Problem For Nonlinear Equation Research Articles

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

[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 towards 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.

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 differential-algebraic 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.

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.

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.

Bifurcation from interval and multiple positive solutions for a nonlinear fractional differential equations with integral boundary conditions

This paper studies the bifurcation from interval and Multiple positive solutions for a boundary-value problem of nonlinear fractional differential equations with integral boundary conditions. Using the bifurcation technique and the topological degree theory, the existence of Multiple positive solutions is investigated and some sufficient conditions are obtained.

One-Dimensional Inverse Problem for Nonlinear Equations of Electrodynamics

For the system of nonlinear electrodynamics equations, we consider the problem of determining the medium conductivity coefficient multiplying the nonlinearity. It is assumed that the permittivity and permeability are constant and the conductivity depends only on one spatial variable, with this conductivity being zero on the half-line x . For a mode in which only two electromagnetic field components participate, the wave propagation process caused by the incidence of a plane wave with a constant amplitude from the domain x0 onto an inhomogeneity localized on the half-line x0 is considered. With a given conductivity coefficient, the conditions for the solvability of the direct problem and the properties of its solution are studied. To solve the inverse problem, the trace of the electrical component of the solution of the direct problem is specified on a finite segment of the axis x=0. A theorem on the local existence and uniqueness of the solution of the inverse problem is established, and a global estimate of the conditional stability of its solutions is found.

Biorthogonal flatlet multiwavelet collocation method for solving the singular nonlinear system with initial and boundary conditions

PurposeThe numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.Design/methodology/approachThe approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.FindingsAccording to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.Originality/valueThe key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

Observer-Based Boundary Fuzzy Control Design of Nonlinear Parabolic PDE Systems Using Mobile Sensors

This paper studies the boundary fuzzy control problem for nonlinear parabolic partial differential equation (PDE) systems under spatially noncollocated mobile sensors. In a real setup, sensors and actuators can never be placed at the same location, and the noncollocated setting may be beneficial in some application scenarios. The control design is very difficult due to the noncollocated mobile observation, which can be solved by an observer-based technique. At first, a Takagi-Sugeno fuzzy PDE model is devoted to accurately representing the nonlinear parabolic PDE system. Next, we present a state estimation scheme including fuzzy Luenberger-type PDE state observer plus mobile sensor guidance. Then, an observer-based boundary fuzzy controller is posed to render the resulting closedloop system exponentially stable, and the exponential decay rate is increased by the designed mobile sensor guidance laws. At last, two examples verify the proposed method.

One-Dimensional Inverse Problem for Nonlinear Equations of Electrodynamics

One-Dimensional Inverse Problem for Nonlinear Equations of Electrodynamics

Solving nonlinear equations with the convex combination of two positive spectral coefficients

We present a spectral algorithm based on the convex combination of two modified spectral coefficients for solving systems of nonlinear equations. The proposed algorithm does not require the exact or approximated directional derivative for its implementation. By employing a derivative-free line search, the global convergence of the sequence generated by the algorithm is supported. Numerical experiments are given to demonstrate the performance of the algorithm compared with a similar algorithm in the literature for solving nonlinear equations problems.

On Finite-Time Blow-Up Problem for Nonlinear Fractional Reaction Diffusion Equation: Analytical Results and Numerical Simulations

The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand on some previous blow-up results for the explicit values and numerical simulation of finite-time blow-up solutions for a semilinear fractional partial differential problem involving a positive power of the solution. We show the behavior solution of the fractional problem, and the numerical solution of the finite-time blow-up solution is also considered. Finally, some illustrative examples and comparisons with the classical problem with integer order are presented, and the validity of the results is demonstrated.

Local Solvability, Blow-Up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: III. Cauchy Problems

Local Solvability, Blow-Up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: III. Cauchy Problems

Local Solvability, Blow-Up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: III. Cauchy Problems

Three Cauchy problems for Sobolev-type equations with a common linear part from the theory of ion acoustic and drift waves in a plasma are considered. The problems are reduced to equivalent integral equations. We prove the existence of unextendable solutions for two problems and the existence of a local-in-time solution for the third problem. For one of the problems, by applying a modified method of Kh.A. Levin, sufficient conditions for finite time blow-up of solutions are obtained and an upper bound for the solution blow-up time is found. For another problem, S.I. Pohozaev’s nonlinear capacity method is used to obtain a finite time blow-up result and two results concerning the nonexistence of even local solutions, and an upper bound for the solution blow-up time is obtained as well.

Study on the Multi-Point Boundary Value Problem for Second-Order Nonlinear Impulsive Integro-Differential Equation

In the field of biological control, there are a large number of systems that gradually evolve over a certain period of time. However, due to some natural or human intervention behavior, the system state will be subjected to some relatively short time interference, so that the system state changes in an instant. This sudden change of state makes the system not simply described by continuous or discrete dynamical systems, but by means of impulse dynamical systems. In this paper, the multi-point boundary value problem of a class of second-order nonlinear impulsive integral differential equations is studied on the basis of impulsive differential equation theory. The main results of this kind of equation are obtained by using the fixed point theorem of strict set contraction operator. Under certain assumptions, the existence of the solution of the equation is proved by constructing the operator on the special cone. Finally, combined with the practical application, the theory was applied to the stability prediction of biological ecosystem, and the correctness of the conclusion was verified.

Well-posedness results for a class of nonlinear reaction-diffusion equations with memory

In this paper, we are interested to study the initial value problem for nonlinear heat equation with memory.  For our problem, we show that the local existence theory related to the finite time blow-up is also obtained for the problem with logarithmic nonlinearity. We also obtain the blowup continuation property of the mild solution. The principal techniques based on some Sobolev embeddings with \(L^p\) spaces.