Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization ...Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization (QL) but its applicability is locally in time. Nonetheless, consecutive applications of it can form a new method which is applicable globally in time. Dividing the control problem equivalently into many finite consecutive control subproblems they can be solved consecutively by a QL method. The proposed QL method for each subproblem constructs an infinite sequence of linear-quadratic optimal boundary control problems. These problems have solutions which converge to any optimal solutions of the subproblem. This implies the uniqueness of optimal solution to the subproblem. Merging solutions to the subproblems the solution of original control problem is obtained and its uniqueness is concluded. This uniqueness result is new. The proposed consecutive quasilinearization method is numerically stable with convergence order at least linear. Its consecutive feature prevents large scale computations and increases machine applicability. Its applicability for globalization of locally convergent methods makes it attractive for designing fast hybrid solution methods with global convergence.展开更多
Objective of our paper is to present the Haar wavelet based solutions of boundary value problems by Haar collocation method and utilizing Quasilinearization technique to resolve quadratic nonlinearity in y. More accur...Objective of our paper is to present the Haar wavelet based solutions of boundary value problems by Haar collocation method and utilizing Quasilinearization technique to resolve quadratic nonlinearity in y. More accurate solutions are obtained by wavelet decomposition in the form of a multiresolution analysis of the function which represents solution of boundary value problems. Through this analysis, solutions are found on the coarse grid points and refined towards higher accuracy by increasing the level of the Haar wavelets. A distinctive feature of the proposed method is its simplicity and applicability for a variety of boundary conditions. Numerical tests are performed to check the applicability and efficiency. C++ program is developed to find the wavelet solution.展开更多
Diffusions of multiple components have numerous applications such as underground water flow, pollutant movement, stratospheric warming, and food processing. Particularly, liquid hydrogen is used in the cooling process...Diffusions of multiple components have numerous applications such as underground water flow, pollutant movement, stratospheric warming, and food processing. Particularly, liquid hydrogen is used in the cooling process of the aeroplane. Further, liquid nitrogen can find applications in cooling equipment or electronic devices, i.e., high temperature superconducting(HTS) cables. So, herein, we have analysed the entropy generation(EG), nonlinear thermal radiation and unsteady(time-dependent) nature of the flow on quadratic combined convective flow over a permeable slender cylinder with diffusions of liquid hydrogen and nitrogen. The governing equations for flow and heat transfer characteristics are expressed in terms of nonlinear coupled partial differential equations. The solutions of these equations are attempted numerically by employing the quasilinearization technique with the implicit finite difference approximation. It is found that EG is minimum for double diffusion(liquid hydrogen and heat diffusion)than triple diffusion(diffusion of liquid hydrogen, nitrogen and heat). The enhancing values of the radiation parameter R_(d) and temperature ratio θ_(w) augment the fluid temperature for steady and unsteady cases as well as the local Nusselt number. Because, the fluid absorbs the heat energy released due to radiation, and in turn releases the heat energy from the cylinder to the surrounding surface.展开更多
We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model.The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation.In this paper,we w...We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model.The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation.In this paper,we will construct an example with a finite time cusp singularity due to the quasilinearity of the wave equation,extended from an earlier resultonaspecial case.展开更多
In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi...In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi-global C^(2)solution to this second-order quasilinear hyperbolic system.After then,the general constructive framework is utilized to obtain the local exact boundary controllability for this second-order system.展开更多
In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on R<sup>N</sup> × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a sm...In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on R<sup>N</sup> × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a small parameter, I<sub>α</sub> is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique. .展开更多
In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilin...In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.展开更多
In this paper, based on the Kirchhoff transformation and the natural boundary element method, a coupled natural boundary element and curved edge finite element is applied to solve the anisotropic quasi-linear problem ...In this paper, based on the Kirchhoff transformation and the natural boundary element method, a coupled natural boundary element and curved edge finite element is applied to solve the anisotropic quasi-linear problem in an unbounded domain with a concave angle. By using the principle of the natural boundary reduction, we obtain the natural integral equation on the artificial boundary of circular arc boundary, and get the coupled variational problem and its numerical method. Then the error and convergence of coupling solution are analyzed. Finally, some numerical examples are verified to show the feasibility of our method.展开更多
In this paper, we prove some Δ-convergence and strong convergence results for the sequence generated by a new algorithm to a minimizer of two convex functions and a common fixed point for quasi-pseudo-contractive map...In this paper, we prove some Δ-convergence and strong convergence results for the sequence generated by a new algorithm to a minimizer of two convex functions and a common fixed point for quasi-pseudo-contractive mappings in Hadamard spaces. Our theorems improve and generalize some recent results in the literature.展开更多
In this paper,we study the following generalized quasilinear Schrdinger equations with critical or supercritical growths-div(g^2(u)▽u) + g(u)g′(u)|▽u|~2+ V(x)u = f(x,u) + λ|u|^(p-2)u,x∈R^N,where λ>0,N≥3,g...In this paper,we study the following generalized quasilinear Schrdinger equations with critical or supercritical growths-div(g^2(u)▽u) + g(u)g′(u)|▽u|~2+ V(x)u = f(x,u) + λ|u|^(p-2)u,x∈R^N,where λ>0,N≥3,g:R → R^+ is a C^1 even function,g(0) = 1,g′(s) ≥ 0 for all s ≥ 0,lim_(|s|→+∞)g(s)/|s|^(α-1):= β > 0 for some α≥ 1 and(α-1)g(s) > g′(s)s for all s > 0 and p ≥α2*.Under some suitable conditions,we prove that the equation has a nontrivial solution for smallλ > 0 using a change of variables and variational method.展开更多
The existence and asymptotic behavior of solution for a class of quasilinear singularly perturbed boundary value problems are discussed under suitable conditions by the theory of differential inequalities and matching...The existence and asymptotic behavior of solution for a class of quasilinear singularly perturbed boundary value problems are discussed under suitable conditions by the theory of differential inequalities and matching principle.展开更多
This paper obtain C1,α- regularity of weak solutions to the p-Laplacian obstacle problem with C1,β-obstacle function under controllable growth conditions.
In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the mon...In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the monotonicity of σi(s) (i= 1,…, n). The nonexistence of global solutions to the initial-boundary value problem of the equation is also discussed, a blowup theorem is proved and a concrete example is given.展开更多
This paper considers the quasilinear elliptic equation where , and 0 < m < p-1 < q < +∞, Ω is a bounded domain in RN(N 3).λ is a positive number. Object is to estimate exactly the magnitute of λ* such ...This paper considers the quasilinear elliptic equation where , and 0 < m < p-1 < q < +∞, Ω is a bounded domain in RN(N 3).λ is a positive number. Object is to estimate exactly the magnitute of λ* such that (1)λ has at least one positive solution if λ ∈ (0, λ*) and no positive solutions if λ > λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ ∈ (0, λ*) and . Finally, the authors obtain a multiplicity result with positive energy of (1)λ when 0 < m < p - 1 < q = (Np)/(N-p) - 1.展开更多
We study a quasilinear elliptic equation with polynomial growth coefficients. The existence of infinitely many solutions is obtained by a dual method and a nonsmooth critical point theory.
Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically inter...Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physically interesting cases are found by using the partial Lagrangian approach.展开更多
In this paper, a quasilinear elliptic system is investigated, which involves concave-convex nonlinearities and nonlinear boundary condition. By Nehari manifold, fibering method and analytic techniques, the existence o...In this paper, a quasilinear elliptic system is investigated, which involves concave-convex nonlinearities and nonlinear boundary condition. By Nehari manifold, fibering method and analytic techniques, the existence of multiple nontrivial nonnegative solutions to this equation is verified.展开更多
文摘Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization (QL) but its applicability is locally in time. Nonetheless, consecutive applications of it can form a new method which is applicable globally in time. Dividing the control problem equivalently into many finite consecutive control subproblems they can be solved consecutively by a QL method. The proposed QL method for each subproblem constructs an infinite sequence of linear-quadratic optimal boundary control problems. These problems have solutions which converge to any optimal solutions of the subproblem. This implies the uniqueness of optimal solution to the subproblem. Merging solutions to the subproblems the solution of original control problem is obtained and its uniqueness is concluded. This uniqueness result is new. The proposed consecutive quasilinearization method is numerically stable with convergence order at least linear. Its consecutive feature prevents large scale computations and increases machine applicability. Its applicability for globalization of locally convergent methods makes it attractive for designing fast hybrid solution methods with global convergence.
文摘Objective of our paper is to present the Haar wavelet based solutions of boundary value problems by Haar collocation method and utilizing Quasilinearization technique to resolve quadratic nonlinearity in y. More accurate solutions are obtained by wavelet decomposition in the form of a multiresolution analysis of the function which represents solution of boundary value problems. Through this analysis, solutions are found on the coarse grid points and refined towards higher accuracy by increasing the level of the Haar wavelets. A distinctive feature of the proposed method is its simplicity and applicability for a variety of boundary conditions. Numerical tests are performed to check the applicability and efficiency. C++ program is developed to find the wavelet solution.
文摘Diffusions of multiple components have numerous applications such as underground water flow, pollutant movement, stratospheric warming, and food processing. Particularly, liquid hydrogen is used in the cooling process of the aeroplane. Further, liquid nitrogen can find applications in cooling equipment or electronic devices, i.e., high temperature superconducting(HTS) cables. So, herein, we have analysed the entropy generation(EG), nonlinear thermal radiation and unsteady(time-dependent) nature of the flow on quadratic combined convective flow over a permeable slender cylinder with diffusions of liquid hydrogen and nitrogen. The governing equations for flow and heat transfer characteristics are expressed in terms of nonlinear coupled partial differential equations. The solutions of these equations are attempted numerically by employing the quasilinearization technique with the implicit finite difference approximation. It is found that EG is minimum for double diffusion(liquid hydrogen and heat diffusion)than triple diffusion(diffusion of liquid hydrogen, nitrogen and heat). The enhancing values of the radiation parameter R_(d) and temperature ratio θ_(w) augment the fluid temperature for steady and unsteady cases as well as the local Nusselt number. Because, the fluid absorbs the heat energy released due to radiation, and in turn releases the heat energy from the cylinder to the surrounding surface.
文摘We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model.The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation.In this paper,we will construct an example with a finite time cusp singularity due to the quasilinearity of the wave equation,extended from an earlier resultonaspecial case.
基金Supported by the Science and Technology Commission of Shanghai Municipality (Grant No.23ZR1402100)the Fundamental Research Funds for the Central Universities (Grant Nos. 2232022G-13 and 2232023G-13)
文摘In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi-global C^(2)solution to this second-order quasilinear hyperbolic system.After then,the general constructive framework is utilized to obtain the local exact boundary controllability for this second-order system.
文摘In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on R<sup>N</sup> × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a small parameter, I<sub>α</sub> is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique. .
文摘In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.
文摘In this paper, based on the Kirchhoff transformation and the natural boundary element method, a coupled natural boundary element and curved edge finite element is applied to solve the anisotropic quasi-linear problem in an unbounded domain with a concave angle. By using the principle of the natural boundary reduction, we obtain the natural integral equation on the artificial boundary of circular arc boundary, and get the coupled variational problem and its numerical method. Then the error and convergence of coupling solution are analyzed. Finally, some numerical examples are verified to show the feasibility of our method.
文摘In this paper, we prove some Δ-convergence and strong convergence results for the sequence generated by a new algorithm to a minimizer of two convex functions and a common fixed point for quasi-pseudo-contractive mappings in Hadamard spaces. Our theorems improve and generalize some recent results in the literature.
基金supported in part by the National Natural Science Foundation of China(1150140311461023)the Shanxi Province Science Foundation for Youths under grant 2013021001-3
文摘In this paper,we study the following generalized quasilinear Schrdinger equations with critical or supercritical growths-div(g^2(u)▽u) + g(u)g′(u)|▽u|~2+ V(x)u = f(x,u) + λ|u|^(p-2)u,x∈R^N,where λ>0,N≥3,g:R → R^+ is a C^1 even function,g(0) = 1,g′(s) ≥ 0 for all s ≥ 0,lim_(|s|→+∞)g(s)/|s|^(α-1):= β > 0 for some α≥ 1 and(α-1)g(s) > g′(s)s for all s > 0 and p ≥α2*.Under some suitable conditions,we prove that the equation has a nontrivial solution for smallλ > 0 using a change of variables and variational method.
基金Supported by the NNSF of China(10901003) Supported by the Natural Science Foundation from the Education Bureau of Anhui Province(KJ2011A135)
文摘The existence and asymptotic behavior of solution for a class of quasilinear singularly perturbed boundary value problems are discussed under suitable conditions by the theory of differential inequalities and matching principle.
文摘This paper obtain C1,α- regularity of weak solutions to the p-Laplacian obstacle problem with C1,β-obstacle function under controllable growth conditions.
基金Natural Science Foundation of Henan Province!(Grant No.98405070) National Natural Science Foundation of China (Grant No.19
文摘In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the monotonicity of σi(s) (i= 1,…, n). The nonexistence of global solutions to the initial-boundary value problem of the equation is also discussed, a blowup theorem is proved and a concrete example is given.
文摘This paper considers the quasilinear elliptic equation where , and 0 < m < p-1 < q < +∞, Ω is a bounded domain in RN(N 3).λ is a positive number. Object is to estimate exactly the magnitute of λ* such that (1)λ has at least one positive solution if λ ∈ (0, λ*) and no positive solutions if λ > λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ ∈ (0, λ*) and . Finally, the authors obtain a multiplicity result with positive energy of (1)λ when 0 < m < p - 1 < q = (Np)/(N-p) - 1.
基金supported in part by the National Natural Science Foundation of China(11261070)
文摘We study a quasilinear elliptic equation with polynomial growth coefficients. The existence of infinitely many solutions is obtained by a dual method and a nonsmooth critical point theory.
文摘Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physically interesting cases are found by using the partial Lagrangian approach.
文摘In this paper, a quasilinear elliptic system is investigated, which involves concave-convex nonlinearities and nonlinear boundary condition. By Nehari manifold, fibering method and analytic techniques, the existence of multiple nontrivial nonnegative solutions to this equation is verified.