Nonlinear Variable Coefficient Research Articles

Soliton cluster solutions of nonlinear Schrödinger equations with variable coefficients in Bessel lattice

The goal of this article is to obtain soliton cluster solutions of (2 + 1)-dimensional nonlinear variable coefficient Schrödinger equations through computerized symbolic computation. By applying the self-similarity method, one soliton solution is constructed for the NLS equations with variable coefficient, which provide a model of the propagation of the soliton waves. Consequently, the solitonary cluster solution is achieved in different structures. Additionally, the propagation of the obtaining solitonary cluster solutions is analyzed and discussed. The results are useful to explain the soliton phenomena in nonlinear optics.

Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations

This paper proposes a temporal second-order fully discrete two-grid finite element method (FEM) for solving nonlinear time-fractional variable coefficient diffusion-wave equations. The method involves introducing an auxiliary variable and utilizing a reducing order technique for the fractional derivative. This reduces the original fractional wave equations to a coupled system consisting of two equations with lower order derivative in time. The time-fractional derivative is discretized using the L2-1σ formula, while a second-order scheme is used for discretizing the first-order time derivative. The spatial direction is discretized using an efficient two-grid Galerkin FEM. The paper demonstrates that the optimal error estimates in L2-norm and H1-norm of the two-grid methods can achieve optimal convergence order when the mesh size satisfies H=h12 and H=hr2r+2, respectively. To handle the initial singularity of the solution and the historical memory of the fractional derivative term, the paper presents the nonuniform and fast two-grid algorithms. The numerical results validate the theoretical analysis and demonstrate the effectiveness of the proposed two-grid methods.

Buckling and Post-Buckling of Bidirectional Porous Beam Under Bidirectional Hygrothermal Environment

In this paper, buckling and nonlinear post-buckling behaviors of a bidirectional porous (BDP) beam are investigated under bidirectional hygrothermal environment. Euler–Bernoulli beam theory with the von Kármán nonlinearity is employed to derive the nonlinear variable coefficient governing differential equations based on Rayleigh quotient method. Analytical solutions of critical buckling load and load–deflection equilibrium path in post-buckling are deduced for the single directional varying (SDV) porous beam. The general numerical solutions for bidirectional varying (BDV) porous beam are obtained by differential quadrature finite element method (DQFEM) with Newton–Raphson iteration method based on the variation principle. The high accuracy of the present numerical method with higher computing efficiency is verified by comparison with published reports and the analytical results in this work. Parametric analysis on effects of the porosity bidirectional distributions, porosity coefficients, distributions of hygrothermal environment and boundary conditions on buckling load and post-buckling response is carried out to enhance the buckling and deformation resistances in design, manufacture and usage of porous structures. The results show that the bidirectional porosity pattern, linear and nonlinear hygrothermal distribution and boundary conditions play a significant role on buckling critical external load and critical hygrothermal increments, buckling form and post-buckling path.

Qade: solving differential equations on quantum annealers

We present a general method, called Qade, for solving differential equations using a quantum annealer. One of the main advantages of this method is its flexibility and reliability. On current devices, Qade can solve systems of coupled partial differential equations that depend linearly on the solution and its derivatives, with non-linear variable coefficients and arbitrary inhomogeneous terms. We test this through several examples that we implement in state-of-the-art quantum annealers. The examples include a partial differential equation and a system of coupled equations. This is the first time that equations of these types have been solved in such devices. We find that the solution can be obtained accurately for problems requiring a small enough function basis. We provide a Python package implementing the method at gitlab.com/jccriado/qade.

Novel and accurate Gegenbauer spectral tau algorithms for distributed order nonlinear time-fractional telegraph models in multi-dimensions

This paper sheds light on the numerical solutions of the multi-dimensional distributed-order (DO) nonlinear time-fractional telegraph equations (TFTEs) models. We present novel and accurate shifted Gegenbauer spectral tau schemes for solving the aforementioned models. In the proposed schemes, the known and unknown functions are approximated in terms of the shifted Gegenbauer polynomials (SGPs) by using the Kronecker product structure in time and space. The matrix representation of the DO fractional derivative of SGPs in the Caputo sense has been derived. Also, novel operational matrices of the multiplication of shifted Gegenbauer functions in multi-dimension have been constructed to remove the obstacle that facing the use of the tau method in the presence of nonlinear term or variable coefficients or both of them. The proposed method takes full advantage of the nonlocal nature of DO fractional differential operators. The presented method is applied on six miscellaneous test examples to illustrate its robustness and effectiveness for smooth and non smooth solutions.

Linearized compact difference schemes applied to nonlinear variable coefficient parabolic equations with distributed delay

AbstractIn this article, combined with the compound trapezoidal formula, two linearized compact difference methods are presented for solving nonlinear variable coefficient parabolic equations with distributed delay. It is proved under some appropriate conditions that these linearized compact schemes are uniquely solvable and convergent of order two in time and order four in space. Finally, with several numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.

The Galerkin-Implicit Method for Solving Nonlinear Variable Coefficients Hyperbolic Boundary Value Problem

This paper has the interest of finding the approximate solution (APPS) of a nonlinear variable coefficients hyperbolic boundary value problem (NOLVCHBVP). The given boundary value problem is written in its discrete weak form (WEFM) and proved have a unique solution, which is obtained via the mixed Galerkin finite element with implicit method that reduces the problem to solve the Galerkin nonlinear algebraic system (GNAS). In this part, the predictor and the corrector techniques (PT and CT, respectively) are proved at first convergence and then are used to transform the obtained GNAS to a linear GLAS . Then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are studied. Some illustrative examples are used, where the results are given by figures that show the efficiency and accuracy for the method.

Dynamics of optical solitons and nonautonomous complex wave solutions to the nonlinear Schrodinger equation with variable coefficients

Variable coefficients nonlinear evolution equations offer us with more real aspects in the inhomogeneities of media and nonuniformities of boundaries than their counter constant coefficients in some real-world problems. Under consideration is a nonlinear variable coefficients Schrodinger’s equation with spatio-temporal dispersion in the Kerr law media. We are aimed at constructing novel solutions to the equation under consideration. Bright and combined dark–bright optical solitons are successfully revealed with aid of the complex amplitude ansatz scheme. Using two test functions, two nonautonomous complex wave solutions in dark and bright optical solitons forms are successfully revealed. The effect of the variable coefficients on the reported results can be clearly seen on the 3-dimensional and contour graphs.

Some exact solutions of a variable coefficients fractional biological population model

In this paper, we investigate the exact solutions of a nonlinear variable coefficients time fractional biological population model using the invariant subspace method. The subspaces with dimensions one, two, and three are derived for certain cases of the variable coefficients. The exact solutions of the nonlinear time fractional biological population model are obtained in some cases.

A rapidly convergent approximation scheme for nonlinear autonomous and non-autonomous wave-like equations

In this work, an efficient approximation scheme has been proposed for getting accurate approximate solution of nonlinear partial differential equations with constant or variable coefficients satisfying initial conditions in a series of exponential instead of an algebraic function of independent variables. As a consequence: i) the convergence of the series found to be faster than the same obtained by few other methods and ii) the exact analytic solution can be obtained from the first few terms of the series of the approximate solution, in cases the equation is integrable. The convergence of the sum of the successive correction terms has been established and an estimate of the error in the approximation has also been presented. The efficiency of the present method has been illustrated through some examples with a variety of nonlinear terms present in the equation.

Generalized symmetries and conservation laws of (3+1)-dimensional variable coefficient Zakharov-Kuznetsov equation

The nonlinear variable coefficient Zakharov-Kuznetsov (Vc-ZK) equation is derived using reductive perturbation technique for ion-acoustic solitary waves in magnetized three-component dusty plasma having negatively charged dust particles, isothermal ions, and electrons. The equation is investigated for generalized symmetries using a recently proposed compatibility method. Some more general symmetries are obtained and group invariant solutions are also constructed for these symmetries. Besides this, the equation is also investigated for nontrivial local conservation laws

An extrapolated CN-WSGD OSC method for a nonlinear time fractional reaction-diffusion equation

In this paper, a new method is proposed to solve the time-fractional reaction-diffusion equation with nonlinear variable coefficients. Based on the second-order weighted and shifted Grünwald difference (WSGD) method for time Riemann-Liouville fractional derivative, we formulate and analyze the extrapolated Crank-Nicolson (ECN) method in time and orthogonal spline collocation (OSC) method in space. We prove the optimal global convergence order by a novel choice of the first step solution uh1. Finally, some numerical results are given to verify the convergence rate in the theoretical analysis.

A (n +1)-dimensional KP Equation with Variable Coefficients with Self-consistent Source

In this paper, the variable coefficient (n+1)-dimensional KP equation is studied with the Hirota bilinear method. Firstly, the logarithmic transformation is used to transform the nonlinear variable coefficient (n+1)-dimensional KP equation into a bilinear form, and then variable coefficient (n+1)-dimensional KP equation with self-consistent source was constructed with the source generation procedure.

Steady-state analysis of heat–moisture–salt coupling in unsaturated layered soil

Based on the theory of porous media, the multifield coupling problem of heat–moisture–salt in unsaturated layered soil under the steady condition is studied considering that the pore of unsaturated soil is filled with liquid water, dissolved salt, water vapour and dry gas. Based on the conservation of mass and energy equations, the paper discusses the mass conservation equations and energy conservation equations for water, gas and salt in unsaturated soils. Considering the one-dimensional steady-state problem, the state equations are obtained by selecting temperature, pore air pressure, pore water pressure and salt solution concentration and their derivatives as the state variables. Under the condition of the boundary values of adjacent soil layers in a given unsaturated soil, the non-linear variable coefficient differential equations are solved by using the shooting method and the validity of the model is verified by comparing with existing steady-state experimental results. Based on numerical examples, the differences in temperature field, pressure field and salinity field between unsaturated layered soil and homogeneous soil are analysed.

Classes of new exact solutions for nonlinear Schrödinger equations with variable coefficients arising in optical fiber

Abstract In this paper, a new modified direct similarity reduction method is considered to find new classes of Jacobi, hyperbolic and periodic wave solutions for both (1 + 1)-dimensional and (2 + 1)-dimensional nonlinear variable coefficients Schrodinger equations (vcNLSE) arising in optical fiber. Additionally, as a conclusion we have arrived in a theorem for both direct reduction and exact solutions to (n + 1)-dimensional vcNLSE. Finally, some physical applications for optical soliton propagation is given joint with figures.

A fast, spectrally accurate homotopy based numerical method for solving nonlinear differential equations

We present an algorithm for constructing numerical solutions to one-dimensional nonlinear, variable coefficient boundary value problems. This scheme is based upon applying the Homotopy Analysis Method (HAM) to decompose a nonlinear differential equation into a series of linear differential equations that can be solved using a sparse, spectrally accurate Gegenbauer discretisation. Uniquely for nonlinear methods, our scheme involves constructing a single, sparse matrix operator that is repeatedly solved in order to solve the full nonlinear problem. As such, the resulting scheme scales quasi-linearly with respect to the grid resolution. We demonstrate the accuracy, and computational scaling of this method by examining a fourth-order nonlinear variable coefficient boundary value problem by comparing the scheme to Newton-Iteration and the Spectral Homotopy Analysis Method, which is the most commonly used implementation of the HAM.

Numerical Study of Some Intelligent Robot Systems Governed by the Fractional Differential Equations

We provide a high order numerical methods for solving some intelligent Robot Systems, which are governed by the fractional nonlinear variable coefficient reaction diffusion equations with delay. The convergence of this numerical algorithm is discussed via the energy method. Besides, this algorithm can also be extended to solve the reaction diffusion equation with multi-delays. Finally, we carry on the numerical analysis, and the results of the numerical tests are coincide with the theoretical results.

Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations

It is recognized that the theory of boundary value problems for fractional order-differential equations is one of the rapidly developing branches of the general theory of differential equations. As far as we know, most of the papers studied the fractional Riemann-Liouville derivative with respect to boundary values that are zero. However, for the purpose of this study, we concern ourselves with Captou type derivative of the order α∈(2, 3), with respect to boundary values that are nonzero. We establish sufficient conditions for the existence of solutions for boundary value problem of nonlinear variable coefficient of fractional order. On the other hand, the boundary value problem is formulated as follows: <i>^cD^αu</i>(<i>t</i>) + <i>p</i>(<i>t</i>)<i>f</i>(<i>t</i>, <i>u</i>(<i>t</i>)) + <i>q</i>(<i>t</i>) = 0, <i>u</i>(0) = <i>a</i>, <i>u'</i>(0) = <i>b</i>, <i>u</i>(1) = <i>d</i>. Where <i>a, b, d</i> ∈ <i>R</i> are constants. In this paper, we investigate the existence and uniqueness of solutions for a class of boundary value problem of the nonlinear variable coefficient of fractional differential equations. The existence of solutions involving Captuo fractional derivatives is discussed under the assumption that the bounded conditions are constants. By means of the Banach contraction mapping principle and Larry- Schauder alternative, the existence of solutions are obtained. Finally, some examples are discussed to illustrate the results, which are generalized to nonlinear fractional derivatives with variable coefficients.

Geometric grid network and third-order compact scheme for solving nonlinear variable coefficients 3D elliptic PDEs

By using nonuniform (geometric) grid network, a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type. Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions. The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values. As an experiment, applications of the compact scheme to Schrödinger equations, sine-Gordon equations, elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values. The results corroborate the reliability and efficiency of the scheme.