering two cases: A heat flux or a convective boundary conditions at the fixed face, and the necessary and sufficient conditions on data are also given in order to have an instantaneous solidification process. We also show that all solutions for the binary‐alloy solidification problem are equivalent under some restrictions for data. Moreover, this implies that the coefficient that characterizes the solidification front for the Rubinstein solution must verify an inequality as a function of all thermal and boundary conditions.
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Top-cited authors
Dumitru Baleanu
  • Institute of Space Sciences
Helmut Neunzert
  • Fraunhofer Institute for Industrial Mathematics
Salim A. Messaoudi
  • University of Sharjah
Dr. Sunil Kumar
  • National Institute of Technology, Jamshedpur
Patrick Ciarlet
  • ENSTA Paris