Problem For Nonlinear Equation Research Articles

Applications of the Stone–Weierstrass theorem in the Calderón problem

We give examples on the use of the Stone–Weierstrass theorem in inverse problems. We show uniqueness in the linearized Calderón problem on holomorphically separable Kähler manifolds and in the Calderón problem for nonlinear equations on conformally transversally anisotropic manifolds. We also study the holomorphic separability condition in terms of plurisubharmonic functions. The Stone–Weierstrass theorem allows us to generalize and simplify earlier results. It also makes it possible to circumvent the use of complex geometrical optics solutions and inversion of explicit transforms in certain cases.

Mild solutions to fourth-order parabolic equations modeling thin film growth with time fractional derivative

In this article, we study initial-boundary problems for fourth-order nonlinear parabolic equations modeling thin film growth with Caputo-type time fractional derivative. By means of the theory of abstract fractional calculus and \(L^p-L^q\) estimates, we establish the existence and uniqueness of local mild solutions in the spaces \(C([0,T]; L^{\frac{\beta N}{2-\beta}}(\Omega))\) with \(1<\beta<2\). Moreover, the local solutions can be extended globally if the initial data is sufficiently small. For more information see https://ejde.math.txstate.edu/Volumes/2024/58/abstr.html

Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного рівняння теплопровідності з експоненціально нелінійним коефіцієнтом теплопровідності

The problem of heat conduction in nonlinear media reduces to solving boundary value problems for nonlinear heat conduction equations, where the coefficients of the equation or the function of the power of heat sources depend on temperature according to some law. Among the numerical methods for solving problems for nonlinear equations of mathematical physics, one can distinguish finite difference methods, finite element methods, variational and projection methods, as well as iterative methods. Among the latter group, the two-sided approximation method is particularly attractive due to its ability to provide a convenient estimate for the error of the approximate solution and to prove the existence of a solution to the original problem. The theory of linear partially ordered spaces was developed by L. V. Kantorovich in the second half of the 1930s. Further development of this theory is associated with the works of M. A. Krasnosel’skii, H. Amann, V. I. Opojtsev, N. S. Kurpel, B. A. Shuvar, A. I. Kolosov etc. The aim of this article is to develop a two-sided approximation method based on the use of Green's functions for solving the first boundary value problem for a one-dimensional nonlinear heat conduction equation and to investigate its performance when solving test problems. To achieve this goal, the unknown function was replaced, and the boundary value problem was reduced to a Hammerstein integral equation, which was considered as a nonlinear operator equation in a partially ordered Banach space. Conditions for the existence of a unique positive solution to the problem and conditions for two-sided convergence of successive approximations to it were obtained. The developed method was implemented in software and tested on solving test problems. The results of the computational experiment are illustrated with graphical and tabular information

Inverse scattering transform for the higher-order integrable discrete nonlinear Schrödinger equation with nonzero boundary conditions

The inverse scattering transform is a tool for solving the initial value problem of nonlinear equations, and the solution of the initial-value problem by inverse scattering transform proceeds in three steps: direct scattering, time evolution and inverse problem. In this paper, we discuss the higher-order integrable discrete nonlinear Schrödinger equation which is subjected to inverse scattering transform under nonzero boundary conditions, and the corresponding soliton solutions are obtained and illustrated.

Euler-Maruyama approximation for stochastic fractional neutral integro-differential equations with weakly singular kernel

This manuscript examines the problem of nonlinear stochastic fractional neutral integro-differential equations with weakly singular kernels. Our focus is on obtaining precise estimates to cover all possible cases of Abel-type singular kernels. Initially, we establish the existence, uniqueness, and continuous dependence on the initial value of the true solution, assuming a local Lipschitz condition and linear growth condition. Additionally, we develop the Euler-Maruyama method for numerical solution of the equation and prove its strong convergence under the same conditions as the well-posedness. Moreover, we determine the accurate convergence rate of this method under global Lipschitz conditions and linear growth conditions.

Dirichlet Problem for Nonlinear Higher-Order Equations in Upper Half Plane

In this article, we consider the Dirichlet problem for nonlinear higher-order equations in upper half plane. Firstly we introduce the solutions of inhomogeneous polyanalytic equation in upper half plane. Then we investigate the properties of relevant integral operators. Lastly we transform the Dirichlet problem for nonlinear higher-order equations in upper half plane into the system of integro-differential equations and we obtain the existence of unique solution using Banach fixed point theorem.

The Analitical Solution of Linear and Non-Linear Differential- Algebraic Equations (DAEs) with Laplace-Padé Series Method

In this paper, we apply Laplace-Padé Series method to solve linear and non-linear differentialalgebraic equations (DAEs). Firstly, The basic properties of the Laplace-Padé Series method are given. Secondly, we calculate the arbitrary order power series of differential-algebraic equations (DAEs), then convert it to the series form Laplace-Padé. Then, the three differential-algebraic equations (DAEs) are solved by Laplace-Padé Series method. It was seen that the method gave effective and fast results. Therefore, the method can be easily applied to linear and non-linear differential-algebraic equations (DAEs) problems in different fields.

Global existence of nonlinear elastic waves: Non-hyperelastic case

For the Cauchy problem of nonlinear elastic wave equations for three-dimensional isotropic, homogeneous and hyperelastic materials with null condition, global existence of classical solutions with small initial data was proved in Agemi (2000) [1] and Sideris (2000) [17] independently. How to treat the more general non-hyperelastic case is suggested as an open problem in Agemi (2005) [2]. The aim of this paper is to resolve this problem.

A regularization strategy to a backward problem for general nonlinear parabolic equations in Sobolev spaces

This article is devoted to solving a backward problem for general nonlinear parabolic equations in Sobolev spaces. The problem is hardly solved by computation since it is severely ill‐posed in Hadamard's sense. Using the Fourier transform and the truncation of high‐frequency components, we construct a regularized solution for the problem from the data given inexactly. As usual for ill‐posed problems, the rate of convergence can be obtained under additional smoothness assumptions on the regularity of the exact solution, the so‐called source conditions. By using a source condition in a generalized form, we prove a convergence estimate in a Sobolev space . Corresponding to different levels of the source condition, the convergence rates are improved gradually. We apply our theory to the backward problem with a convection–diffusion operator and a cubic nonlinearity. Finally, the numerical results are presented to test the validity of the proposed regularization method.

Blow-up of solutions to the wave equations with memory terms in Schwarzschild spacetime

The main goal of this work is to discuss formation of singularities of solutions to the Cauchy problems for single wave equation and coupled system of wave equations with nonlinear memory terms in Schwarzschild spacetime. For the initial value problem of nonlinear wave equations, blow-up results of sign changing solutions are derived by employing the suitable test function technique which is related to the Riemann-Liouville fractional derivative and contradiction argument. Additionally, upper bound lifespan estimates of solutions to the small initial value problem of wave equations are established by imposing certain conditions on the initial values. It is worth to mention that crucial difficulty is the setting of test function framework, which has not been applied to the Schwarzschild spacetime in the previous literatures. To the best of our knowledge, the results in Theorems 1.1-1.4 are new.

Rigidity of inverse problems for nonlinear elliptic equations on manifolds

Rigidity of inverse problems for nonlinear elliptic equations on manifolds

[formula omitted] sparsity regularization for nonlinear ill-posed problems

In this paper, the α‖⋅‖ℓ1−β‖⋅‖ℓ2 sparsity regularization with parameters α≥β≥0 is studied for nonlinear ill-posed inverse problems. The well-posedness of the regularization is investigated. Compared to the case where α>β≥0, the results for the case α=β>0 are weaker due to the lack of coercivity and Radon-Riesz property of the regularization term. Under certain conditions on the nonlinearity of F, sparsity is shown for every minimizer of the α‖⋅‖ℓ1−β‖⋅‖ℓ2 regularized inverse problem. Moreover, for the case α>β≥0, convergence rates O(δ12) and O(δ) are proved for the regularized solution toward a sparse exact solution, under different yet commonly adopted conditions on the nonlinearity of F. The iterative soft thresholding algorithm is shown to be useful to solve the α‖⋅‖ℓ1−β‖⋅‖ℓ2 regularized problem for nonlinear ill-posed equations. Numerical results illustrate the efficiency of the proposed method.

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

Numerical Analysis for Sturm–Liouville Problems with Nonlocal Generalized Boundary Conditions

For the generalized Sturm–Liouville problem (GSLP), a new formulation is undertaken to reduce the number of unknowns from two to one in the target equation for the determination of eigenvalue. The eigenparameter-dependent shape functions are derived for using in a variable transformation, such that the GSLP becomes an initial value problem for a new variable. For the uniqueness of eigenfunction an extra condition is imposed, which renders the right-end value of the new variable available; a derived implicit nonlinear equation is solved by an iterative method without using the differential; we can achieve highly precise eigenvalues. For the nonlocal Sturm–Liouville problem (NSLP), we consider two types of integral boundary conditions on the right end. For the first type of NSLP we can prove sufficient conditions for the positiveness of the eigenvalue. Negative eigenvalues and multiple solutions may exist for the second type of NSLP. We propose a boundary shape function method, a two-dimensional fixed-quasi-Newton method and a combination of them to solve the NSLP with fast convergence and high accuracy. From the aspect of numerical analysis the initial value problem of ordinary differential equations and scalar nonlinear equations are more easily treated than the original GSLP and NSLP, which is the main novelty of the paper to provide the mathematically equivalent and simpler mediums to determine the eigenvalues and eigenfunctions.

Modern technique to study Cauchy-type problem of fractional variable order differential equations with infinite delay via phase space

In this paper, we use a novel approach to study the existence, uniqueness, and stability of solutions (EUSS) to a Cauchy-type problem of nonlinear fractional differential equations (FrDiEq) of variable order with infinite delay (CPNFDEVOID). Contrary to the techniques taken in the literature, which were centered on the usage of the concept of generalized intervals and the idea of piecewise constant functions, our approach is straightforward and based on a novel fractional operator that is more appropriate and demonstrates the solvability and stability of the main problem under less restrictive presumptions. The results are achieved in this paper by using Fixed Point Theory (FPT). The application, which includes an example and supporting images, concludes the paper.

Positional embeddings for solving PDEs with evolutional deep neural networks

This work extends the paradigm of evolutional deep neural networks (EDNNs) to solving parametric time-dependent partial differential equations (PDEs) on domains with geometric structure. By introducing positional embeddings based on eigenfunctions of the Laplace-Beltrami operator, geometric properties are encoded intrinsically and Dirichlet, Neumann and periodic boundary conditions of the PDE solution are enforced directly through the neural network architecture. The proposed embeddings lead to improved error convergence for static PDEs and extend EDNNs towards computational domains of realistic complexity.Several steps are taken to improve performance of EDNNs: Solving the EDNN update equation with a Krylov solver avoids the explicit assembly of Jacobians and enables scaling to larger neural networks. Computational efficiency is further improved by an ad-hoc active sampling scheme that uses the PDE dynamics to effectively sample collocation points. A modified linearly implicit Rosenbrock method is proposed to alleviate the time step requirements of stiff PDEs. Lastly, a completely training-free approach, which automatically enforces initial conditions and only requires time integration, is compared against EDNNs that are trained on the initial conditions. We report results for the Korteweg-de Vries equation, a nonlinear heat equation and (nonlinear) advection-diffusion problems on domains with and without holes and various boundary conditions, to demonstrate the effectiveness of the method. The numerical results highlight EDNNs as a promising surrogate model for parametrized PDEs with slow decaying Kolmogorov n-width.

Numerical solution of fuzzy fractional volterra integro differential equations with boundary conditions

In this study, the boundary value problem of fuzzy fractional nonlinear Volterra integro differential equations of order 1 < ϱ ≤ 2 is addressed. Fuzzy fractional derivatives are defined in the Caputo sense. To show the existence result, the Krasnoselkii theorem from the theory of fixed points is used, where as the well-known contraction mapping concept is utilized in order to show the solution is unique to the proposed problem. Moreover, a novel Adomian decomposition method is utilized to get numerical solution; the approach behind deriving the solution is from Adomian polynomials, and it is organized according to the recursive relation that is obtained. The proposed method significantly decreases the numerical computations by obtaining solutions without the need of discretization or constrictive assumptions. According to the results, there is substantial agreement between the series solutions produced by the fuzzy Adomian decomposition method. Finally, using MATLAB, the symmetry between the lower and upper-cut representations of the fuzzy solutions is demonstrated in the numerical result.

Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space

This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of differential equations with mixed Volterra-Fredholm integral equations using fixed point theory. This work have two important results, the first result was the discussion of the existence of solutions using the Krasnoselskii fixed point theorem after transforming the problem into integral equation firstly then into operator problem suitable for the fixed point theory. The second result will be the existence and uniqueness of solution, this result was obtained by the use of Banach fixed point theorem after the same transformation used in the first one. This work give as conclusion that the boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredholm integral equation has a unique solution in Banach space. Finally, this work was ended with an example to illustrate the results obtained.

The Cauchy problem for general nonlinear wave equations with doubly dispersive

The Cauchy problem for general nonlinear wave equations with doubly dispersive

LANDAU LEGENDRE WAVELET GALERKIN METHOD APPLIED TO STUDY TWO-PHASE MOVING BOUNDARY PROBLEM OF HEAT TRANSFER IN FINITE REGION

In this paper, we developed a mathematical model of solidification where specific heat and thermal conductivity are temperature dependent. This model is a two-phase moving boundary problem (MBP) of heat transfer in finite region and represents as MBP of system of parabolic nonlinear second-order partial differential equations (PDEs). We developed a Landau Legendre wavelet Galerkin method for finding the solution of the problem. The MBP of a system of PDEs is transformed into a variable boundary value problem of nonlinear ordinary defferential equations (ODEs) by the use of dimensionless variables and the Landau transform. The problem is converted into a system of algebraic equations with the application of Legendre wavelet Galerkin method. In particular case, we compared present solution with Laplace transform solution and found approximately the same. The whole investigation has been done in dimensionless form. When the specific heat and thermal conductivity exponentially vary in temperatures, it is the effect of dimensionless parameters: thermal diffusivity (&amp;alpha;&lt;sub&gt;12&lt;/sub&gt;), ratio of thermal conductivity (&lt;i&gt;k&lt;/i&gt;&lt;sub&gt;12&lt;/sub&gt;), dimensionless temperature (&amp;theta;&lt;sub&gt;&lt;i&gt; f&lt;/i&gt;&lt;/sub&gt;), Fourier number (F&lt;sub&gt;0&lt;/sub&gt;), Stefan number (Ste), and ratio of densities (&amp;rho;&lt;sub&gt;1&lt;/sub&gt; / &amp;rho;&lt;sub&gt;2&lt;/sub&gt;) are discussed in detail.