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This paper studies the asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in ℝn. Firstly, the global existence and uniqueness of classical solutions for small initial data are established. Then, we obtain the Lp, 2 ⩽ p ⩽ +℞ decay rate of solutions. The approach is based on detailed analysis of the Green function of the linearized equation with the technique of long wave-short wave decomposition and the Fourier analysis.

In this study, we propose a simple direct meshless scheme based on the Gaussian radial basis function for the one-dimensional linear and nonlinear convection–diffusion problems, which frequently occur in physical phenomena. This is fulfilled by constructing a simple ‘anisotropic’ space–time Gaussian radial basis function. According to the proposed scheme, there is no need to remove time-dependent variables during the whole solution process, which leads it to a really meshless method. The suggested meshless method is implemented to the challenging convection–diffusion problems in a direct way with ease. Numerical results show that the proposed meshless method is simple, accurate, stable, easy-to-program and efficient for both linear and nonlinear convection–diffusion equation with different values of Péclet number. To assess the accuracy absolute error, average absolute error and root-mean-square error are used.

In this paper, we discuss the virtual element formulation for the nonlinear convection–diffusion–reaction equation. We consider the Streamline upwind Petrov–Galerkin stabilization to reduce the non...

This paper is concerned with traveling curved fronts of bistable reaction–diffusion equations with nonlinear convection in a two-dimensional space. By constructing super- and subsolutions, we establish the existence of traveling curved fronts. Furthermore, we show that the traveling curved front is globally asymptotically stable.

In this paper, we consider the free boundary problem of a reaction diffusion equation with nonlinear convection term in one dimensional space. Our study contains three parts: in the first part we establish the existence and uniqueness of global solution, in the second part we obtain the spreading–vanishing dichotomy, and in third part, we obtain some estimations of the asymptotic speed of free boundaries when spreading happens.

In this article, a two-level meshless local Petrov Galerkin method (MLPG) is proposed to analyze nonlinear convection–diffusion equation based on the radial basis function (RBF) collocation method....

In this article, the polynomial based differential quadrature method is used to develop a numerical scheme for solving convection–diffusion–reaction model with a general reaction term. The ...

In this paper, we analyze the blow-up rates and uniqueness of entire large solutions to the equation $\Delta u=a(x)f(u)+ \mu b(x) |\nabla u|^{q}$ , $x\in \mathbb{R}^{N}$ ( $N\geq 3$ ), where $\mu > 0$ , $q > 0$ and $a, b\in \mathrm {C}^{\alpha }_{\mathrm{loc}}(\mathbb{R}^{N})$ ( $\alpha \in (0, 1)$ ). The weight a is nonnegative, b is able to change sign in $\mathbb{R}^{N}$ , and $f\in C^{1}[0, \infty )$ is positive and nondecreasing on $(0, \infty )$ and rapidly or regularly varying at infinity. Additionally, we investigate the uniqueness of entire large solutions.

The inverse coefficient problem for the nonlinear convection-diffusion-reaction equation is considered. A velocity vector and a mass-transfer coefficient are considered as the unknown coefficients and are recovered with the help of the additional information about the boundary value problem’s solution. The inverse coefficient problem is reduced to a two-parameter problem of multiplicative control, the solvability of which is proved in a general form. For a cubic reaction coefficient the local stability estimates of the control problem’s solutions are obtained regarding to a rather small perturbation of either the cost functional or the specified functions of the boundary value problem.

This paper considers the problem of estimating the permeability in a nonlinear convection-diffusion equation. To overcome the large calculation burden of conventional methods, we apply an adaptive multigrid conjugate gradient method to solve this inverse problem. This new method combines the multigrid multiscale idea with the conjugate gradient method, and adopts the necessary condition that the optimum solution should be the fixed point of the multigrid inversion method. Some numerical results verify that the proposed method both dramatically reduces the required computations and improves the inversion quality.

In this article, we propose a meshless local Petrov Galerkin (MLPG) method based on least square radial basis function partition of unity method (LS-RBF-PUM), which is applied to the nonlinear conv...

This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one-dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.

We propose a weak Galerkin (WG) finite element method for solving one-dimensional nonlinear convection–diffusion problems. Based on a weak form, the semi-discrete WG finite element scheme is established and analyzed. We prove the stability of the semi-discrete solution and derive the optimal order error estimate in the discrete $$H^1$$ -norm and $$L^2$$ -norm, respectively. Numerical experiment is conducted to confirm the theoretical results.

Consider an anisotropic parabolic equation with a nonlinear convection term depending on the spatial variable. If the diffusion coefficients are degenerate, in general, the boundary trace cannot be defined for the weak solution. The existence and the uniqueness of weak solution are researched without the boundary value condition. Moreover, a general method to prove stability of weak solutions independent of the boundary value condition is introduced for the first time.

In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for a general nonlinear convection-diffusion equation (NCDE) and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and we find that the numerical results are in good agreement with analytical solutions and that the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, the DUGKS can also adopt the nonuniform mesh. Besides, we perform some comparisons among the DUGKS, the finite-volume lattice Boltzmann model (FV-LBM), the single-relaxation-time lattice Boltzmann model (SLBM), and the multiple-relaxation-time lattice Boltzmann model (MRT-LBM). The results show that the present DUGKS is more accurate than the FV-LBM, more stable than the SLBM, and almost has the same accuracy as the MRT-LBM. Moreover, the use of nonuniform mesh may make the DUGKS model more flexible.

Consider a non-Newtonian fluid equation with a nonlinear convection term and a source term. The existence of the weak solution is proved by Simon’s compactness theorem. By the Hölder inequality, if both the diffusion coefficient and the convection term are degenerate on the boundary, then the stability of the weak solutions may be proved without the boundary value condition. If the diffusion coefficient is only degenerate on a part of the boundary value, then a partial boundary value condition is required. Based on this partial boundary, the stability of the weak solutions is proved. Moreover, the uniqueness of the weak solution is proved based on the optimal boundary value condition.

The solvability of the boundary value and extremum problems for the convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of substances is proven. The role of the control in the extremum problem is played by the boundary function in the Dirichlet condition. For a particular reaction coefficient in the extremum problem, the optimality system and estimates of the local stability of its solution to small perturbations of the quality functional and one of specified functions is established.

In the existence of an aligned magnetic field over the inclined shrinking/stretching stratified sheet in a non-Darcy porous medium, the two-dimensional boundary layer flow of an upper-convected Maxwell fluid is analyzed. The heat transfer effects are acknowledged by using the nonlinear convection. The system of partial differential equations, which administrates the distinctive properties of flow and heat transfer, is depleted into ordinary differential equations with the use of similarity variables. The governing equations are determined numerically by utilizing the shooting technique. The response of varied implicated parameters on velocity, skin friction, and temperature accounts is inspected graphically and displayed in the table. It is noted that local inertia coefficient is accountable for the reduction in the velocity profile and the aligned magnetic field has the opposite relation for the shrinking and stretching sheet.

This paper is devoted to developing finite-element methods for solving a two-dimensional boundary value problem for a singularly perturbed time-dependent convection–diffusion–reaction equation. The solution to this problem may vary rapidly in thin layers. As a result, spurious oscillations occur in the solution if the standard Galerkin method is used. In the multiscale finite-element methods, the original problem is split into the grid-scale and subgrid-scale problems, which allows capturing the problem features at a scale smaller than the element mesh size. In this study two methods are considered: the variational multiscale method with algebraic sub-scale approximation (VMM-ASA) and the residual-free bubbles (RFB) method. In the first method the subgrid-scale problem is simulated by the residual of the grid-scale equation and intrinsic time scales. In the second method the subgrid-scale problem is approximated by special functions. The grid-scale and subgrid-scale problems are formulated via the linearization procedure on the subgrid component applied to the original problem. The computer implementation of the methods was carried out using a commercial finite-element package. The efficiency of the developed methods is evaluated by solving a test boundary value problem for the nonlinear equation. Cases with different values of the diffusion coefficient have been analyzed. Based on the numerical investigation, it is shown that the multiscale methods enable improving the stability of the numerical solution and decreasing the quantity and the amplitude of oscillations compared to the standard Galerkin method. In the case of a small diffusion coefficient, the developed methods can yield a satisfactory numerical solution on a sufficiently coarse mesh.

Radiative effect has appealed consideration of investigators since radiative effect has numerous uses in engineering and industry. Specific uses encompass glass production, polymer processing, solar power, gas cooled, satellites, nuclear reactors and space technology. Keeping the above-mentioned practicality in perspective, here nonlinear convected non-Newtonian (Eyring–Powell) material flow subjected to radiative effect is described. Double stratification, chemical reaction and Robin’s type condition are accounted for modeling. Boundary-layer idea yields nonlinear governing expressions which are transfigured to ordinary differential equations via transformation procedure. Significant aspects of dimensionless factors are inspected via graphical description. We witnessed lower thermal and solutal fields subjected to stratification factors (thermal and solutal stratification).

We consider an identification problem for a stationary nonlinear convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by an optimization method. The solvability of the boundary value problem and the extremal problem is proved. In the case that the reaction coefficient is quadratic, when the equation acquires cubic nonlinearity, we deduce an optimality system. Analyzing it, we establish some estimates of the local stability of solutions to the extremal problem under small perturbations both of the quality functional and the given velocity vector which occurs multiplicatively in the convection–diffusion–reaction equation.

The nonlinear convective flow of an Oldroyd-B fluid due to a nonlinear stretching sheet with varying thickness is examined. The salient features of the random movement and thermophoresis are described. Formulation is made with the nonlinear thermal radiation and heat generation/absorption. Further, the convective conditions and double stratification are taken into account. The resulting flow problems are tackled by the optimal homotopy analysis method (OHAM). The resulting nonlinear problems are solved for the velocity, temperature, and concentration fields. The temperature and concentration gradients are numerically discussed. The total residual error is calculated. The Nusselt number is an increasing function of the radiation parameter. The Sherwood number increases with the increase in the solutal stratification or the Schmidt number. The main outcomes are presented in conclusions. This study has a wide range of applications such as thermal stratification of oceans, reservoirs, and rivers, density stratification of atmosphere, hydraulic lifts, and polymer processing.

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of nonstationary nonlinear convection-diffusion initial- boundary value problem in a time-dependent domain. The problem is reformulated using the arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so-called ALE derivative and an additional convective term. The problem is discretized with the use of the ALE- space time discontinuous Galerkin method (ALE-STDGM). In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty. The nonlinear convection terms are discretized with the aid of a numerical flux. The main attention is paid to the proof of the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties.

We solve a twofold ill-posed inverse problem of a nonlinear convection-diffusion equation, endowed with unknown source term and unknown moving boundary. When the solution is expanded through the superposition of a family of modified homogenization functions in a reduced domain, the unknown space-time-dependent source function can be obtained by solving a small scale linear system, with the help from the over-specified data of left flux, the measured final time data and some inner data, which are very saving. Simultaneously, the unknown moving boundary can be detected by solving nonlinear equations using the Newton iterative method. Numerical examples are given to confirm that the superposition of homogenization functions method (SHFM) can recover the unknown source function and moving boundary very well.

In this paper, we study superconvergence properties of the local discontinuous Galerkin (LDG) methods for solving nonlinear convection-diffusion equations in one space dimension. The main technicality is an elaborate estimate to terms involving projection errors. By introducing a new projection and constructing some correction functions, we prove the $$(2k+1)$$ th order superconvergence for the cell averages and the numerical flux in the discrete $$L^2$$ norm with polynomials of degree $$k\ge 1$$ , no matter whether the flow direction $$f'(u)$$ changes or not. Superconvergence of order $$k +2$$ ( $$k +1$$ ) is obtained for the LDG error (its derivative) at interior right (left) Radau points, and the convergence order for the error derivative at Radau points can be improved to $$k+2$$ when the direction of the flow doesn’t change. Finally, a supercloseness result of order $$k+2$$ towards a special Gauss–Radau projection of the exact solution is shown. The superconvergence analysis can be extended to the generalized numerical fluxes and the mixed boundary conditions. All theoretical findings are confirmed by numerical experiments.

We analyze an iterative procedure for solving nonlinear algebraic systems arising from the discretization of nonlinear, non-steady reaction-convection-diffusion equations with non-constant (and, in general, nonlinear) velocity terms. The basic idea underlying the procedure consists in lagging the diffusion and the velocity terms of the discretized system, which is thus partly linearized. After analyzing the discretized system and proving some results on the monotonicity of the operators and on the uniqueness of the solution, we prove sufficient conditions that ensure the convergence of this lagged method. We also describe the inner iteration and show how the weakly nonlinear systems arising at each lagged iteration can be solved efficiently. Finally, we analyze numerically the entire solution process by several numerical experiments.

This research contemplates the flow and heat transport of MHD rheological Eyring–Powell fluid embedded with dust and graphene nanoparticles (GP) in an ethylene–glycol (EG) mixture in the presence of nonlinear convection, Cattaneo–Christov heat flux, and thermal radiation. Primarily existing PDEs (fluid and dust phase) are transferred to non-dimensional form by invoking similarity transformations then solved numerically through RKF-45 method. The graphene particles are significantly used in energy transmission in aerospace, power and propulsion generation etc. Through graphical illustrations, velocity and temperature profiles (fluid and dust phases) converse for various prominent parameters. The results of friction factor and heat transfer rate are presented and analyzed. Validation of the present result is made with the existing data. Results demonstrate that increasing nonlinear convection parameter has an inverse relationship with the Nusselt number and the velocity in the dust and fluid phases. This may happen due to the domination of unsteadiness in the flow.

We study the metastable dynamics of solutions to nonlinear evolutive equations of parabolic type, with a particular attention to the case of the viscous scalar Burgers equation with small viscosity $$\varepsilon $$ . In order to describe rigorously such slow motion, we adapt the strategy firstly proposed in Mascia and Strani (SIAM J Math Anal 45:3084–3113, 2013) by linearizing the original equation around a metastable state and by studying the system obtained for the couple $$(\xi ,v)$$ , where $$\xi $$ is the position of the internal shock layer and v is a perturbative term. The main result of this paper provides estimates for the speed of the shock layer and for the error v; in particular, in the case of the viscous Burgers equation, we prove they are exponentially small in $$\varepsilon $$ . As a consequence, the time taken for the solution to reach the unique stable steady state is exponentially large, and we have exponentially slow motion.

In this paper, we present the optimal L2-norm error estimate of the local discontinuous Galerkin method based on the generalized alternating numerical flux for nonlinear convection–diffusion problem. We obtain the optimal (k + 1)-th error estimate for the piecewise tensor product polynomials space of degree at most k defined on any quasi-uniform Cartesian meshes. The highlight of the analysis is the use of some suitable designed global projections. Numerical experiments are given to verify the sharpness of the theoretical results.

In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the $$L^2$$ -norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve $$p+1$$ order of convergence for the solution and its spatial derivative in the $$L^2$$ -norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order $$p+1$$ towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order $$p+3/2$$ towards a special Gauss–Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the $$(p+1)$$ -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the $$L^2$$ -norm at $$\mathcal {O}(h^{p+3/2})$$ rate. Finally, we prove that the global effectivity index in the $$L^2$$ -norm converge to unity at $$\mathcal {O}(h^{1/2})$$ rate. Our proofs are valid for arbitrary regular meshes using $$P^p$$ polynomials with $$p\ge 1$$ . Finally, several numerical examples are given to validate the theoretical results.

This paper is concerned with the problem of identifying the diffusion parameters in a nonlinear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. The forward problem is discretized using finite-difference methods and the inverse problem is formulated as a minimization problem with regularization terms. In order to overcome disturbance of local minimum, a wavelet multiscale method is applied to solve this parameter identification inverse problem. This method works by decomposing the inverse problem into multiple scales with wavelet transform so that the original inverse problem is reformulated to a set of sub-inverse problems relying on scale variables, and successively solving these sub-inverse problems according to the size of scale from the smallest to the largest. The stable and fast regularized Gauss–Newton method is applied to each scale. Numerical simulations show that the proposed algorithm is widely convergent, computationally efficient, and has the anti-noise and de-noising abilities.

Purpose This paper aims to provide a well-behaved nonlinear scheme and accelerating iteration for the nonlinear convection diffusion equation with fundamental properties illustrated. Design/methodology/approach A nonlinear finite difference scheme is studied with fully implicit (FI) discretization used to acquire accurate simulation. A Picard–Newton (PN) iteration with a quadratic convergent ratio is designed to realize fast solution. Theoretical analysis is performed using the discrete function analysis technique. By adopting a novel induction hypothesis reasoning technique, the L∞ (H1) convergence of the scheme is proved despite the difficulty because of the combination of conservative diffusion and convection operator. Other properties are established consequently. Furthermore, the algorithm is extended from first-order temporal accuracy to second-order temporal accuracy. Findings Theoretical analysis shows that each of the two FI schemes is stable, its solution exists uniquely and has second-order spatial and first/second-order temporal accuracy. The corresponding PN iteration has the same order of accuracy and quadratic convergent speed. Numerical tests verify the conclusions and demonstrate the high accuracy and efficiency of the algorithms. Remarkable acceleration is gained. Practical implications The numerical method provides theoretical and technical support to accelerate resolving convection diffusion, non-equilibrium radiation diffusion and radiation transport problems. Originality/value The FI schemes and iterations for the convection diffusion problem are proposed with their properties rigorously analyzed. The induction hypothesis reasoning method here differs with those for linearization schemes and is applicable to other nonlinear problems.

Recently, Buchmuller and Helzel proposed a modified dimension-by-dimension finite-volume (FV) WENO method on Cartesian grids for multidimensional nonlinear conservation laws which can retain the full order of accuracy of the underlying one-dimensional (1D) reconstruction. In this work, we extend this method to multidimensional convection–diffusion equations. The 1D sixth-order central reconstruction of the conserved quantity is utilized for discretizing the diffusion terms in which the diffusion coefficients may be nonlinear functions of the conserved quantity. Using high-order accurate conversions between edge-averaged values and edge center values of any sufficiently smooth quantity, high-order accurate convective and viscous numerical fluxes at cell interfaces are computed. The present modified FV method uses fourth-order accurate conversions for the diffusive fluxes. Numerical examples show that the present method achieves fourth-order accuracy for multidimensional smooth problems, and is suitable for the numerical simulation of viscous shocked flows.

The combined effects of nonlinear thermal convection and radiation in 3D boundary layer flow of non-Newtonian nanofluid are scrutinized numerically. The flow is induced by the stretching of a flat plate in two lateral directions. The mechanism of heat and mass transport under thermophoretic and Brownian motion is elaborated via implementation of the thermal convective condition. The prevailing two-point nonlinear boundary value problem is reduced to a two-point ordinary differential problem by employing suitable similarity transformations. The solutions are computed by the implementation of homotopic scheme. At the end, a comprehensive parametric study has been conducted to analyze the typical trend of the solutions. It is found that the nanoparticle volume fraction and temperature profiles are stronger for the case of solar radiation in comparison with problem without radiation. Keywords: Non-linear convection, Nonlinear density temperature (NDT), Nonlinear thermal radiation, Three dimensional flow, Maxwell fluid, Nanoparticles

We investigate the onset of average convection and its nonlinear regimes in a single-component fluid layer overlying a fluid-saturated porous layer. A heated from below cavity with a superposed fluid and a porous medium undergoes high-frequency and small-amplitude vertical vibrations in the gravitational field. Porosity of the medium decreases linearly with depth at a positive porosity gradient. Thermal vibrational convection equations are obtained by the averaging method and solved numerically. The shooting method, Galerkin method and finite-difference method are applied. It is shown that for small vibration accelerations, a convective flow is generated as short-wave rolls in the fluid layer overlying a porous medium. The heat flux undergoes abrupt changes as the supercriticality increases. It is due to the fluid flow penetrating into pores. A magnitude of the jump grows with the growth of vibration intensity. For sufficiently large vibration accelerations, the average convection is excited in the form of long-wave rolls that penetrate both layers. Here, the Nusselt number is 2–3 times higher than its value in the static gravity field.

Main objective of present analysis is to study the magnetohydrodynamic (MHD) nonlinear convective flow of thixotropic nanofluid. Flow is due to nonlinear stretching surface with variable thickness. Nonlinear thermal radiation and heat generation/absorption are utilized in the energy expression. Convective conditions and zero mass flux at sheet are considered. Intention in present analysis is to develop a model for nanomaterial comprising Brownian motion and thermophoresis phenomena. Appropriate transformations are implemented for the conversion of partial differential systems into a sets of ordinary differential equations. The transformed expressions have been scrutinized through homotopic algorithm. Behavior of various sundry variables on velocity, temperature, nanoparticle concentration, skin friction coefficient and local Nusselt number are displayed through graphs. It is concluded that qualitative behaviors of temperature and thermal layer thickness are similar for radiation and temperature ratio variables. Moreover an enhancement in heat generation/absorption show rise to thermal field.

Motivated by the applications from chemical engineering, in this paper we present a rigorous derivation of the effective model describing the convection-diffusion-reaction process in a thin domain. The problem is described by a nonlinear elliptic problem with nonlinearity appearing both in the governing equation as well in the boundary condition. Using rigorous analysis in appropriate functional setting, we show that the starting singular problem posed in a two- dimensional region can be approximated with one which is regular, one-dimensional and captures the effects of all physical processes which are relevant for the original problem.

In this paper, two classes of implicit–explicit (IMEX) multistep finite-element methods are constructed to solve the nonlinear delay convection-diffusion-reaction equations. This equations are discretized with finite-element method in space and IMEX multistep methods (including IMEX one-leg methods and IMEX linear multistep methods) in time. The resulting schemes are efficient because it can reduce the coupling degree, and avoid solving the large-scale systems of nonlinear algebraic equations. The L2-norm error estimates with handling the breaking point and stability results for the proposed methods are obtained. Numerical results demonstrate the efficiency and accuracy of the approaches.

In this paper we study the following Cauchy problem: u t = u x x + ( u n ) x , ( x , t ) ∈ R × ( 0 , ∞ ) , u ( x , 0 ) = 0 , x ≠ 0 , where parameter n ≥ 0 . Its nonnegative solution is called singular solution when u ( x , t ) satisfies the equation in the sense of distribution, initial conditions in the classical sense and also u ( x , t ) exhibits a singularity at the origin ( 0 , 0 ) . As we know, the singular solution is called source-type solution if the initial is M δ ( x ) , where δ ( x ) is Dirac measure and constant M > 0 . The solution is called very singular solution if it possesses more singularity than that of source-type solution at the origin. Here we focus on what happens in the interactive effect between the diffusion and convection in a whole physical process. We find critical values n 2 n 1 n 0 such that there exists unique source-type solution in the exponent range of 0 n n 0 , while there exists no nonnegative singular solution if n ≥ n 0 . Only in the case of n 2 n n 1 there exists a very singular solution, but in the case of n ≥ n 1 or n ≤ n 2 there is no solution that exhibits more singular than source-type solution at the origin. Furthermore we describe the short time asymptotic behavior of the singular solutions when such Cauchy problem is solvability for source-type solution or very singular solution.

In this paper, the meshless Generalized Finite Difference Method (GFDM) in conjunction with the second-order explicit Runge-Kutta method (RK2 method) is presented to solve coupled unsteady nonlinear convection-diffusion equations (CDEs). Compared with the conventional Euler method, the RK2 method not only has higher accuracy but also reduces the possibility of numerical oscillation in time discretization, especially for the nonlinear and coupled cases. The generalized finite difference method, which is a localized collocation method, is famous for its simplicity and adaptability in the numerical solution of partial differential equations. Benefiting from Taylor series and moving least squares, its partial derivatives can be formed by a series of surrounding space points. In comparison with traditional finite difference methods, the proposed GFDM is free of mesh and available for irregular discretization nodes. In this study, the stencil selection algorithms are introduced to choose the stencil support of a certain node from the whole discretization nodes. Error analysis and numerical investigations are presented to demonstrate the effectiveness of the proposed GFDM for solving the coupled linear and nonlinear unsteady convection-diffusion equations. Then it is successfully applied to three benchmark examples of the coupled unsteady nonlinear CDEs encountered in the double-diffusive natural convection process, chemotaxis-haptotaxis model of cancer invasion, and thermo-hygro coupling model of concrete.

A class of nonlinear convection-diffusion equation is studied in this paper. The partial differential equation is converted into nonlinear ordinary differential equation by introducing a similarity transformation. The asymptotic analytical solutions are obtained by using double-parameter transformation perturbation expansion method (DPTPEM). The influences of convection functional coefficientk(z)and power law indexnon the heat transport characteristics are discussed and shown graphically. The comparison with the numerical results is presented and it is found to be in excellent agreement. The method and technique used in this paper have the significance in studying other engineering problems.

By Karamata regular variation theory, we first derived the exact asymptotic behavior of the local solution to the problem -φʹʹ(s) = g(φ(s)), φ(s) > 0, s ∈ (0, a) and φ(0) = 0. Then, by a perturbation method and constructing comparison functions, we derived the exact asymptotic behavior of the unique classical solution near the boundary to a singular Dirichlet problem -Δu = b(x)g(u) + λ|▽u|q, u > 0, x ∈ Ω, u|aΩ = 0, where Ω is a bounded domain with smooth boundary, λ ∈ ℝ, q ∈ [0, 2]; g ∈ C1((0, ∞), (0, ∞)), is decreasing in (0, ∞) with lims→0 + g(s) = +∞; the weight b is positive in Ω and singular on the boundary.

In this paper, a finite-difference lattice Boltzmann (LB) model for nonlinear isotropic and anisotropic convection-diffusion equations is proposed. In this model, the equilibrium distribution function is delicately designed in order to recover the convection-diffusion equation exactly. Different from the standard LB model, the temporal and spatial steps in this model are decoupled such that it is convenient to study convection-diffusion problem with the non-uniform grid. In addition, it also preserves the advantage of standard LB model that the complex-valued convection-diffusion equation can be solved directly. The von Neumann stability analysis is conducted to discuss the stability region which can be used to determine the free parameters appeared in the model. To test the performance of the model, a series of numerical simulations of some classical problems, including the diffusion equation, the nonlinear heat conduction equation, the Sine-Gordon equation, the Gaussian hill problem, the BurgersFisher equation, and the nonlinear Schrdinger equation, have also been carried out. The results show that the present model has a second-order convergence rate in space, and generally it is also more accurate than the standard LB model.

The present article investigates the magnetohydrodynamic (MHD) three-dimensional nonlinear convective flow of an Oldroyd-B nanofluid over a stretching sheet. Heat transfer analysis is reported in the presence of nonlinear thermal radiation and heat generation/absorption. The effects of Brownian motion and thermophoresis are considered in energy and concentration expressions. Meaningful solutions are established for the velocity, temperature and concentration. It is observed that both components of velocity show opposite behavior for mixed convection parameter, ratio of concentration to thermal buoyancy forces and ratio parameter. Local Nusselt and Sherwood numbers are analyzed for the pertinent parameters. Temperature and heat transfer rate are enhanced for thermal radiation and temperature ratio parameters. It is observed that the impact of Brownian motion on the temperature and nanoparticle concentration is quite reverse.

Nonlinear convection–diffusion equations are widely used for the description of various processes and phenomena in physics, mechanics and biology. In this work we consider a family of nonlinear ordinary differential equations which is a traveling wave reduction of a nonlinear convection–diffusion equation with a polynomial source. We study a question about integrability of this family of nonlinear ordinary differential equations. We consider both stationary and non–stationary cases of this equation with and without convection. In order to construct general analytical solutions of equations from this family we use an approach based on nonlocal transformations which generalize the Sundman transformations. We show that in the stationary case without convection the general analytical solution of the considered family of equations can be constructed without any constraints on its parameters and can be expressed via the Weierstrass elliptic function. Since in the general case this solution has a cumbersome form we find some correlations on the parameters which allow us to construct the general solution in the explicit form. We show that in the non–stationary case both with and without convection we can find a general analytical solution of the considered equation only imposing some correlation on the parameters. To this aim we use criteria for the integrability of the Lienard equation which have recently been obtained. We find explicit expressions in terms of exponential and elliptic functions for the corresponding analytical solutions.

The present paper examines the effect of nonlinear convection on the stagnation point flow of an incompressible viscous fluid over a shrinking sheet embedded in porous medium. The governing boundary layer equations are transformed into ordinary differential equations using the similarity transformations. The reduced equations are then solved numerically using the implicit finite difference scheme also known as Keller box method. The physical features of the associated flow parameters are analyzed with the help of graphs and tables. The skin friction and wall temperature gradient are also calculated and discussed. It is found that the solution range increases significantly with nonlinear convection and porous medium parameters.

In this work we prove that for some class of initial data the solution of the Cauchy problem ut = (um)xx + a(um)x + u(1 - um-1), x ∈ ℝ; t > 0 u(0; x) = u0(x), u0(x) ≥ 0 approaches the travelling solution, spreading either to the right or to the left, or two travelling waves moving in opposite directions.