Nonlinear Source Term Research Articles

On Quasilinear Anisotropic Parabolic Equations with Time-Dependent Exponents

The Cauchy-Dirichlet problem for the anisotropic parabolic equation with variable exponents in the presence of a nonlinear source and gradient term is considered. We prove the existence and uniqueness of a weak solution that is Lipschitz continuous in the space variables.

On a coupled system of Kirchhoff beam equations with nonlinear frictional damping and source term

On a coupled system of Kirchhoff beam equations with nonlinear frictional damping and source term

SOLVING AN INVERSE PROBLEM FOR A GENERALIZED TIME-DELAYED BURGERS-FISHER EQUATION BY HAAR WAVELET METHOD

In this paper, a numerical method consists of combining Haar wavelet method and Tikhonov regularization method to determine unknown boundary condition and unknown nonlinear source term for the generalized time-delayed Burgers-Fisher equation using noisy data is presented. A stable numerical solution is determined for the problem. We also show that the rate of convergence of the method is as exponential $ \Bigl(O\left(\frac{1}{2^{J+1}}\right)\Bigr) $, where $ J $ is maximal level of resolution of wavelet. Some numerical results are reported to show the efficiency and robustness of the proposed approach for solving the inverse problems.

A CFD-sectional algorithm for population balance equation coupled with multi-dimensional flow dynamics

A novel CFD-sectional algorithm is developed to address the challenges in the existing sectional simulations coupled with multi-dimensional fluid dynamics, including solution of complex sectional coefficients, serious computational burden for lots of coupled partial differential equations, and nonlinear source terms. The sectional coefficients are specified by a numerical quadrature with adaptive integration limits, which proves to be computationally efficient and accurate. The inter-equation coupling is treated by hybrid-segregated procedures and the source term is linearized by the operator splitting method. The CFD-sectional algorithm is validated against a self-preserving solution of particles undergoing Brownian coagulation. The acoustic agglomeration in a standing wave is simulated as a representative case. It has been demonstrated that the predictions regarding the particle size distribution and agglomeration process agree well with the experimental data, which verifies the capability of the developed CFD-sectional algorithm in simulating the spatially inhomogeneous population balance equation coupled with multi-dimensional flows.

Nonexistence of global solutions for a fractional system of strongly coupledintegro-differential equations

In this paper, we study the nonexistence of nontrivial global solutions for a system of two strongly coupled fractional differential equations. Each equation involves two fractional derivatives and a nonlinear source term. The fractional derivatives are of Caputo type of subfirst orders. The nonlinear sources are nonlocal in time. They have the form of a convolution of a polynomial of the state with a (possibly singular) kernel. The system under consideration is a generalization of many interesting special systems of equations whose solutions do not exist globally in time. We establish some criteria under which no nontrivial global solutions exist. Several integral inequalities and estimations are derived and the test function method is adopted. Special cases and examples are given to illustrate the results.

Global existence, asymptotic stability and blow‐up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

AbstractIn this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow‐up result for solutions with positive and negative initial energy respectively.

A general decay estimate for the nonlinear transmission problem of weak viscoelastic equations with time-varying delay

The transmission of waves over a body composed of two different materials is considered in this paper. One component is a simple elastic part, while the other is a weak viscoelastic component with time-varying delay and damping effect. Besides, one part of the boundary is controlled by a nonlinear memory source term, while the other part is clamped. Under the influence of time-varying delay and damping effect, we derive a general decay estimate of the energy when the memory kernels in the internal weak viscoelasticity and boundary source decay simultaneously.

Blow-up result and energy decay rates for binary mixtures of solids with nonlinear damping and source terms

In this paper we study the long-time behavior of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin.

Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations

Abstract In this paper, we consider the initial boundary value problem for a class of fourth-order wave equations with strong damping term, nonlinear weak damping term, strain term and nonlinear source term in polynomial form. First, the local solution is obtained by using fix point theory. Then, by constructing the potential well structure frame, we get the global existence, asymptotic behavior and blowup of solutions for the subcritical initial energy and critical initial energy respectively. Ultimately, we prove the blowup in finite time of solutions for the arbitrarily positive initial energy case.

Global existence and exponential decay for a viscoelastic Petrovsky system

We investigate the initial-boundary value problem for a viscoelastic Petrovsky system of Kirchhoff-type with nonlinear source term. By applying the standard continuation principle, the existence of global solutions for this problem is proved and the exponential decay estimate of global solutions is obtained.

Properties of global and exponential attractors for nonlinear thermo-diffusion Timoshenko system

In this paper, we consider a new Timoshenko beam model with thermal and mass diffusion effects according to the Gurtin-Pinkin model. Heat and mass exchange with the environment during thermodiffusion in Timoshenko beam, depending on the past history of the temperature and the chemical potential gradients through memory kernels. We analyze the longtime properties for such a model with linear frictional damping and nonlinear source terms. We prove the global well-posedness of the system by using the C0-semigroup theory of linear operators. Then, we show, without assuming the well-known equal wave speeds condition, that the thermal and chemical potential coupling is strong enough to guarantee the quasistability. By showing that the system is gradient and asymptotically compact, we prove the existence of a global attractor with a finite fractal dimension and with a smoothness property. Furthermore, the existence of a fractal exponential attractor is also derived.

Numerical solution of a modified anomalous diffusion equation with nonlinear source term through meshless singular boundary method

In this study, singular boundary method is employed for solving a modified anomalous diffusion process in two dimensional space with nonlinear source term with initial and Dirichlet-type boundary conditions. The process is modeled as a two dimensional nonlinear time-fractional sub-diffusion equation in sense of Riemann-Liouville fractional derivatives. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. We present the numerical operation for calculating the particular solution and homogeneous solution. For gaining approximation particular solution and homogeneous solution we employed MPS method and SBM method, respectively. We use θ-weighted finite difference method as time discretization for time derivatives. We employ Predictor-Corrector Algorithm for the nonlinear source term. A comparison check between SBM and other methods is given to show the accuracy of SBM applying on the equation. Consequently, some numerical examples with different domains is tested and compared with the exact analytical solutions to display the validity and accuracy of the numerical method and compared with other methods.

A 3D‐2D asymptotic analysis of viscoelastic problem with nonlinear dissipative and source terms

The purpose of this article is to study the asymptotic analysis of the solutions of a linear viscoelastic problem with a dissipative and source terms in a three‐dimensional thin domain Ωε. Firstly, we give the strong formulation of the problem and the existence and uniqueness theorem of the weak solution. Then, we establish some estimates independent of the parameter ε. These last will be useful to obtain the limit problem with a specific weak form of the Reynolds equation.

The Neumann problem for a Barenblatt equation with a multiplicative stochastic force and a nonlinear source term

In this paper, we are interested in an existence and uniqueness result for a Barenblatt’s type equation forced by a multiplicative noise, with additionally a nonlinear source term and under Neumann boundary conditions. The idea to show such a well-posedness result is to investigate in a first step the additive case with a linear source term. Through a time-discretization of the equation and thanks to results on maximal monotone operators, one is able to handle the non-linearity of the equation and pass to the limit on the discretization parameter. This allows us to show existence and uniqueness of a solution in the case of an additive noise and a linear source term. In a second step, thanks to a fixed point procedure, one shows the announced result.

A numerical method for a time-fractional advection–dispersion equation with a nonlinear source term

In this paper we propose an implicit finite-difference scheme to approximate the solution of an initial-boundary value problem for a time-fractional advection–dispersion equation with variable coefficients and a nonlinear source term. The time fractional derivative is taken in the sense of Caputo. The method is unconditionally stable and convergent. Some numerical examples are included and the results confirm the theoretical analysis. One of the examples is the fractional Fisher equation of mathematical biology.

Erratum: “Hydrodynamics of Diffusive Processes” [ASME Appl. Mech. Rev., 2001, 54(5), pp. 391–404; DOI: 10.1115/1.1399288

Erratum: “Hydrodynamics of Diffusive Processes” [ASME Appl. Mech. Rev., 2001, 54(5), pp. 391–404; DOI: 10.1115/1.1399288

Mesoscopic Simulation of the (2 + 1)-Dimensional Wave Equation with Nonlinear Damping and Source Terms Using the Lattice Boltzmann BGK Model.

In this work, we develop a mesoscopic lattice Boltzmann Bhatnagar-Gross-Krook (BGK) model to solve (2 + 1)-dimensional wave equation with the nonlinear damping and source terms. Through the Chapman-Enskog multiscale expansion, the macroscopic governing evolution equation can be obtained accurately by choosing appropriate local equilibrium distribution functions. We validate the present mesoscopic model by some related issues where the exact solution is known. It turned out that the numerical solution is in very good agreement with exact one, which shows that the present mesoscopic model is pretty valid, and can be used to solve more similar nonlinear wave equations with nonlinear damping and source terms, and predict and enrich the internal mechanism of nonlinearity and complexity in nonlinear dynamic phenomenon.

Weak Solutions to a System of Nonlinear Fractional Boundary Value Problems Via Variational Form

In this paper, the weak solutions to a class of nonlinear system of fractional boundary value problems are studied. Each equation of the system does have two kinds of nonlinear terms; one of them is in terms of gradient, and the other one is a nonlinear source term with respect to dependent variable. It is proved that there exists at least three distinct weak solutions by using the critical point theory and the variational method. To this aim, we apply the well-known theorem on the construction of the critical set of functionals with a weak compactness condition. Moreover, the main result is demonstrated by some examples to show its validity.

A multiscale view of nonlinear diffusion in biology: From cells to tissues

This paper presents a review on the mathematical tools for the derivation of macroscopic models in biology from the underlying description at the scale of cells as it is delivered by a kinetic theory model. The survey is followed by an overview of research perspectives. The derivation is inspired by the Hilbert’s method, known in classic kinetic theory, which is here applied to a broad class of kinetic equations modeling multicellular dynamics. The main difference between this class of equations with respect to the classical kinetic theory consists in the modeling of cell interactions which is developed by theoretical tools of stochastic game theory rather than by laws of classical mechanics. The survey is focused on the study of nonlinear diffusion and source terms.

The maximum norm analysis of a nonmatching grids method for a class of parabolic equation with nonlinear source terms

Motivated by the idea which has been introduced by M. Haiour and S.Boulaaras (Proc. Indian Acad. Sci. (Math. Sci.) Vol. 121,No. 4, November 2011,pp.481--493), We provide a maximum norm analysis of Euler scheme combined with finite element Schwarz alternating method for a class of parabolic equation with nonlinear source terms on two overlapping subdomains with nonmatching grids. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The two meshes being mutually independent on the overlap region, a triangle belonging to one triangulation does not necessarily belong to the other one. Under a stability analysis on the theta scheme which given by our work in (App. Math. Comp., 217, 6443--6450 (2011).), we establish, on each subdomain, an optimal asymptotic behavior between the discrete Schwarz sequence and the asymptotic solution of parabolic differential equations.