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2. Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities.
- Author
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Cingolani, Silvia, Gallo, Marco, and Tanaka, Kazunaga
- Subjects
NONLINEAR equations ,EQUATIONS ,SCALAR field theory ,MOUNTAIN pass theorem - Abstract
In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (− Δ) s u + μ u = ( I α * F (u)) F ′ (u) in R N , (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, "Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities," Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, "Nonlinear scalar field equations II: existence of infinitely many solutions," Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C
1 -regularity. [ABSTRACT FROM AUTHOR]- Published
- 2024
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3. Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons' conjecture.
- Author
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Guo, Yahong, Ma, Lingwei, and Zhang, Zhenqiu
- Subjects
PARABOLIC operators ,ELLIPTIC operators ,LOGICAL prediction ,EQUATIONS ,NONLINEAR equations ,PROBLEM solving - Abstract
In this paper, we consider the general dual fractional parabolic problem ∂ t α u (x , t) + L u (x , t) = f (t , u (x , t)) in R n × R. We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons' conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L and the Marchaud time derivative ∂ t α , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons' conjecture. It is worth noting that our results are novel even for a special case of L , the fractional Laplacian (−Δ)
s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators. [ABSTRACT FROM AUTHOR]- Published
- 2024
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4. Scattering of an inhomogeneous coupled Schrödinger system in the conformal space.
- Author
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Saanouni, Tarek and Peng, Congming
- Subjects
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CONFORMAL mapping , *NONLINEAR systems , *NONLINEAR equations , *MATHEMATICS , *EXPONENTS - Abstract
This paper studies the inhomogeneous defocusing coupled Schrödinger system i u ˙ j + Δ u j = | x | - ρ (∑ 1 ≤ k ≤ m a j k | u k | p ) | u j | p - 2 u j , ρ > 0 , j ∈ [ 1 , m ] . The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 (ℝ N) such that x f ∈ L 2 (ℝ N) . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x u 0 ∈ L 2 enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 , where p 0 is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets | t | α ∥ u (t) ∥ L r (ℝ N) ≲ 1 for some α > 0 and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ u 0 . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 . This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that p ≥ 2 , which restricts the space dimensions to N ≤ 3 . In a paper in progress, the authors treat the same problem in the complementary case ρ < 0 . [ABSTRACT FROM AUTHOR]
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- 2024
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5. Global sharp gradient estimates for a nonlinear parabolic equation on Riemannian manifolds.
- Author
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Chuan, Le Huy, Dung, Nguyen Thac, and Manh, Nguyen Tien
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RIEMANNIAN manifolds , *HEAT equation , *NONLINEAR equations , *MANURES , *MATHEMATICS - Abstract
In this paper, we employ the techniques in [C. Cavaterra, S. Dipierro, Z. Gao and E. Valdinoci, Global gradient estimates for a general type of nonlinear parabolic equations, J. Geom. Anal. 32 2022, 2, Paper No. 65] and the approach in [H. T. Dung and N. T. Dung, Sharp gradient estimates for a heat equation in Riemannian manifolds, Proc. Amer. Math. Soc. 147 2019, 12, 5329–5338] to derive sharp gradient estimates for a positive solution to the heat equation u t = Δ u + a u log u in a complete noncompact Riemannian manifold (where a is a real constant). This is an extension of the gradient estimates of Dung and Dung. [ABSTRACT FROM AUTHOR]
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- 2024
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6. Chirped optical solitons for the complex Ginzburg–Landau equation with Hamiltonian perturbations and Kerr law nonlinearity.
- Author
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Tang, Ming-Yue and Meng, Tong-Yu
- Subjects
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OPTICAL solitons , *PERTURBATION theory , *NONLINEAR equations - Abstract
What the motivation of this paper is to provide chirped optical solitons for the complex Ginzburg–Landau equation with Hamiltonian perturbations and Kerr law nonlinearity. We get 19 exact chirped solutions by utilizing trial equation method and the complete discriminant system for polynomial method, which are richer than the solutions acquired in existing papers. We draw the two-dimensional graphs of amplitudes and corresponding chirps in order to verify the existence of the solutions and discuss the dynamical properties of the solutions. To our knowledge, this is the first time that comprehensive set of exact chirped solutions of the governing equation in the paper are obtained. The model and the results obtained in this paper may help explain some nonlinear problems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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7. Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system.
- Author
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Ortegón Gallego, Francisco, Rhoudaf, Mohamed, and Talbi, Hajar
- Subjects
NONLINEAR equations ,ELLIPTIC operators ,ELLIPTIC equations ,THERMISTORS ,EQUATIONS - Abstract
In this paper, we analyze the following nonlinear elliptic problem A (u) = ρ (u) | ∇ φ | 2 in Ω , div (ρ (u) ∇ φ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω. where A(u) = −div a(x, u, ∇u) is a Leray-Lions operator of order p. The second member of the first equation is only in L
1 (Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique. [ABSTRACT FROM AUTHOR]- Published
- 2024
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8. Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces.
- Author
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Taheri, Ali and Vahidifar, Vahideh
- Subjects
METRIC spaces ,NONLINEAR equations ,CURVATURE ,HAMILTON-Jacobi equations ,DIFFERENTIAL inequalities - Abstract
In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. On the standing wave in coupled fractional Klein–Gordon equation.
- Author
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Guo, Zhenyu and Zhang, Xin
- Subjects
STANDING waves ,KLEIN-Gordon equation ,POTENTIAL well ,NONLINEAR equations - Abstract
The aim of this paper is to deal with the standing wave problems in coupled nonlinear fractional Klein–Gordon equations. First, we establish the constrained minimizations for a single nonlinear fractional Laplace equation. Then we prove the existence of a standing wave with a ground state using a variational argument. Next, applying the potential well argument and the concavity method, we obtain the sharp criterion for blowing up and global existence. Finally, we show the instability of the standing wave. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. 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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11. Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity.
- Author
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Shi, Shaoguang, Zhai, Zhichun, and Zhang, Lei
- Subjects
NONLINEAR equations ,VISCOSITY solutions ,HAMILTON-Jacobi equations - Abstract
In this paper, when studying the connection between the fractional convexity and the fractional p-Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u (x) is the viscosity sub-solution of the equation if and only if u (x) is so-called (α , p) -convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an (α , p) -convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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12. A new clique polynomial approach for fractional partial differential equations.
- Author
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Adel, Waleed and Srinivasa, Kumbinarasaiah
- Subjects
PARTIAL differential equations ,FRACTIONAL differential equations ,ALGEBRAIC equations ,GRAPH theory ,NONLINEAR equations ,CHEBYSHEV polynomials - Abstract
This paper generates a novel approach called the clique polynomial method (CPM) using the clique polynomials raised in graph theory and used for solving the fractional order PDE. The fractional derivative is defined in terms of the Caputo fractional sense and the fractional partial differential equations (FPDE) are converted into nonlinear algebraic equations and collocated with suitable grid points in the current approach. The convergence analysis for the proposed scheme is constructed and the technique proved to be uniformly convegant. We applied the method for solving four problems to justify the proposed technique. Tables and graphs reveal that this new approach yield better results. Some theorems are discussed with proof. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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13. A study on solvability of the fourth-order nonlinear boundary value problems.
- Author
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Gou, Haide
- Subjects
NONLINEAR boundary value problems ,BOUNDARY value problems ,NONLINEAR equations ,MAXIMUM principles (Mathematics) ,LINEAR equations - Abstract
The purpose of the paper is devoted to proving the solvability of the fourth order boundary value problem. Firstly, we build a maximum principle for the corresponding linear equation, by the use of this maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to solve the nonlinear equation, secondly, the existence and uniqueness results for the problem is obtained. In addition, an example is presented to show the application of our main results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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14. Simplified Levenberg–Marquardt Method in Hilbert Spaces.
- Author
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Mahale, Pallavi and Shaikh, Farheen M.
- Subjects
HILBERT space ,NONLINEAR operators ,COMMERCIAL space ventures ,OPERATOR equations ,NONLINEAR equations - Abstract
In 2010, Qinian Jin considered a regularized Levenberg–Marquardt method in Hilbert spaces for getting stable approximate solution for nonlinear ill-posed operator equation F (x) = y , where F : D (F) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y and obtained rate of convergence results under an appropriate source condition. In this paper, we propose a simplified Levenberg–Marquardt method in Hilbert spaces for solving nonlinear ill-posed equations in which sequence of iteration { x n δ } is defined as x n + 1 δ = x n δ - ( α n I + F ′ (x 0) * F ′ (x 0) ) - 1 F ′ (x 0) * (F (x n δ) - y δ) . Here { α n } is a decreasing sequence of nonnegative numbers which converges to zero, F ′ (x 0) denotes the Fréchet derivative of F at an initial guess x 0 ∈ D (F) for the exact solution x † and (F ′ (x 0)) * denote the adjoint of F ′ (x 0) . In our proposed method, we need to calculate Fréchet derivative of F only at an initial guess x 0 . Hence, it is more economic to use in numerical computations than the Levenberg–Marquardt method used in the literature. We have proved convergence of the method under Morozov-type stopping rule using a general tangential cone condition. In the last section of the paper, numerical examples are presented to demonstrate advantages of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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15. Numerical algorithm for nonlinear fractional equations with nonlocal boundary conditions based on a modified minimum residual method.
- Author
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Jiang, Wei, Yue, Zihan, Chen, Zhong, and Wu, Fei
- Subjects
NONLINEAR equations ,SYSTEMS theory ,LINEAR systems ,ALGORITHMS ,NEWTON-Raphson method - Abstract
In this paper, we will solve a nonlinear time-fractional equation with nonlocal boundary conditions. First, we construct a set of suitable base according to the correlation theory of reproducing kernel space and the nonlocal boundary conditions. Then we introduce F-derivative and Newton iterative to linearize the nonlinear terms of the equation. The appropriate initial iteration value is constructed by boundary conditions. Finally, we use the ɛ-approximate solution theory to solve the system of linear equations and give the corresponding theoretical derivation. In conclusion, several numerical examples illustrate the feasibility and effectiveness of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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16. Closed-form solutions and conserved quantities of a new integrable (2 + 1)-dimensional Boussinesq equation of nonlinear sciences.
- Author
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Adeyemo, Oke Davies and Khalique, Chaudry Masood
- Subjects
BOUSSINESQ equations ,NONLINEAR equations ,CONSERVED quantity ,CONSERVATION laws (Mathematics) ,POWER series ,ELLIPTIC functions - Abstract
In this paper, we investigate a newly introduced integrable (2 + 1)-dimensional Boussinesq equation. Solutions of this equation are obtained by Lie symmetry reductions and direct integration. We achieve diverse solitary wave solutions of the equation among which are non-topological soliton as well as Jacobi elliptic function solutions. Moreover, we generate some closed-form solutions of the equation which are in the form of bright, singular and non-singular periodic solitons. Power series solution of the equation is also generated. In a bid to have a sound understanding of the physical phenomena of the underlying model, we exhibited graphically the motion of the secured results. Besides, we discuss the obtained results as well as their respective graphs. Conclusively, we construct conservation laws of the aforementioned equation by employing the general multiplier approach. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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17. Existence and asymptotic behavior of strictly convex solutions for singular k-Hessian equations with nonlinear gradient terms.
- Author
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He, Xingyue, Gao, Chenghua, Wang, Jingjing, and Yao, Xiaobin
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NONLINEAR equations ,CONVEX functions ,EQUATIONS - Abstract
In this paper, we mainly consider the singular k-Hessian equations S k (λ (D 2 u)) = h (x) f (- u) + g (| D u |) in Ω and S k (λ (D 2 u)) = h (x) f (- u) (1 + g (| D u |) ) in Ω with the Dirichlet boundary condition u = 0 on ∂ Ω , where Ω ⊂ ℝ N ( N ≥ 2 ) is a strictly convex, bounded smooth domain. Using the method of upper and lower solutions and the Karamata regular variation theory, we get new criteria of the existence and asymptotic behavior of strictly convex solutions under different conditions imposed on h, f and g. This problem is more difficult to solve than the k-Hessian problem without gradient terms, and requires additional new conditions in the proof process. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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18. Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire.
- Author
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Gholami, Amin, Ganji, Davood D., Rezazadeh, Hadi, Adel, Waleed, and Bekir, Ahmet
- Subjects
NONLINEAR oscillations ,PENDULUMS ,NONLINEAR equations ,LINEAR equations ,MOTION ,NONLINEAR systems ,NONLINEAR oscillators - Abstract
The paper deals with the application of a strong method called the modified Mickens iteration technique which is used for solving a strongly nonlinear system. The system describes the motion of a simple mathematical pendulum with a particle attached to it through a stretched wire. This model has great applications especially in the area of nonlinear vibrations and oscillation systems. The proposed method depends on determining the frequency and amplitude of the system through the modified Mickens iterative approach which is a modification of the regular Mickens approach. The preliminaries of the proposed technique are present and the application to the model is discussed. The method depends on the Mickens iteration approach which transforms the considered equation into a linear form and then is solving this equation result in the approximate solution. Some examples are given to validate and illustrate the effectiveness and convenience of the method. These results are compared with other relative techniques from the literature in terms of finding the frequency of the two examined models. The method produces more accurate results when compared to these methods and is considered a strong candidate for solving other nonlinear problems with applications in science and engineering. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
19. Borderline regularity for fully nonlinear equations in Dini domains.
- Author
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Adimurthi, Karthik and Banerjee, Agnid
- Subjects
NONLINEAR equations ,PARTIAL differential equations ,BOUNDARY value problems ,MONGE-Ampere equations ,DIFFERENTIAL equations ,MAXIMUM principles (Mathematics) - Abstract
3 L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189-213. 5 L. A. Caffarelli, Interior W2,pW^{2,p} estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135-150. Keywords: Fully nonlinear equations; borderline gradient continuity; Dini \mathrm{Dini}-domains; 35J25; 35J60 EN Fully nonlinear equations borderline gradient continuity Dini \mathrm{Dini}-domains 35J25 35J60 619 634 16 10/03/22 20221001 NES 221001 1 Introduction The aim of this paper is to obtain pointwise gradient continuity estimates up to the boundary for viscosity solutions ted to improving the presentation of the paper. [Extracted from the article]
- Published
- 2022
- Full Text
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20. A modified iteratively regularized Landweber iteration method: Hölder stability and convergence rates.
- Author
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Mittal, Gaurav and Giri, Ankik Kumar
- Subjects
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INVERSE problems , *BANACH spaces , *NONLINEAR equations , *PROBLEM solving - Abstract
In this paper, we formulate the modified iteratively regularized Landweber iteration method in Banach spaces to solve the inverse problems for which the forward operator may be smooth or non-smooth. We study the convergence analysis of the modified method for both the perturbed as well as unperturbed data by utilizing the Hölder stability estimates. In the presence of perturbed data, we terminate the method via a discrepancy principle and show that it is in fact a convergence regularization method that terminates within a few iterations. In the presence of unperturbed data, we show that the iterates converge to the exact solution. Additionally, we deduce the convergence rates in the presence of perturbed as well as unperturbed data. Finally, we discuss two inverse problems on which the method is applicable. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. Adaptive neural network surrogate model for solving the nonlinear elastic inverse problem via Bayesian inference.
- Author
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Huo, Fuchang, Zhang, Kai, Gao, Yu, and Li, Jingzhi
- Subjects
- *
MARKOV chain Monte Carlo , *INVERSE problems , *ELASTICITY , *BAYESIAN field theory , *NONLINEAR equations - Abstract
In this paper, we consider a Bayesian method for nonlinear elastic inverse problems. As a working model, we are interested in the inverse problem of restoring elastic properties from measured tissue displacement. In order to reduce the computational cost, we will use the following multi-fidelity model approach. First, we construct a surrogate low-fidelity DNNs-based model in the prior distribution, then use a certain number of simulations of high fidelity model associated with an adaptive strategy online to update the low-fidelity model locally. Numerical examples show that the proposed method can solve nonlinear elastic inverse problems efficiently and accurately. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. Error Identities for Parabolic Equations with Monotone Spatial Operators.
- Author
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Repin, Sergey I.
- Subjects
MONOTONE operators ,BOUNDARY value problems ,INITIAL value problems ,NONLINEAR equations ,EQUATIONS ,PARABOLIC operators - Abstract
The article studies quantitative relations (error identities) that characterize distances between exact solutions of nonlinear evolutionary problems and functions considered as approximations. The restrictions imposed on such a function are minimal and actually come down to the condition that it belongs to the same functional class as the generalized solution of the problem under consideration. Functional identities of this type reflect the most general relations between deviations from exact solutions of parabolic initial boundary value problems and those data that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class regardless of the method by which it was constructed. Therefore, they can serve as basic tools for deriving fully reliable a posteriori estimates of approximation errors as well as for analysis of modeling errors. The corresponding examples are discussed in the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert Spaces.
- Author
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Mahale, Pallavi and Kumar, Ankush
- Subjects
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NONLINEAR operators , *HILBERT space , *INVERSE problems , *REAL numbers , *OPERATOR equations , *NONLINEAR equations - Abstract
In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ( u ) = y {F(u)=y} , where F : 풟 ( F ) ⊂ 햴 → 햸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 햴 {\mathsf{U}} and 햸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ T 0 + α n I ) - 1 T 0 ∗ ( F ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of
F at an initial guess u 0 ∈ 풟 ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operatorF , we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
24. Another hybrid conjugate gradient method as a convex combination of WYL and CD methods.
- Author
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Guefassa, Imane, Chaib, Yacine, and Bechouat, Tahar
- Subjects
- *
CONJUGATE gradient methods , *NONLINEAR equations , *LINEAR equations , *LINEAR systems , *PROBLEM solving - Abstract
Conjugate gradient (CG) methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems. In this paper, a new hybrid conjugate gradient (CG) method is presented and analyzed for solving unconstrained optimization problems, where the parameter β k \beta_{k} is a convex combination of β k WYL \beta_{k}^{\mathrm{WYL}} and β k CD \beta_{k}^{\mathrm{CD}} . Under the strong Wolfe line search, the new method possesses the sufficient descent condition and the global convergence properties. The preliminary numerical results show the efficiency of our method in comparison with other CG methods. Furthermore, the proposed algorithm HWYLCD was extended to solve the problem of a mode function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. A projected gradient method for nonlinear inverse problems with 훼ℓ1 − 훽ℓ2 sparsity regularization.
- Author
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Zhao, Zhuguang and Ding, Liang
- Subjects
- *
NONLINEAR equations , *THRESHOLDING algorithms , *NONLINEAR functions , *INVERSE problems - Abstract
The non-convex α ∥ ⋅ ∥ ℓ 1 − β ∥ ⋅ ∥ ℓ 2 ( α ≥ β ≥ 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ∥ ⋅ ∥ ℓ 1 − β ∥ ⋅ ∥ ℓ 2 regularized function can be computed by applying the ST-( α ℓ 1 − β ℓ 2 ) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). Unfortunately, It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. Nevertheless, the current applicability of the PG method is limited to linear inverse problems. In this paper, we extend the PG method based on a surrogate function approach to nonlinear inverse problems with the α ∥ ⋅ ∥ ℓ 1 − β ∥ ⋅ ∥ ℓ 2 ( α ≥ β ≥ 0 ) regularization in the finite-dimensional space R n . It is shown that the presented algorithm converges subsequentially to a stationary point of a constrained Tikhonov-type functional for sparsity regularization. Numerical experiments are given in the context of a nonlinear compressive sensing problem to illustrate the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. An extensive study of the regularity of solutions to doubly singular equations.
- Author
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Vespri, Vincenzo and Vestberg, Matias
- Subjects
EQUATIONS ,NONLINEAR equations - Abstract
In recent years, many papers have been devoted to the regularity of doubly nonlinear singular evolution equations. Many of the proofs are unnecessarily complicated, rely on superfluous assumptions or follow an inappropriate approximation procedure. This makes the theory unclear and quite chaotic to a nonspecialist. The aim of this paper is to fix all the misprints, to follow correct procedures, to exhibit, possibly, the shortest and most elegant proofs and to give a complete and self-contained overview of the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
27. A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity.
- Author
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Elmahdi, Emadidin Gahalla Mohmed and Huang, Jianfei
- Subjects
FINITE differences ,NONLINEAR equations ,CAPUTO fractional derivatives - Abstract
This paper presents a linearized finite difference scheme for solving a kind of time-space fractional nonlinear diffusion-wave equations with initial singularity, where the Caputo fractional derivative in time and the Riesz fractional derivative in space are involved. First, the considered problem is equivalently transformed into its partial integro-differential form. Then, the fully discrete scheme is constructed by using the Crank–Nicolson technique, the L1 approximation, and the convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and the fractional central difference formula are applied to approximate the second-order derivative and the Riesz fractional derivative in space, respectively. Moreover, the stability and convergence of the proposed scheme are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
28. New soliton waves and modulation instability analysis for a metamaterials model via the integration schemes.
- Author
-
Gu, Yongyi, Manafian, Jalil, Mahmoud, Mustafa Z., Ghafel, Sukaina Tuama, and Ilhan, Onur Alp
- Subjects
NONLINEAR differential equations ,SOLITONS ,PARTIAL differential equations ,SCHRODINGER equation ,NONLINEAR equations ,TRAVELING waves (Physics) ,DIFFERENTIAL equations - Abstract
In this paper, the exact analytical solutions to the generalized Schrödinger equation are investigated. The Schrodinger type equations bearing nonlinearity are the important models that flourished with the wide-ranging arena concerning plasma physics, nonlinear optics, fluid-flow, and the theory of deep-water waves, etc. In this exploration, the soliton and other traveling wave solutions in an appropriate form to the generalized nonlinear Schrodinger equation by means of the extended sinh-Gordon equation expansion method, tan(Γ(ϖ))-expansion method, and the improved cos(Γ(ϖ)) function method are obtained. The suggested model of the nonlinear Schrodinger equation is turned into a differential ordinary equation of a single variable through executing some operations. One soliton, periodic, and singular wave solutions to this important equation in physics are reached. The periodic solutions are expressed in terms of the rational functions. Soliton solutions are obtained from them as a particular case. The obtained solutions are figured out in the profiles of 2D, density, and 3D plots by assigning suitable values of the involved unknown constants. Modulation instability (MI) is employed to discuss the stability of got solutions. These various graphical appearances enable the researchers to understand the underlying mechanisms of intricate phenomena of the leading equation. The individual performances of the employed methods are praiseworthy which deserves further application to unravel any other nonlinear partial differential equations (NLPDEs) arising in various branches of sciences. The proposed methodologies for resolving NLPDEs have been designed to be effectual, unpretentious, expedient, and manageable. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
29. Higher order Traub–Steffensen type methods and their convergence analysis in Banach spaces.
- Author
-
Kumar, Deepak, Sharma, Janak Raj, and Singh, Harmandeep
- Subjects
BANACH spaces ,NONLINEAR equations ,TAYLOR'S series ,ITERATIVE methods (Mathematics) - Abstract
In this paper, we consider two-step fourth-order and three-step sixth-order derivative free iterative methods and study their convergence in Banach spaces to approximate a locally-unique solution of nonlinear equations. Study of convergence analysis provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions using higher order derivatives. Furthermore, in quest of fast algorithms, a generalized q-step scheme with increasing convergence order 2q + 2 is proposed and analyzed. Novelty of the q-step algorithm is that, in each step, order of convergence is increased by an amount of two at the cost of only one additional function evaluation. To maximize the computational efficiency, the optimal number of steps is calculated. Theoretical results regarding convergence and computational efficiency are verified through numerical experimentation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
30. N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds.
- Author
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Li, Li, Liu, Yiyan, and Yu, Fajun
- Subjects
DARBOUX transformations ,PLANE wavefronts ,LAX pair ,NONLINEAR equations ,EQUATIONS ,SOLITONS - Abstract
In this paper, we propose and investigate the reverse-space–time nonlocal nonlinear Fokas–Lenells equation by the idea of Ablowitz and Musslimani. The reverse-space–time Fokas–Lenells equation, associated with a 2 × 2 matrix Lax pair, is the important integrable system, which can be reduced to the nonlocal Fokas–Lenells equation. Based on its Lax pair, we construct nonlocal version of N-fold Darboux transformation (DT) for the Fokas–Lenells equation, and obtain two kinds of soliton solutions from vanishing and plane wave backgrounds. Further some novel one-soliton and two-soliton are derived with the zero and nonzero seed solutions through complex computations, including the bright soliton, kink soliton and breather wave soliton. Moreover, various graphical analyses on the presented solutions are made to reveal the dynamic behaviors, which display the elastic interactions between two solitons and their amplitudes keeping unchanged after the interactions except for the phase shifts. It is clearly shown that these solutions have new properties which differ from ones of the classical Fokas–Lenells equation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
31. On local and global structures of transmission eigenfunctions and beyond.
- Author
-
Liu, Hongyu
- Subjects
EIGENFUNCTIONS ,INVERSE problems ,SCATTERING (Physics) ,EIGENVALUES ,NONLINEAR equations - Abstract
The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many aspects in a delicate way. The properties of the transmission eigenvalues have been extensively and intensively studied over the years, whereas the intrinsic properties of the transmission eigenfunctions are much less studied. Recently, in a series of papers, several intriguing local and global geometric structures of the transmission eigenfunctions are discovered. Moreover, those longly unveiled geometric properties produce some interesting applications of both theoretical and practical importance to direct and inverse scattering problems. This paper reviews those developments in the literature by summarizing the results obtained so far and discussing the rationales behind them. There are some side results of this paper including the general formulations of several types of transmission eigenvalue problems, some interesting observations on the connection between the transmission eigenvalue problems and several challenging inverse scattering problems, and several conjectures on the spectral properties of transmission eigenvalues and eigenfunctions, with most of them are new to the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
32. On Caputo tempered implicit fractional differential equations in b-metric spaces.
- Author
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Salim, Abdelkrim, Krim, Salim, Lazreg, Jamal Eddine, and Benchohra, Mouffak
- Subjects
NONLINEAR equations ,FRACTIONAL differential equations - Abstract
This paper deals with the existence and uniqueness results for a class of problems for nonlinear Caputo tempered implicit fractional differential equations in b-metric spaces with initial and nonlocal conditions. The arguments are based on some fixed point theorems. Furthermore, two illustrations are presented to demonstrate the plausibility of our results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
33. Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian.
- Author
-
Taheri, Ali
- Subjects
LIOUVILLE'S theorem ,NONLINEAR equations ,METRIC spaces ,RICCI flow ,ELLIPTIC equations ,CURVATURE - Abstract
In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a (k , m) -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
34. Finite-time attractivity of strong solutions for generalized nonlinear abstract Rayleigh–Stokes equations.
- Author
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Tuan, Tran Van
- Subjects
RESOLVENTS (Mathematics) ,HILBERT space ,EQUATIONS ,DIFFERENCE equations ,NONLINEAR equations ,STOKES equations - Abstract
In the present paper, we address the global solvability and finite-time attractivity of strong solutions for an abstract Rayleigh–Stokes-type equation involving nonlinear perturbations in Hilbert spaces. Based on regularity estimates of resolvent operator, local estimates on Hilbert scales and fixed point arguments, we obtain some results on global existence and finite-time attractivity of strong solutions to our problem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
35. Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds.
- Author
-
Goffi, Alessandro and Pediconi, Francesco
- Subjects
NEUMANN boundary conditions ,RIEMANNIAN manifolds ,NONLINEAR equations ,POISSON'S equation ,ELLIPTIC operators ,EUCLIDEAN domains ,ELLIPTIC equations - Abstract
In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue space, under various integrability regimes. Our method is based on an integral refinement of the Bochner identity, and leads to "semilinear Calderón–Zygmund" type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclidean space will also be discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. Blowup and global existence of mild solutions for fractional extended Fisher–Kolmogorov equations.
- Author
-
Chen, Pengyu, Ma, Weifeng, Tao, Shu, and Zhang, Kaibin
- Subjects
BLOWING up (Algebraic geometry) ,NONLINEAR equations ,EQUATIONS ,NONLINEAR functions - Abstract
In this paper, we investigate the blowup, as well as global existence, and uniqueness of mild solutions for the initial-boundary value problem to a class of fractional extended Fisher–Kolmogorov equations with a general nonlinear term. We establish a general framework to find the global mild solutions for fractional extended Fisher–Kolmogorov equations with general nonlinear function, which will provide an effective way to deal with such problems. The results obtained in this paper can be considered as a contribution to this nascent field. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
37. Parallel iterative finite-element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions.
- Author
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Zhou, Kangrui and Shang, Yueqiang
- Subjects
NAVIER-Stokes equations ,NONLINEAR equations ,PARALLEL algorithms ,ALGORITHMS - Abstract
Based on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
38. Secant-type iteration for nonlinear ill-posed equations in Banach space.
- Author
-
George, Santhosh, Sreedeep, C. D., and Argyros, Ioannis K.
- Subjects
BANACH spaces ,NONLINEAR equations ,REGULARIZATION parameter - Abstract
In this paper, we study secant-type iteration for nonlinear ill-posed equations involving 푚-accretive mappings in Banach spaces. We prove that the proposed iterative scheme has a convergence order at least 2.20557 using assumptions only on the first Fréchet derivative of the operator. Further, using a general Hölder-type source condition, we obtain an optimal error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
39. Reconstruction of local volatility surface from American options.
- Author
-
Gong, Wenxiu and Xu, Zuoliang
- Subjects
FINITE differences ,OPTIONS (Finance) ,MARKET prices ,PRICES ,NONLINEAR equations - Abstract
In this paper, we discuss the reconstruction of a local volatility surface from American option prices. First of all, the American option prices are calculated by an accurate and fast finite difference scheme. Then the local volatility is obtained by minimizing the distance between theoretical prices and market option prices, which yields an optimization problem. The Bicubic spline regularization technique is used to overcome the ill-posedness of the reconstruction problem. We solve the nonlinear optimization problem by using a gradient-based optimization algorithm. Finally, we test our model with numerical examples and real market American put option data. The results show the good performance of our method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
40. Optimal conditions of solvability of periodic problem for systems of differential equations with argument deviation.
- Author
-
Mukhigulashvili, Sulkhan
- Subjects
DIFFERENTIAL equations ,NONLINEAR equations - Abstract
In this paper, we study, on the interval I = [ a , b ] , the problem of the solvability and unique solvability of systems of higher-order differential equations with argument deviation u i (m i) (t) = p i (t) u i + 1 (τ i (t)) + q i (t) (i = 1 , ... , n) and u i (m i) (t) = f i (t , u i + 1 (τ i (t))) + q 0 i (t) (i = 1 , ... , n) , under the periodic boundary conditions u i (j) (b) - u i (j) (a) = c i j (i = 1 , ... , n , j = 0 , ... , m i - 1) , where u n + 1 = u 1 , n ≥ 2 , m i ≥ 1 , p i ∈ L ∞ (I ; R) , q 0 i ∈ L (I ; R) , f i : I × R → R are Carathéodory class functions, and τ i : I → I are measurable functions. The optimal conditions are obtained, which guarantee the unique solvability of the linear problem and take into account the effect of argument deviation. Based on these results, the optimal conditions of the solvability and unique solvability are proved for the nonlinear problem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
41. Arbitrary High-Order Unconditionally Stable Methods for Reaction-Diffusion Equations with inhomogeneous Boundary Condition via Deferred Correction.
- Author
-
Koyaguerebo-Imé, Saint-Cyr Elvi Rodrigue and Bourgault, Yves
- Subjects
REACTION-diffusion equations ,INITIAL value problems ,BOUNDARY value problems ,ELLIPTIC equations ,NONLINEAR equations ,FINITE element method - Abstract
In this paper, we analyse full discretizations of an initial boundary value problem (IBVP) related to reaction-diffusion equations. To avoid possible order reduction, the IBVP is first transformed into an IBVP with homogeneous boundary conditions (IBVPHBC) via a lifting of inhomogeneous Dirichlet, Neumann or mixed Dirichlet–Neumann boundary conditions. The IBVPHBC is discretized in time via the deferred correction method for the implicit midpoint rule and leads to a time-stepping scheme of order 2 p + 2 of accuracy at the stage p = 0 , 1 , 2 , ... of the correction. Each semi-discretized scheme results in a nonlinear elliptic equation for which the existence of a solution is proven using the Schaefer fixed point theorem. The elliptic equation corresponding to the stage 푝 of the correction is discretized by the Galerkin finite element method and gives a full discretization of the IBVPHBC. This fully discretized scheme is unconditionally stable with order 2 p + 2 of accuracy in time. The order of accuracy in space is equal to the degree of the finite element used when the family of meshes considered is shape-regular, while an increment of one order is proven for a quasi-uniform family of meshes. Numerical tests with a bistable reaction-diffusion equation having a strong stiffness ratio, a Fisher equation, a linear reaction-diffusion equation addressing order reduction and two linear IBVPs in two dimensions are performed and demonstrate the unconditional convergence of the method. The orders 2, 4, 6, 8 and 10 of accuracy in time are achieved. Except for some linear problems, the accuracy of DC methods is better than that of BDF methods of same order. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
42. The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow.
- Author
-
Azami, Shahroud
- Subjects
HEAT equation ,NONLINEAR equations ,SMOOTHNESS of functions - Abstract
Let (M
n , F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow ∂ g (x , t) ∂ t = 2 h (x , t) , where g(t) is the symmetric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. In this paper, we consider local Li-Yau type gradient estimates for positive solutions of the following nonlinear heat equation with potential ∂ t u (x , t) = Δ m u (x , t) − R (x , t) u (x , t) − a u (x , t) log u (x , t) , (x , t) ∈ M × [ 0 , T ] , along the Finsler-geometric flow, where 퓡 is a smooth function, and a is a real nonpositive constant. As an application we obtain a global estimate and a Harnack estimate. Our results are also natural extension of similar results on Riemannian-geometric flow. [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
- View/download PDF
43. Wavelet collocation methods for solving neutral delay differential equations.
- Author
-
Faheem, Mo, Raza, Akmal, and Khan, Arshad
- Subjects
COLLOCATION methods ,NONLINEAR differential equations ,DELAY differential equations ,NONLINEAR equations - Abstract
In this paper, we proposed wavelet based collocation methods for solving neutral delay differential equations. We use Legendre wavelet, Hermite wavelet, Chebyshev wavelet and Laguerre wavelet to solve the neutral delay differential equations numerically. We solved five linear and one nonlinear problem to demonstrate the accuracy of wavelet series solution. Wavelet series solution converges fast and gives more accurate results in comparison to other methods present in literature. We compare our results with Runge–Kutta-type methods by Wang et al. (Stability of continuous Runge–Kutta-type methods for nonlinear neutral delay-differential equations," Appl. Math. Model, vol. 33, no. 8, pp. 3319–3329, 2009) and one-leg θ methods by Wang et al. (Stability of one-leg θ method for nonlinear neutral differential equations with proportional delay," Appl. Math. Comput., vol. 213, no. 1, pp. 177–183, 2009) and observe that our results are more accurate. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
44. A projected homotopy perturbation method for nonlinear inverse problems in Banach spaces.
- Author
-
Xia, Yuxin, Han, Bo, and Wang, Wei
- Subjects
- *
INVERSE problems , *BANACH spaces , *NONLINEAR equations , *PARAMETER identification - Abstract
In this paper, we propose and analyze a projected homotopy perturbation method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. To expedite convergence, the approach uses two search directions given by homotopy perturbation iteration, and the new iteration is calculated as the projection of the current iteration onto the intersection of stripes decided by above directions. The method allows to use L 1 -like penalty terms, which is significant to reconstruct sparsity solutions. Under reasonable conditions, we establish the convergence and regularization properties of the method. Finally, two parameter identification problems are presented to indicate the effectiveness of capturing the property of the sparsity solutions and the acceleration effect of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
45. A new class of exact solutions of von Karman's equation in the nonlinear theory of gas dynamics.
- Author
-
Jokhadze, Otar
- Subjects
VON Karman equations ,GAS dynamics ,NONLINEAR theories ,NONLINEAR equations ,POISSON'S equation ,VON Neumann algebras - Abstract
In this paper, by using invariants of Riemann and general solutions of Euler–Poisson–Darboux–Riemann equations, a new class of exact solutions of von Karman's equation in the nonlinear theory of gas dynamics is given. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
46. Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations.
- Author
-
Grunau, Hans-Christoph, Miyake, Nobuhito, and Okabe, Shinya
- Subjects
BIHARMONIC equations ,CAUCHY problem ,HEAT equation ,NONLINEAR equations ,OPERATOR equations - Abstract
This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
47. Reliable Computer Simulation Methods for electrostatic Biomolecular Models Based on the Poisson–Boltzmann Equation.
- Author
-
Kraus, Johannes, Nakov, Svetoslav, and Repin, Sergey
- Subjects
POISSON'S equation ,COMPUTER simulation ,SOBOLEV spaces ,NONLINEAR equations ,EQUATIONS - Abstract
The paper is concerned with the reliable numerical solution of a class of nonlinear interface problems governed by the Poisson–Boltzmann equation. Arising in electrostatic biomolecular models these problems typically contain measure-type source terms and their solution often exposes drastically different behaviour in different subdomains. The interface conditions reflect the requirement that the potential and its normal derivative must be continuous. In the first part of the paper, we discuss an appropriate weak formulation of the problem that guarantees existence and uniqueness of the generalized solution. In the context of the considered class of nonlinear equations, this question is not trivial and requires additional analysis, which is based on a special splitting of the problem into simpler subproblems whose weak solutions can be defined in standard Sobolev spaces. This splitting also suggests a rational numerical solution strategy and a way of deriving fully guaranteed error bounds. These bounds (error majorants) are derived for each subproblem separately and, finally, yield a fully computable majorant of the difference between the exact solution of the original problem and any energy-type approximation of it. The efficiency of the suggested computational method is verified in a series of numerical tests related to real-life biophysical systems. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
48. A derivative-free iterative method for nonlinear ill-posed equations with monotone operators.
- Author
-
George, Santhosh and Thamban Nair, M.
- Subjects
ITERATIVE methods (Mathematics) ,NUMERICAL analysis ,NONLINEAR equations ,MONOTONE operators ,OPERATOR theory - Abstract
Recently, Semenova [12] considered a derivative free iterative method for nonlinear ill-posed operator equations with a monotone operator. In this paper, a modified form of Semenova's method is considered providing simple convergence analysis under more realistic nonlinearity assumptions. The paper also provides a stopping rule for the iteration based on an a priori choice of the regularization parameter and also under the adaptive procedure considered by Pereverzev and Schock [11]. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
49. An Improvement on a Class of Fixed Point Iterative Methods for Solving Absolute Value Equations.
- Author
-
Shams, Nafiseh Nasseri and Beik, Fatemeh Panjeh Ali
- Subjects
ABSOLUTE value ,NONLINEAR equations ,EQUATIONS ,SHIFT systems - Abstract
We consider a class of iterative methods based on block splitting (BBS) to solve absolute value equations A x - | x | = b . Recently, several works were devoted to deriving sufficient conditions for the convergence of iterative methods of this type under certain assumptions including ν := ∥ A - 1 ∥ < 1 . However, the BBS-type iterative methods tend to converge slowly when 휈 is very close to one (i.e., ν ≈ 1 ). In this paper, using an auxiliary matrix, we develop a new approach by first rewriting the main problem into a new equivalent block system having shifted (1 , 1) -block and then constructing a fixed point iteration. The exploited strategy can significantly improve the convergence speed of the BBS-type iterative methods when ν ≈ 1 . Numerical experiments are reported to demonstrate the superiority of the new modified iterative scheme over the existing original form of BBS-type methods in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
50. The boundary value problem for one class of higher-order nonlinear partial differential equations.
- Author
-
Kharibegashvili, Sergo and Midodashvili, Bidzina
- Subjects
NONLINEAR differential equations ,BOUNDARY value problems ,EXISTENCE theorems ,NONLINEAR equations - Abstract
In this paper, we consider the boundary value problem for one class of higher-order nonlinear partial differential equations. We prove theorems on existence, uniqueness and nonexistence of solutions of this problem. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
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