74 results
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2. Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators.
- Author
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Della Pietra, Francesco and Piscitelli, Gianpaolo
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ELLIPTIC operators , *NONLINEAR operators , *EIGENVALUES , *CONVEX domains , *NONLINEAR equations , *ELLIPTIC equations - Abstract
The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p -Laplace operator, namely: λ 1 (β , Ω) = min ψ ∈ W 1 , p (Ω) ∖ { 0 } ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N − 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, anisotropic mean convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ 1 in the case β > 0 , and a upper bound in the case β < 0. As a consequence, we prove, for β > 0 , a lower bound for λ 1 (β , Ω) in terms of the anisotropic inradius of Ω and, for β < 0 , an upper bound of λ 1 (β , Ω) in terms of β. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. On an Iterative Method of Solving Direct and Inverse Problems for Parabolic Equations.
- Author
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Boykov, I. V. and Ryazantsev, V. A.
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NUMERICAL integration , *NONLINEAR equations , *NONLINEAR operators , *EQUATIONS , *OPERATOR equations , *INVERSE problems - Abstract
This paper is devoted to approximate methods of solving direct and inverse problems for parabolic equations. An approximate method to solve the initial problem of a multidimensional nonlinear parabolic equation has been proposed. It is based on reducing the initial problem to a nonlinear multidimensional Fredholm intergral equation of the second kind, which is approximated by a system of nonlinear algebraic equatiions using a method of mechanical quadratures. In constructing a computational scheme, the points of local splines have been applied for optimal with respect to order approximation of a functional class that contains the solutions of parabolic equations. For the numerical implementation of the computational scheme, we have used the generalization of a continuous method of solving nonlinear operator equations that is described in the paper. In addition, the inverse problem of a parabolic equation with a fractional order derivative with respect to a time variable has been studied. Approximate methods of determining the fractional order of the time derivative and a coefficient at a spatial derivative have been proposed. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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4. The extreme solutions for a σ‐Hessian equation with a nonlinear operator.
- Author
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Zhang, Xinguang, Chen, Peng, Wu, Yonghong, and Wiwatanapataphee, Benchawan
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NONLINEAR operators , *OPERATOR equations , *NONLINEAR equations - Abstract
The aim of this paper is to study the existence of extreme solutions and their properties for a general σ$$ \sigma $$‐Hessian equation involving a nonlinear operator. By introducing a suitable growth condition and developing a iterative technique, some new results on existence and asymptotic estimates of minimum and maximum solutions are derived. Moreover, we also establish the iterative sequences that converge uniformly to the extreme solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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5. Applications of Quadratic Stochastic Operators to Nonlinear Consensus Problems.
- Author
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Saburov, M. and Saburov, Kh.
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NONLINEAR equations , *MULTIAGENT systems , *AUTOMATIC control systems , *NONLINEAR operators - Abstract
Historically, an idea of reaching consensus through repeated averaging was introduced by DeGroot for a structured time-invariant and synchronous environment. Since that time, the consensus which is the most ubiquitous phenomenon of multiagent systems becomes popular in the various scientific fields such as biology, physics, control engineering, and social science. In this paper, we overview the recent development of applications of quadratic stochastic operators on nonlinear consensus problems. We also present some refinement and improvement of the previous results. [ABSTRACT FROM AUTHOR]
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- 2024
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6. Properties of the free boundaries for the obstacle problem of the porous medium equations.
- Author
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Kim, Sunghoon, Lee, Ki-Ahm, and Park, Jinwan
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POROUS materials , *NONLINEAR operators , *EQUATIONS , *NONLINEAR equations - Abstract
In this paper, we study the existence and interior W 2 , p -regularity of the solution, and the regularity of the free boundary ∂ { u > ϕ } to the obstacle problem of the porous medium equation, u t = Δ u m ( m > 1 ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ { u > ϕ } and ∂ { u > 0 } , we consider two cases on the initial data which make the free boundary ∂ { u > ϕ } separate from the free boundary ∂ { u > 0 } . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 -regularity of the free boundary ∂ { u > ϕ } is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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7. Data-driven moving horizon state estimation of nonlinear processes using Koopman operator.
- Author
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Yin, Xunyuan, Qin, Yan, Liu, Jinfeng, and Huang, Biao
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NONLINEAR estimation , *COORDINATE transformations , *NONLINEAR operators , *KALMAN filtering , *NONLINEAR equations - Abstract
In this paper, a data-driven constrained state estimation method is proposed for nonlinear processes. Within the Koopman operator framework, we propose a data-driven model identification procedure for state estimation based on the algorithm of extended dynamic mode decomposition, which seeks an optimal approximation of the Koopman operator for a nonlinear process in a higher-dimensional space that correlates with the original process state-space via a prescribed nonlinear coordinate transformation. By implementing the proposed procedure, a linear state-space model can be established based on historic process data to describe the dynamics of a nonlinear process and the nonlinear dependence of the sensor measurements on process states. Based on the identified Koopman operator, a linear moving horizon estimation (MHE) algorithm that explicitly addresses constraints on the original process states is formulated to efficiently estimate the states in the higher-dimensional space. The states of the treated nonlinear process are recovered based on the state estimates provided by the MHE estimator designed in the higher-dimensional space. Two process examples are utilized to demonstrate the effectiveness and superiority of the proposed framework. • We propose a Koopman-based data-driven modeling method for general nonlinear processes for the state estimation purpose, which can provide infinite- step-ahead state predictions. • We develop a linear moving horizon estimation scheme to handle constrained nonlinear state estimation problems for general nonlinear processes. • We present two case studies, including an experimental study on a water-tank process consisting of four interconnected water tanks. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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8. A new sufficiently descent algorithm for pseudomonotone nonlinear operator equations and signal reconstruction.
- Author
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Awwal, Aliyu Muhammed and Botmart, Thongchai
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OPERATOR equations , *SIGNAL reconstruction , *NONLINEAR equations , *LIPSCHITZ continuity , *NONLINEAR operators , *ALGORITHMS - Abstract
This paper presents a new sufficiently descent algorithm for system of nonlinear equations where the underlying operator is pseudomonotone. The conditions imposed on the proposed algorithm to achieve convergence are Lipschitz continuity and pseudomonotonicity which is weaker than monotonicity assumption forced upon many algorithms in this area found in the literature. Numerical experiments on selected test problems taken from the literature validate the efficiency of the new algorithm. Moreover, the new algorithm demonstrates superior performance in comparison with some existing algorithms. Furthermore, the proposed algorithm is applied to reconstruct some disturbed signals. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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9. Solution estimates for one class of elliptic and parabolic nonlinear equations.
- Author
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Kakharman, N. and Otelbaev, M.
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NONLINEAR equations , *ELLIPTIC equations , *NONLINEAR operators , *PARABOLIC operators , *SEPARATION of variables , *ELLIPTIC operators - Abstract
In this paper, we provide an alternative proof of the separation theorem, which can be easily extended to a certain class of systems of elliptic and parabolic nonlinear equations in non-reflexive spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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10. Decay estimates for the time-fractional evolution equations with time-dependent coefficients.
- Author
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Smadiyeva, Asselya G. and Torebek, Berikbol T.
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NONLINEAR evolution equations , *NONLINEAR operators , *NONLINEAR equations , *OPERATOR equations , *POROUS materials , *EVOLUTION equations - Abstract
In this paper, the initial-boundary value problems for the time-fractional degenerate evolution equations are considered. Firstly, in the linear case, we obtain the optimal rates of decay estimates of the solutions. The decay estimates are also established for the time-fractional evolution equations with nonlinear operators such as: p-Laplacian, the porous medium operator, degenerate operator, mean curvature operator and Kirchhoff operator. At the end, some applications of the obtained results are given to derive the decay estimates of global solutions for the time-fractional Fisher-KPP-type equation and the time-fractional porous medium equation with the nonlinear source. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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11. Periodic Homogenization of the Principal Eigenvalue of Second-Order Elliptic Operators.
- Author
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Dávila, Gonzalo, Rodríguez-Paredes, Andrei, and Topp, Erwin
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EIGENVALUES , *NONLINEAR operators , *ASYMPTOTIC homogenization , *ELLIPTIC operators , *NONLINEAR equations , *EIGENFUNCTIONS , *ELLIPTIC equations - Abstract
In this paper we investigate homogenization results for the principal eigenvalue problem associated to 1-homogeneous, uniformly elliptic, second-order operators. Under rather general assumptions, we prove that the principal eigenpair associated to an oscillatory operator converges to the eigenpair associated to the effective one. This includes the case of fully nonlinear operators. Rates of convergence for the eigenvalues are provided for linear and nonlinear problems, under extra regularity/convexity assumptions. Finally, a linear rate of convergence (in terms of the oscillation parameter) of suitably normalized eigenfunctions is obtained for linear problems. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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12. Research on Some Problems for Nonlinear Operators in the Z-Z-B Space.
- Author
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Liu, Yiping and Zhu, Chuanxi
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NONLINEAR operators , *FIXED point theory , *NONLINEAR equations , *OPERATOR equations - Abstract
In this paper, we first propose a new concept of Z-Z-B spaces, which is a generalization of Z-C-X spaces. Meanwhile, the new concept of the superior cone is introduced. Secondly, we study some new problems for semi-closed 1-set-contractive operators in the Z-Z-B space and obtain some new results. These new theorems are proven by combining partial order theory with fixed point index theory. Regarding these theorems, in the latter part of the paper, the proofs are omitted since the methods of proving these theorems are similar. Moreover, two important inequality lemmas are proven. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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13. Approximating Multiple Roots of Applied Mathematical Problems Using Iterative Techniques.
- Author
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Behl, Ramandeep, Arora, Himani, Martínez, Eulalia, and Singh, Tajinder
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NONLINEAR operators , *NONLINEAR equations , *MULTIPLICITY (Mathematics) , *ITERATIVE methods (Mathematics) - Abstract
In this study, we suggest a new iterative family of iterative methods for approximating the roots with multiplicity in nonlinear equations. We found a lack in the approximation of multiple roots in the case that the nonlinear operator be non-differentiable. So, we present, in this paper, iterative methods that do not use the derivative of the non-linear operator in their iterative expression. With our new iterative technique, we find better numerical results of Planck's radiation, Van Der Waals, Beam designing, and Isothermal continuous stirred tank reactor problems. Divided difference and weight function approaches are adopted for the construction of our schemes. The convergence order is studied thoroughly in the Theorems 1 and 2, for the case when multiplicity p ≥ 2 . The obtained numerical results illustrate the preferable outcomes as compared to the existing ones in terms of absolute residual errors, number of iterations, computational order of convergence (COC), and absolute error difference between two consecutive iterations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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14. Liouville-Type Theorem for Nonlinear Elliptic Equations Involving Generalized Greiner Operator.
- Author
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Shi, Wei
- Subjects
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ELLIPTIC equations , *NONLINEAR equations , *NONLINEAR operators - Abstract
In this paper, we study the Liouville-type behaviors of the generalized Greiner operators with nonlinear boundary value conditions. We use the technique based upon the method of moving planes. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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15. On an Approximate Method for Recovering a Function from Its Autocorrelation Function.
- Author
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Boykov, I. V. and Pivkina, A. A.
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NONLINEAR operators , *OPERATOR equations , *SIGNAL reconstruction , *NONLINEAR equations , *OPERATOR functions , *FUNCTIONALS - Abstract
When solving many physical and technical problems, a situation arises when only operators (functionals) from the objects under study (signals, images, etc.) are available for observations (measurements). It is required to restore the object from its known operator (functional). In many cases, the correlation (autocorrelation) function acts as an operator. A large number of papers have been devoted to the study of the existence of a solution to the problem of signal reconstruction from its autocorrelation function and the uniqueness of this solution. Since the solution to the problem of restoring a function from its autocorrelation function is not known in an analytical form, the problem of developing approximate methods arises. This is relevant not only in the problems of signal and image recovery, but also in solving the phase problem. From the above, the relevance of the problem of restoring a function (images) from the autocorrelation function follows. The article is devoted to approximate methods for solving this problem. Materials and methods. The construction and justification of the computing scheme is based on a continuous method for solving nonlinear operator equations, based on the theory of stability of solutions to ordinary differential equation systems. The method is stable under perturbations of the parameters of the mathematical model and, when solving nonlinear operator equations, does not require the reversibility of the Gateaux (or Freshet) derivatives of nonlinear operators. Results. In this work, an approximate method of signal reconstruction from its autocorrelation function and calculation of the phase of its spectrum from the reconstructed signal is constructed and substantiated. Conclusions. An approximate method for reconstructing a signal from its autocorrelation function and calculating the phase of its spectrum from the recovered signal is constructed and substantiated. The method does not require additional information about the signal under study. The results of the work can be used in solving a number of problems in optics, crystallography, and biology. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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16. Hypersingular Integral Equations of Prandtl's Type: Theory, Numerical Methods, and Applications.
- Author
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Boykov, Ilya, Roudnev, Vladimir, and Boykova, Alla
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INTEGRAL equations , *NONLINEAR integral equations , *NONLINEAR operators , *OPERATOR equations , *NONLINEAR equations , *COLLOCATION methods , *SINGULAR integrals - Abstract
In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl's type. We obtained the estimates of the convergence rate and the method error. The constructed computational scheme includes a continuous method for solving nonlinear operator equations, which is stable for perturbations of the coefficients and the right-hand sides of equations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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17. Modified Adomian Method through Efficient Inverse Integral Operators to Solve Nonlinear Initial-Value Problems for Ordinary Differential Equations.
- Author
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AL-Mazmumy, Mariam, Alsulami, Aishah A., Bakodah, Huda O., and Alzaid, Nawal
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INTEGRAL operators , *NONLINEAR operators , *NONLINEAR equations , *DECOMPOSITION method , *ORDINARY differential equations , *ANALYTICAL solutions , *INITIAL value problems - Abstract
The present manuscript examines different forms of Initial-Value Problems (IVPs) featuring various types of Ordinary Differential Equations (ODEs) by proposing a proficient modification to the famous standard Adomian decomposition method (ADM). The present paper collected different forms of inverse integral operators and further successfully demonstrated their applicability on dissimilar nonlinear singular and nonsingular ODEs. Furthermore, we surveyed most cases in this very new method, and it was found to have a fast convergence rate and, on the other hand, have high precision whenever exact analytical solutions are reachable. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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18. EXACT SOLITON SOLUTIONS FOR CONFORMABLE FRACTIONAL SIX WAVE INTERACTION EQUATIONS BY THE ANSATZ METHOD.
- Author
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ALQARALEH, SAHAR M., TALAFHA, ADEEB G., MOMANI, SHAHER, AL-OMARI, SHRIDEH, and AL-SMADI, MOHAMMED
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NONLINEAR equations , *WAVE equation , *PARTIAL differential equations , *NONLINEAR operators , *HYPERBOLIC functions , *NONLINEAR evolution equations , *EVOLUTION equations - Abstract
In this paper, a conformable fractional time derivative of order α ∈ (0 , 1 ] is considered in view of the Lax-pair of nonlinear operators to derive a fractional nonlinear evolution system of partial differential equations, called the Fractional-Six-Wave-Interaction-Equations, which is derived in terms of one temporal plus one and two spatial dimensions. Further, an ansatz consisting of linear combinations of hyperbolic functions with complex coefficients is utilized to obtain an infinite set of exact soliton solutions for this system. Certain numerical examples are introduced to show the effectiveness of the ansatz method in obtaining exact solutions for similar systems of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
19. Properties of a quasi-uniformly monotone operator and its application to the electromagnetic p-curl systems.
- Author
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Song, Chang-Ho, Ri, Yong-Gon, and Sin, Cholmin
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NONLINEAR operators , *OPERATOR equations , *NONLINEAR equations - Abstract
In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation Au = b. We prove that if A is a quasi-uniformly monotone and hemi-continuous operator, then A−1 is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence of Galerkin approximations to the solution of steady-state electromagnetic p-curl systems. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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20. Iterative Methods of Solving Ambartsumian Equations. Part 1.
- Author
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Boykov, I. V. and Shaldaeva, A. A.
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NONLINEAR operators , *NUMERICAL integration , *OPERATOR equations , *NONLINEAR equations , *EQUATIONS , *ITERATIVE methods (Mathematics) - Abstract
Ambartsumian equation and its generalizations are some of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. An analytical solution to this equation is currently unknown, and the development of approximate methods is urgent. To solve the Ambartsumian equation, several iterative methods are proposed that are used in solving practical problems. Methods of collocations and mechanical quadratures have also been constructed and substantiated under rather severe conditions. It is of considerable interest to construct an iterative method adapted to the coefficients and kernels of the equation. This paper is devoted to the construction of such method. The construction of the iterative method is based on a continuous method for solving nonlinear operator equations. The method is based on the Lyapunov stability theory and is stable against perturbation of the initial conditions, coefficients, and kernels of the equations being solved. An additional advantage of the continuous method for solving nonlinear operator equations is that its implementation does not require the reversibility of the Gateaux derivative of the nonlinear operator. An iterative method for solving the Ambartsumian equation is constructed and substantiated. Model examples were solved to illustrate the effectiveness of the method. Equations generalizing the classical Ambartsumian equation are considered. To solve them, computational schemes of collocation and mechanical quadrature methods are constructed, which are implemented by a continuous method for solving nonlinear operator equations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
21. Maximum principles and monotonicity of solutions for fractional p-equations in unbounded domains.
- Author
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Liu, Zhao
- Subjects
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SINGULAR integrals , *NONLINEAR operators , *NONLINEAR equations , *STANDARD & Poor's 500 Index , *MAXIMUM principles (Mathematics) , *LINEAR operators - Abstract
In this paper, we consider the following non-linear equations in unbounded domains Ω with exterior Dirichlet condition: { (− Δ) p s u (x) = f (u (x)) , x ∈ Ω , u (x) > 0 , x ∈ Ω , u (x) = 0 , x ∈ R n ∖ Ω , where (− Δ) p s is the fractional p -Laplacian defined as (0.1) (− Δ) p s u (x) = C n , s , p P. V. ∫ R n | u (x) − u (y) | p − 2 [ u (x) − u (y) ] | x − y | n + s p d y with 0 < s < 1 and p ≥ 2. We first establish a maximum principle in unbounded domains involving the fractional p -Laplacian by estimating the singular integral in (0.1) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p -Laplacians and apply it to derive the monotonicity and uniqueness of solutions. There have been similar results for the classical Laplacian [3] and for the fractional Laplacian [39] , which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p -Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on f (⋅) and on the domain Ω. We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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22. STABILITY OF SOLUTIONS TO NONLINEAR EVOLUTION PROBLEMS.
- Author
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RAMM, ALEXANDER G.
- Subjects
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NONLINEAR equations , *NONLINEAR operators , *LYAPUNOV stability , *LINEAR operators - Abstract
Let u0 = F(u; t); u(0) = u0; (1), u 2 H, H is a Hilbert space, F(u; t) is a nonlinear operator in H. If F(u; t) = A(t)u + B(u; t), where A(t) is a linear operator, B(u; t) = O(kuk2) for kuk ! 0, then problem (1) has a solution u = 0 if u0 = 0. If ku0k is small then the stability problem is: will the solution to (1) exist for all t > 0 and be small for all t > 0, A.M. Lyapunov gave in 1892 sufficient conditions for this to happen. In our paper a new technical tool is given for answering the above question. This tool (a nonlinear inequality) allows one to give old and new results on Lyapunov stability. One of such results is proved in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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23. A dual-mixed approximation for a Huber regularization of generalized p-Stokes viscoplastic flow problems.
- Author
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González-Andrade, Sergio and Méndez, Paul E.
- Subjects
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BINGHAM flow , *NONLINEAR operators , *OPERATOR equations , *NEWTON-Raphson method , *NONLINEAR equations , *DIFFERENTIABLE dynamical systems , *VISCOSITY - Abstract
In this paper, we propose a dual-mixed formulation for stationary viscoplastic flows with yield, such as the Bingham or the Herschel-Bulkley flow. The approach is based on a Huber regularization of the viscosity term and a two-fold saddle point nonlinear operator equation for the resulting weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of slantly differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments in two and three dimensions to investigate the behavior and efficiency of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
24. Existence results for a system of nonlinear operator equations and block operator matrices in locally convex spaces.
- Author
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Bahidi, Fatima, Krichen, Bilel, and Mefteh, Bilel
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NONLINEAR equations , *OPERATOR equations , *NONLINEAR integral equations , *NONLINEAR operators , *MATRICES (Mathematics) , *COMPOSITION operators - Abstract
The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space (X , { | ⋅ | p } p ∈ Λ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 휏-Krein–Šmulian property, where 휏 is a weaker Hausdorff locally convex topology of 푋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 휏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C (R +) × C (R +) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
25. Tykhonov well-posedness of a mixed variational problem.
- Author
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Cai, Dong-ling, Sofonea, Mircea, and Xiao, Yi-bin
- Subjects
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NONLINEAR operators , *LAGRANGE multiplier , *NONLINEAR equations - Abstract
We consider a mixed variational problem governed by a nonlinear operator and a set of constraints. Existence, uniqueness and convergence results for this problem have already been obtained in the literature. In this current paper we complete these results by proving the well-posedness of the problem, in the sense of Tykhonov. To this end we introduce a family of approximating problems for which we state and prove various equivalence and convergence results. We illustrate these abstract results in the study of a frictionless contact model with elastic materials. The process is assumed to be static and the contact is with unilateral constraints. We derive a weak formulation of the model which is in the form of a mixed variational problem with unknowns being the displacement field and the Lagrange multiplier. Then, we prove various results on the corresponding mixed problem, including its well-posedness in the sense of Tykhonov, under various assumptions on the data. Finally, we provide mechanical interpretation of our results. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
26. Controllability for abstract semilinear control systems with homogeneous properties.
- Author
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Jeong, Jin-Mun, Son, Sang-Jin, and Park, Ah-Ran
- Subjects
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NONLINEAR operators , *SET-valued maps , *OPERATOR equations , *NONLINEAR equations , *SURJECTIONS , *CARLEMAN theorem - Abstract
This paper considers the approximate controllability for a class of abstract semilinear control systems using a method called surjectivity theorems for nonlinear operator equations under restrictive assumptions. The sufficient conditions for approximate controllability are derived under the natural assumptions on the nonlinear terms, which is an odd homogeneous and the nonlinear inverse considered as a multivalued mapping is bounded. An example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
27. Nonlinear eigenvalue problems for nonhomogeneous Leray–Lions operators.
- Author
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Abdelwahed, Mohamed and Chorfi, Nejmeddine
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NONLINEAR equations , *CRITICAL point theory , *MATHEMATICAL analysis , *VARIATIONAL principles , *NONLINEAR analysis , *NONLINEAR operators - Abstract
This paper deals with the mathematical analysis of a class of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator. We are concerned both with the coercive and the noncoercive (and nonresonant) cases, which are in relationship with two associated Rayleigh quotients. The proof combines critical point theory arguments and the dual variational principle. The arguments developed in this paper can be extended to other classes of nonlinear eigenvalue problems with nonstandard growth. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
28. Existence results for a class of nonlinear singular transport equations in bounded spatial domains.
- Author
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Latrach, Khalid, Oummi, Hssaine, and Zeghal, Ahmed
- Subjects
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TRANSPORT equation , *NONLINEAR operators , *NONLINEAR equations , *CASE studies - Abstract
In this paper, we prove the existence of solutions to a nonlinear singular transport equation (ie, transport equation with unbounded collision frequency and unbounded collision operator) with vacuum boundary conditions in bounded spatial domain on Lp‐spaces with 1 ≤ p<+∞. This problem was already considered in6,8,9 under the hypothesis that the collision frequency σ(·) and the collision operator are bounded. In this work, we show that these hypotheses are not necessary; it suffices to assume that σ(·) is locally bounded, and the collision operator is bounded between Xpσ (a weighted space) and Xp (cf Section 2). Although the analysis for p∈(1,+∞) is standard in the sense that it uses the Schauder fixed point theorem, the compactness of the involved operator is not easy to derive. However, the analysis in the case p=1 uses the concept of Dunford‐Pettis operators and a new version of the Darbo fixed point theorem for a measure of weak noncompactness introduced in the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
29. The localized method of fundamental solutions for 2D and 3D second-order nonlinear boundary value problems.
- Author
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Zhao, Shengdong, Gu, Yan, Fan, Chia-Ming, and Wang, Xiao
- Subjects
- *
NONLINEAR boundary value problems , *NEWTON-Raphson method , *NONLINEAR equations , *NONLINEAR analysis , *NONLINEAR operators , *RADIAL basis functions , *SPARSE matrices - Abstract
In this paper, a new framework for the numerical solutions of general nonlinear problems is presented. By employing the analog equation method, the actual problem governed by a nonlinear differential operator is converted into an equivalent problem described by a simple linear equation with unknown fictitious body forces. The solution of the substitute problem is then obtained by using the localized method of fundamental solutions, where the fictitious nonhomogeneous term is approximated using the dual reciprocity method using the radial basis functions. The main difference between the classical and the present localized method of fundamental solutions is that the latter produces sparse and banded stiffness matrix which makes the method very suitable for large-scale nonlinear simulations, since sparse matrices are much cheaper to inverse at each iterative step of the Newton's method. The present method is simple in derivation, efficient in calculation, and may be viewed as a completive alternative for nonlinear analysis, especially for large-scale problems with complex-shape geometries. Preliminary numerical experiments involving second-order nonlinear boundary value problems in both two- and three-dimensions are presented to demonstrate the accuracy and efficiency of the present method. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
30. Fixed Point, Data Dependence, and Well-Posed Problems for Multivalued Nonlinear Contractions.
- Author
-
Iqbal, Iram, Hussain, Nawab, Al-Sulami, Hamed H., and Hassan, Shanza
- Subjects
- *
NONLINEAR equations , *NONLINEAR operators , *METRIC spaces , *CONTRACTIONS (Topology) - Abstract
The aim of the paper is to discuss data dependence, existence of fixed points, strict fixed points, and well posedness of some multivalued generalized contractions in the setting of complete metric spaces. Using auxiliary functions, we introduce Wardowski type multivalued nonlinear operators that satisfy a novel class of contractive requirements. Furthermore, the existence and data dependence findings for these multivalued operators are obtained. A nontrivial example is also provided to support the results. The results generalize, improve, and extend existing results in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
31. FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations.
- Author
-
Abubakar, Auwal Bala, Muangchoo, Kanikar, Ibrahim, Abdulkarim Hassan, Abubakar, Jamilu, and Rano, Sadiya Ali
- Subjects
- *
OPERATOR equations , *NONLINEAR operators , *ALGORITHMS , *MONOTONE operators , *CONJUGATE gradient methods , *NONLINEAR equations - Abstract
This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
32. ANALYTICAL METHODS FOR NON-LINEAR FRACTIONAL KOLMOGOROV-PETROVSKII-PISKUNOV EQUATION Soliton Solution and Operator Solution.
- Author
-
XU, Bo, ZHANG, Yufeng, and ZHANG, Sheng
- Subjects
- *
NONLINEAR equations , *NONLINEAR operators , *EQUATIONS - Abstract
Kolmogorov-Petrovskii-Piskunov equation can be regarded as a generalized form of the Fitzhugh-Nagumo, Fisher and Huxley equations which have many applications in physics, chemistry and biology. In this paper, two fractional extended versions of the non-linear Kolmogorov-Petrovskii-Piskunov equation are solved by analytical methods. Firstly, a new and more general fractional derivative is defined and some properties of it are given. Secondly, a solution in the form of operator representation of the non-linear Kolmogorov-Petrovskii- Piskunov equation with the defined fractional derivative is obtained. Finally, some exact solutions including kink-soliton solution and other solutions of the non-linear Kolmogorov-Petrovskii-Piskunov equation with Khalil et al.'s fractional derivative and variable coefficients are obtained. It is shown that the fractional- order affects the propagation velocity of the obtained kink-soliton solution. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
33. Extrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations.
- Author
-
Meinlschmidt, Hannes and Rehberg, Joachim
- Subjects
- *
EQUATIONS , *SEMICONDUCTORS , *DIFFERENTIAL operators , *NONLINEAR equations , *PARABOLIC operators , *NONLINEAR analysis , *NONLINEAR operators - Abstract
In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order X D s − 1 , q (Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
34. A dynamical system approach to a class of radial weighted fully nonlinear equations.
- Author
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Maia, Liliane, Nornberg, Gabrielle, and Pacella, Filomena
- Subjects
- *
NONLINEAR equations , *DYNAMICAL systems , *EXTREMAL problems (Mathematics) , *QUADRATIC differentials , *CRITICAL exponents , *ENERGY consumption , *NONLINEAR operators - Abstract
In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator M −. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
35. Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold.
- Author
-
Gasiński, Leszek and Winkert, Patrick
- Subjects
- *
NONLINEAR equations , *ELLIPTIC equations , *NONLINEAR operators - Abstract
In this paper we study quasilinear elliptic equations driven by the so-called double phase operator and with a nonlinear boundary condition. Due to the lack of regularity, we prove the existence of multiple solutions by applying the Nehari manifold method along with truncation and comparison techniques and critical point theory. In addition, we can also determine the sign of the solutions (one positive, one negative, one nodal). Moreover, as a result of independent interest, we prove for a general class of such problems the boundedness of weak solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
36. On the strong maximum principle.
- Author
-
Mohammed, Ahmed and Vitolo, Antonio
- Subjects
- *
MAXIMUM principles (Mathematics) , *NONLINEAR operators , *ELLIPTIC operators , *NONLINEAR equations , *ELLIPTIC equations , *VISCOSITY solutions , *EQUATIONS - Abstract
In this paper we study the strong maximum principle for equations of the form F [ u ] = H (u , | D u |) where F is either a fully nonlinear elliptic operator or is the p-Laplace operator. We give sufficient conditions on H to ensure that the strong maximum principle (SMP) holds. The condition is also necessary for SMP to hold for the the equation F [ u ] = g (| D u |). [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
37. Solutions of Nonlinear Operator Equations by Viscosity Iterative Methods.
- Author
-
Aibinu, Mathew, Thakur, Surendra, and Moyo, Sibusiso
- Subjects
- *
OPERATOR equations , *NONLINEAR equations , *INTEGRAL equations , *FREDHOLM equations , *ALGORITHMS , *NONLINEAR operators , *ITERATIVE methods (Mathematics) - Abstract
Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a fixed point of a nonexpansive mapping in Banach spaces. Our technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The scheme is effective for obtaining the solutions of various nonlinear operator equations as it involves the generalized contraction. The results are applied to obtain a fixed point of λ -strictly pseudocontractive mappings, solution of α -inverse-strongly monotone mappings, and solution of integral equations of Fredholm type. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
38. A revisit on Landweber iteration.
- Author
-
Real, Rommel and Jin, Qinian
- Subjects
- *
INVERSE problems , *BANACH spaces , *NONLINEAR operators , *NONLINEAR equations , *DISCREPANCY theorem , *REFLEXIVITY - Abstract
In this paper we revisit the discrepancy principle for Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces and prove a new convergence result which requires neither the Gâteaux differentiability of the forward operator nor the reflexivity of the image space. Therefore, we expand the applied range of the discrepancy principle for Landweber iteration to cover non-smooth ill-posed inverse problems and to handle the situation that the data is contaminated by various types of noise. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
39. Langevin equation in terms of conformable differential operators.
- Author
-
AHMAD, Bashir, AGARWAL, Ravi P., ALGHANMI, Madeaha, and ALSAEDI, Ahmed
- Subjects
- *
DIFFERENTIAL operators , *LANGEVIN equations , *NONLINEAR equations , *FUNCTIONAL analysis , *NONLINEAR operators - Abstract
In this paper, we establish sufficient criteria for the existence of solutions for a new kind of nonlinear Langevin equation involving conformable differential operators of different orders and equipped with integral boundary conditions. We apply the modern tools of functional analysis to derive the desired results for the problem at hand. Examples are constructed for the illustration of the obtained results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
40. Fixed-Point Theorems for Multivalued Operator Matrix Under Weak Topology with an Application.
- Author
-
Jeribi, Aref, Kaddachi, Najib, and Krichen, Bilel
- Subjects
- *
SET-valued maps , *TOPOLOGY , *BANACH spaces , *NONLINEAR equations , *NONLINEAR operators - Abstract
In the present paper, we establish some fixed-point theorems for a 2 × 2 block operator matrix involving multivalued maps acting on Banach spaces. These results are formulated in terms of weak sequential continuity and the technique of measures of weak noncompactness. The results obtained are then applied to a coupled system of nonlinear equations. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
41. Application of a new accelerated algorithm to regression problems.
- Author
-
Dixit, Avinash, Sahu, D. R., Singh, Amit Kumar, and Som, T.
- Subjects
- *
NONLINEAR operators , *HILBERT space , *ALGORITHMS , *NONLINEAR equations , *NONEXPANSIVE mappings - Abstract
Many iterative algorithms like Picard, Mann, Ishikawa are very useful to solve fixed point problems of nonlinear operators in real Hilbert spaces. The recent trend is to enhance their convergence rate abruptly by using inertial terms. The purpose of this paper is to investigate a new inertial iterative algorithm for finding the fixed points of nonexpansive operators in the framework of Hilbert spaces. We study the weak convergence of the proposed algorithm under mild assumptions. We apply our algorithm to design a new accelerated proximal gradient method. This new proximal gradient technique is applied to regression problems. Numerical experiments have been conducted for regression problems with several publicly available high-dimensional datasets and compare the proposed algorithm with already existing algorithms on the basis of their performance for accuracy and objective function values. Results show that the performance of our proposed algorithm overreaches the other algorithms, while keeping the iteration parameters unchanged. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
42. Inverse problems for nonlinear Navier–Stokes–Voigt system with memory.
- Author
-
Khompysh, Kh., Shakir, A.G., and Kabidoldanova, A.A.
- Subjects
- *
NONLINEAR equations , *NONLINEAR systems , *NON-Newtonian fluids , *VISCOELASTIC materials , *NONLINEAR operators , *INVERSE problems - Abstract
This paper deals with the unique solvability of some inverse problems for nonlinear Navier–Stokes–Voigt (Kelvin–Voigt) system with memory that governs the flow of incompressible viscoelastic non-Newtonian fluids. The inverse problems that study here, consist of determining a time dependent intensity of the density of external forces, along with a velocity and a pressure of fluids. As an additional information, two types of integral overdetermination conditions over space domain are considered. The system supplemented also with an initial and one of the boundary conditions: stick and slip boundary conditions. For all inverse problems, under suitable assumptions on the data, the global and local in time existence and uniqueness of weak and strong solutions were established. • The inverse problems are equivalent to the direct problems for a nonlinear parabolic equation with nonlinear nonlocal operator of the function u. • The inverse problems have unique weak and strong solutions in local time. • The following inverse problems have unique weak and strong solutions in global time in particular case. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
43. Nonlinear elliptic problems in weighted variable exponent Sobolev spaces by topological degree.
- Author
-
Hammou, Mustapha Ait and Azroul, El Houssine
- Subjects
- *
SOBOLEV spaces , *TOPOLOGICAL spaces , *NONLINEAR equations , *TOPOLOGICAL degree , *NONLINEAR operators , *EXPONENTS - Abstract
In this paper, we prove the existence of solutions for the nonlinear p(.) degenerate problems involving nonlinear operators of the form - div a(x, ∇u) = f(x, u, ∇u) where a and fare Caratheodoryfunctions satisfying some nonstandard growth conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
44. The Convergence Estimation of the Parallel Algorithm of the Linear Cauchy Problem Solution for Large Systems of First-Order Ordinary Differential Equations Using the Solution as Expansion over Orthogonal Polynomials.
- Author
-
Moryakov, A. V.
- Subjects
- *
ORTHOGONAL polynomials , *CAUCHY problem , *NONLINEAR operators , *ORDINARY differential equations , *NONLINEAR equations , *PARALLEL algorithms - Abstract
This paper is devoted to the algorithm of the linear Cauchy problem solution for large systems of first-order ordinary differential equations using parallel calculations. The proof of the convergence of the iteration process using the solution as expansion over orthogonal polynomials for the interval [0,1] is presented. The features of this algorithm are its simplicity, the opportunity to get a solution by parallel calculations, and also the possibility to get a solution for nonlinear problems by changing the operator using the solution from the iteration process. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
45. An asymptotic treatment for non-convex fully nonlinear elliptic equations: Global Sobolev and BMO type estimates.
- Author
-
da Silva, J. V. and Ricarte, G. C.
- Subjects
- *
ELLIPTIC equations , *NONLINEAR equations , *NONLINEAR operators , *ELLIPTIC operators , *GEOMETRIC analysis , *ESTIMATES , *GEOMETRIC approach - Abstract
In this paper, we establish global Sobolev a priori estimates for L p -viscosity solutions of fully nonlinear elliptic equations as follows: F (D 2 u , D u , u , x) = f (x) in Ω u (x) = φ (x) on ∂ Ω by considering minimal integrability condition on the data, i.e. f ∈ L p (Ω) , φ ∈ W 2 , p (Ω) for n < p < ∞ and a regular domain Ω ⊂ ℝ n , and relaxed structural assumptions (weaker than convexity) on the governing operator. Our approach makes use of techniques from geometric tangential analysis, which consists in transporting "fine" regularity estimates from a limiting operator, the Recession profile, associated to F to the original operator via compactness methods. We devote special attention to the borderline case, i.e. when f ∈ p − BMO ⊋ L ∞ . In such a scenery, we show that solutions admit BMO type estimates for their second derivatives. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
46. An improved verification algorithm for nonlinear systems of equations based on Krawczyk operator.
- Author
-
Hou, Guoliang and Zhang, Shugong
- Subjects
- *
NONLINEAR operators , *NONLINEAR equations , *DIFFERENTIAL inclusions , *ALGORITHMS - Abstract
In this paper an improved version of a verification algorithm for solving nonlinear systems of equations based on Krawczyk operator is presented. Compared with the original algorithm, the improved verification algorithm not only saves computing time, but also computes a narrower (or at least the same) inclusion of the solution to nonlinear systems of equations for certain classes of problems. Numerical results demonstrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
47. On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators.
- Author
-
Della Pietra, Francesco, Gavitone, Nunzia, and Piscitelli, Gianpaolo
- Subjects
- *
NONLINEAR operators , *ELLIPTIC operators , *MEASURE theory , *NONLINEAR equations - Abstract
Let Ω be a bounded open set of R n , n ≥ 2. In this paper we mainly study some properties of the second Dirichlet eigenvalue λ 2 (p , Ω) of the anisotropic p -Laplacian − Q p u : = − div (F p − 1 (∇ u) F ξ (∇ u)) , where F is a suitable smooth norm of R n and p ∈ ] 1 , + ∞ [. We provide a lower bound of λ 2 (p , Ω) among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as p → + ∞. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
48. Two-grid economical algorithms for parabolic integro-differential equations with nonlinear memory.
- Author
-
Wang, Wansheng and Hong, Qingguo
- Subjects
- *
INTEGRO-differential equations , *NONLINEAR equations , *PARABOLIC differential equations , *NONLINEAR operators , *ALGORITHMS , *MEMORY - Abstract
Abstract In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is shown that these algorithms are as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size H and the fine grid size h satisfy H = O (h r − 1 r ). Especially for PIDEs with nonlinear memory defined by a lower order nonlinear operator, our two-grid algorithm can save significant storage and computing time. Numerical experiments are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
49. The regularity theory for the double obstacle problem for fully nonlinear operator.
- Author
-
Lee, Ki-Ahm and Park, Jinwan
- Subjects
- *
NONLINEAR operators , *NONLINEAR equations - Abstract
In this paper, we prove the existence and uniqueness of W 2 , p (n < p < ∞) solutions of a double obstacle problem. Moreover, we show the optimal regularity of the solution and the local C 1 regularity of the free boundary under a thickness assumption at the free boundary point on the intersection of two free boundaries. In the study of the regularity of the free boundary, we deal with a general problem, the no-sign reduced double obstacle problem with an upper obstacle ψ , F (D 2 u , x) = f χ Ω (u) ∩ { u < ψ } + F (D 2 ψ , x) χ Ω (u) ∩ { u = ψ } , u ≤ ψ in B 1 , where Ω (u) = B 1 ∖ { u = 0 } ∩ { ∇ u = 0 } . [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
50. Dual reciprocity hybrid boundary node method for nonlinear problems.
- Author
-
Yan, Fei, Jiang, Quan, Bai, Guo-Feng, Li, Shao-Jun, Li, Yun, and Qiao, Zhi-Bin
- Subjects
- *
NONLINEAR equations , *RADIAL basis functions , *POISSON'S equation , *NONLINEAR operators , *LAPLACIAN operator - Abstract
In this paper, a boundary type meshless method of dual reciprocity hybrid boundary node method (DHBNM) is proposed to solve complicate Poisson type linear and nonlinear problems. Firstly, the solutions are divided into the complementary solutions related to homogeneous equation and the particular solutions solved by nonhomogeneous terms, for the latter, they are approximated by the radial basis function interpolation based on dual reciprocity method, and the complementary solutions are obtained based on simple Poisson's equation by hybrid boundary node method, by which a simple fundamental solution of the Laplacian operator is employed instead of some other complicated ones; then a function of field functions and their derivatives on any point can be easily obtained, employing the concept of the analog equation of Katsikadelis, the field functions and their derivatives can be expressed as the function of unknown series of coefficients, and a series of nonlinear equivalent equations can be established by collocating the original governing equation at discrete points in the interior and on boundary of the domain. As a result, a new meshless method of dual reciprocity hybrid boundary node method is proposed to solve nonlinear Poisson type problems, because of the usage of those techniques, the boundary type meshless properties can be kept for any type of nonlinear equations. Different types of classical nonlinear problems are presented to validate the effectiveness and the accuracy of the present method. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
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