In this paper we consider the following quasi-linear parabolic equations Lu = u t ā div A(x, t, u, u x) +B(x, t, u, u x = 0 , where ā¦
In this paper we consider the following quasi-linear parabolic equations Lu = u t ā div A(x, t, u, u x) +B(x, t, u, u x = 0 , where A is a given vector function of the variables x,t,u,u X , and B is a given scalar function of the some variables. We assume that they are difined in the rectangle
Aim of this paper is the qualitative analysis of the solution of a boundary value problem for a third-order non linear parabolic equation which describes several dissipative models. When the ā¦
Aim of this paper is the qualitative analysis of the solution of a boundary value problem for a third-order non linear parabolic equation which describes several dissipative models. When the source term is linear, the problem is explictly solved by means of a Fourier series with properties of rapid convergence. In the non linear case,appropriate estimates of this series allow to deduce the asymptotic behaviour of the solution.
Aim of this paper is the qualitative analysis of the solution of a boundary value problem for a third-order non linear parabolic equation which describes several dissipative models. When the ā¦
Aim of this paper is the qualitative analysis of the solution of a boundary value problem for a third-order non linear parabolic equation which describes several dissipative models. When the source term is linear, the problem is explictly solved by means of a Fourier series with properties of rapid convergence. In the non linear case,appropriate estimates of this series allow to deduce the asymptotic behaviour of the solution.
Aim of this paper is the qualitative analysis of the solution of a boundary value problem for a third-order non linear parabolic equation which describes several dissipative models. When the ā¦
Aim of this paper is the qualitative analysis of the solution of a boundary value problem for a third-order non linear parabolic equation which describes several dissipative models. When the source term is linear, the problem is explictly solved by means of a Fourier series with properties of rapid convergence. In the non linear case,appropriate estimates of this series allow to deduce the asymptotic behaviour of the solution.
A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a ā¦
A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions.
In this paper, we study the stability property and asymptotic behavior for a quasi-linear parabolic flow in the whole line. We first show the existence and uniqueness of global solutions ā¦
In this paper, we study the stability property and asymptotic behavior for a quasi-linear parabolic flow in the whole line. We first show the existence and uniqueness of global solutions of the problem. Then we study the stability of the solution to the straight line. We prove the asymptotic behavior or the convergence of the global solution. Similar to the behavior of solutions to heat equation, we prove that the stationary line attracts the graphical curves which surround it.
where $Q_{T}\equiv\{(x, t):0<x<1,0<t<T\}$ .Assume that the following conditions for $a$ and $f$ are satisfied: (i) for any finite $M>0,$ $a(x, z)\in C^{1}([0,1]\times[-M, M])$ , (ii) for fixed, positive constantsv and ā¦
where $Q_{T}\equiv\{(x, t):0<x<1,0<t<T\}$ .Assume that the following conditions for $a$ and $f$ are satisfied: (i) for any finite $M>0,$ $a(x, z)\in C^{1}([0,1]\times[-M, M])$ , (ii) for fixed, positive constantsv and $\mu,$ $ 0<v\leq a(x, z)\leq\mu$ on $[0,1]\times[-M, M]$ , (iii) for fixed, positive constant $C,$ $|\partial_{x}a(x, z)|+|\partial_{z}a(x, z)|\leq C$ on $From the conditions $(i)-(v)$ applying Theorem 5.2 and Remark 5.1 in [5], we see that there exists a unique solution $u(x, t)\in H^{2+\beta,1+\beta}([0,1]\times[0, T])$ to the initial boundary value problem $(1.1)-(1.3)$ .So we may define D-N map as follows:$\Lambda(a, f):u(O, t)=f(t)\mapsto u_{x}(0, t)$ on $[0, T]$ .We are interested in uniqueness results for $a(x, u)$ of the equation (1.1) from $\Lambda(a, f)$ .Isakov [4] proved the uniqueness for $a(x, u)$ in the case that the spatial dimension is greater than or equal to 2 by using the completeness of products of solutions for linear parabolic equations.But in the case that the spatial dimension is one, the completeness of products of solutions has not been proved yet.So we need another method for proving the uniqueness for $a(x, u)$ .In [1] it was shown that the coefficient $\kappa$ of the equation $a(u)u_{t}=\kappa(a(u)u_{x})_{x}$ was uniquely determined from overspecified boundary data
Aļ¢ļ³ļ“ļ²ļ”ļ£ļ“.Our aim is to examine the nonlinear parabolic differential equation u xxg(t, x) f (u t , u x ) = 0. We present three examples for the solution of ā¦
Aļ¢ļ³ļ“ļ²ļ”ļ£ļ“.Our aim is to examine the nonlinear parabolic differential equation u xxg(t, x) f (u t , u x ) = 0. We present three examples for the solution of the equation of some special forms.A maximum principle and some uniqueness results are given.Moreover, the approximate solution of the equation with g(t, x) = 1, obtained by the difference method is investigated.
A class of quasi -linear parabolic eq uations is discussed.The existence and uniqueness of global clas-sical solutions are obtained to this problem.
A class of quasi -linear parabolic eq uations is discussed.The existence and uniqueness of global clas-sical solutions are obtained to this problem.
In this thesis, we study the pseudo-parabolic equation [] where is a continuous function satisfying [] for some constants are constants and is nontrivial, nonnegative as well as has a ā¦
In this thesis, we study the pseudo-parabolic equation [] where is a continuous function satisfying [] for some constants are constants and is nontrivial, nonnegative as well as has a compact support. The characteristics of solutions depend on the value of and the initial condition . The nonnegative classical solution blows up in a finite time when [] regardless of the initial condition.
Abstract A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of ā¦
Abstract A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions.
is considered. This is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but one side is inaccessible to measurements. ā¦
is considered. This is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but one side is inaccessible to measurements. This problem is well known to be severely ill-posed: a small perturbation in the data, g, may cause dramatically large errors in the solution. The results available in the literature are mainly devoted to the case of constant coefficients, where one can find an explicit representation for the solution of the problem. In this paper a stability estimate of the Hƶlder type for the solution of this general problem is established, it is also shown how to apply the mollification method recently proposed by Dinh Nho HƔo to solve the problem in a stable way.
Abstract We study the homogeneous Dirichlet problem for the fully nonlinear equation u_{t}=|\Delta u|^{m-2}\Delta u-d|u|^{\sigma-2}u+f\quad\text{in ${Q_{T}=\Omega% \times(0,T)}$,} with the parameters {m>1} , {\sigma>1} and {d\geq 0} . At the points ā¦
Abstract We study the homogeneous Dirichlet problem for the fully nonlinear equation u_{t}=|\Delta u|^{m-2}\Delta u-d|u|^{\sigma-2}u+f\quad\text{in ${Q_{T}=\Omega% \times(0,T)}$,} with the parameters {m>1} , {\sigma>1} and {d\geq 0} . At the points where {\Delta u=0} , the equation degenerates if {m>2} , or becomes singular if {m\in(1,2)} . We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as {t\to\infty} . Sufficient conditions for exponential or power decay of {\|\nabla u(t)\|_{2,\Omega}} are derived. It is proved that for certain ranges of the exponents m and Ļ, every strong solution vanishes in a finite time.
Introduction Preliminaries Green's functions for systems with constant coefficients Green's function for systems linearized along shock profiles Estimates on Green's function Estimates on crossing of initial layer Estimates on truncation ā¦
Introduction Preliminaries Green's functions for systems with constant coefficients Green's function for systems linearized along shock profiles Estimates on Green's function Estimates on crossing of initial layer Estimates on truncation error Energy type estimates Wave interaction Stability analysis Application to magnetohydrodynamics Bibliography
In this paper, we consider the stochastic Burgers' equation, which is forced by multiplicative noise in the Stratonovich sense. To get a new trigonometric and hyperbolic stochastic solutions, we apply ā¦
In this paper, we consider the stochastic Burgers' equation, which is forced by multiplicative noise in the Stratonovich sense. To get a new trigonometric and hyperbolic stochastic solutions, we apply expā”(āĻ(Ī¼))-expansion method. In addition, we demonstrate the effect of multiplicative noise on the exact solutions of the Burgers' equation by introducing some graphic representations.
We study directed polymers subject to a quenched random potential in d transversal dimensions. This system is closely related to the Kardar-Parisi-Zhang equation of nonlinear stochastic growth. By a careful ā¦
We study directed polymers subject to a quenched random potential in d transversal dimensions. This system is closely related to the Kardar-Parisi-Zhang equation of nonlinear stochastic growth. By a careful analysis of the perturbation theory we show that physical quantities develop singular behavior for d to 4. For example, the universal finite size amplitude of the free energy at the roughening transition is proportional to (4-d)^(1/2). This shows that the dimension d=4 plays a special role for this system and points towards d=4 as the upper critical dimension of the Kardar-Parisi-Zhang problem.
Abstract In this work, we design, analyze, and test the multiwavelets Galerkin method to solve the twoādimensional Burgers equation. Using CrankāNicolson scheme, time is discretized and a PDE is obtained ā¦
Abstract In this work, we design, analyze, and test the multiwavelets Galerkin method to solve the twoādimensional Burgers equation. Using CrankāNicolson scheme, time is discretized and a PDE is obtained for each time step. We use the multiwavelets Galerkin method for solving these PDEs. Multiwavelets Galerkin method reduces these PDEs to sparse systems of algebraic equations. The cost of this method is proportional to the number of nonzero coefficients at each time step. The results illustrate, by selecting the appropriate threshold while the number of nonzero coefficients reduces, the error will not be less than a certain amount. The L 2 stability and convergence of the scheme have been investigated by the energy method. Illustrative examples are provided to verify the efficiency and applicability of the proposed method.
This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In ā¦
This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.
In this paper, we use the natural decomposition method (NDM) for solving inviscid Burger equation (BE). The NDM is associated with the Adomain decomposition method (ADM) and the natural transform ā¦
In this paper, we use the natural decomposition method (NDM) for solving inviscid Burger equation (BE). The NDM is associated with the Adomain decomposition method (ADM) and the natural transform method. Applying the analytic method, we solved successfully both lin-ear and non-linear partial differential equations. By applying the NDM, we compute the best approximation solution of linear and non-linear par-tial differential equations. In our experiments, we report comparisons with the exact solution.
We present a mean-field theory for the dynamics of driven flow with exclusion in graphenelike structures, and numerically check its predictions. We treat first a specific combination of bond transmissivity ā¦
We present a mean-field theory for the dynamics of driven flow with exclusion in graphenelike structures, and numerically check its predictions. We treat first a specific combination of bond transmissivity rates, where mean field predicts, and numerics to a large extent confirms, that the sublattice structure characteristic of honeycomb networks becomes irrelevant. Dynamics, in the various regions of the phase diagram set by open boundary injection and ejection rates, is then in general identical to that of one-dimensional systems, although some discrepancies remain between mean-field theory and numerical results, in similar ways for both geometries. However, at the critical point for which the characteristic exponent is z = 3/2 in one dimension, the mean-field value z = 2 is approached for very large systems with constant (finite) aspect ratio. We also treat a second combination of bond (and boundary) rates where, more typically, sublattice distinction persists. For the two rate combinations, in continuum or late-time limits, respectively, the coupled sets of mean-field dynamical equations become tractable with various techniques and give a two-band spectrum, gapless in the critical phase. While for the second rate combination quantitative discrepancies between mean-field theory and simulations increase for most properties and boundary rates investigated, theory still is qualitatively correct in general, and gives a fairly good quantitative account of features such as the late-time evolution of density profile differences from their steady-state values.
From particular polynomials, we construct rational solutions to the Burgers' equation as a quotient of a polynomial of degree n -1 in x and n -1 -n 2 in t, ā¦
From particular polynomials, we construct rational solutions to the Burgers' equation as a quotient of a polynomial of degree n -1 in x and n -1 -n 2 in t, by a polynomial of degree n in x and n 2 in t, |n| being the greater integer less or equal to n.We call these solutions, solutions of order n.We construct explicitly these solutions for orders 1 until 20.
In this research, the Differential Transformation Method (DTM) has been utilized to solve the hyperbolic Telegraph equation. This method can be used to obtain the exact solutions of this equation. ā¦
In this research, the Differential Transformation Method (DTM) has been utilized to solve the hyperbolic Telegraph equation. This method can be used to obtain the exact solutions of this equation. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method. Mathematical subject classification: 35Lxx, 35Qxx.
This paper examines the properties of a regularization of Burgers equation in one and multiple dimensions using a filtered convective velocity, which we have dubbed as convectively filtered Burgers (CFB) ā¦
This paper examines the properties of a regularization of Burgers equation in one and multiple dimensions using a filtered convective velocity, which we have dubbed as convectively filtered Burgers (CFB) equation. A physical motivation behind the filtering technique is presented. An existence and uniqueness theorem for multiple dimensions and a general class of filters is proven. Multiple invariants of motion are found for the CFB equation and are compared with those found in viscous and inviscid Burgers equation. Traveling wave solutions are found for a general class of filters and are shown to converge to weak solutions of inviscid Burgers equation with the correct wave speed. Accurate numerical simulations are conducted in 1D and 2D cases where the shock behavior, shock thickness, and kinetic energy decay are examined. Energy spectrum are also examined and are shown to be related to the smoothness of the solutions.
Abstract In this paper, we focus on the wellāposedness, blowāup phenomena, and continuity of the dataātoāsolution map of the Cauchy problem for a twoācomponent higher order CamassaāHolm (CH) system. The ā¦
Abstract In this paper, we focus on the wellāposedness, blowāup phenomena, and continuity of the dataātoāsolution map of the Cauchy problem for a twoācomponent higher order CamassaāHolm (CH) system. The local wellāposedness is established in Besov spaces with , which improves the local wellāposedness result proved before in Tang and Liu [Z. Angew. Math. Phys. 66 (2015), 1559ā1580], Ye and Yin [arXiv preprint arXiv:2109.00948 (2021)], Zhang and Li [Nonlinear Anal. Real World Appl. 35 (2017), 414ā440], and Zhou [Math. Nachr. 291 (2018), no. 10, 1595ā1619]. Next, we consider the continuity of the solutionātoādata map, that is, the illāposedness is derived in Besov space with and . Then, the nonuniform continuous and Hƶlder continuous dependence on initial data for this system are also presented in Besov spaces with and . Finally, the precise blowāup criteria for the strong solutions of the twoācomponent higher order CH system is determined in the lowest Sobolev space with , which improves the blowāup criteria result established before in He and Yin [Discrete Contin. Dyn. Syst. 37 (2016), no. 3, 1509ā1537] and Zhou [Math. Nachr. 291 (2018), no. 10, 1595ā1619].
Abstract The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function ā¦
Abstract The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation ; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.
Physical applications involving time-fractional derivatives are reflecting some memory characteristics. These inherited memories have been identified as a homotopy mapping of the fractional-solution into the integer-solution preserving its physical shapes. ā¦
Physical applications involving time-fractional derivatives are reflecting some memory characteristics. These inherited memories have been identified as a homotopy mapping of the fractional-solution into the integer-solution preserving its physical shapes. The aim of the current work is threefold. First, we present a new technique which is constructed by combining the Laplace transform tool with the residual power series method. Precisely, we provide the details of implementing the proposed method to treat time-fractional nonlinear problems. Second, we test the validity and the efficiency of the method on the temporal-fractional NewellāWhiteheadāSegel model. Then, we implement this new methodology to study the temporal-fractional (1+1)-dimensional Burger's equation and the DrinfeldāSokolovāWilson system. Further, for accuracy and reliability purposes, we compare our findings with other methods being used in the literature. Finally, we provide 2-D and 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.
Abstract In this article, we use the weak Galerkin (WG) finite element method to study a class of time fractional generalized Burgers' equation. The existence of numerical solutions and the ā¦
Abstract In this article, we use the weak Galerkin (WG) finite element method to study a class of time fractional generalized Burgers' equation. The existence of numerical solutions and the stability of fully discrete scheme are proved. Meanwhile, by applying the energy method, an optimal order error estimate in discrete L 2 norm is established. Numerical experiments are presented to validate the theoretical analysis.
Numerical solutions for Burgersā equation based on the Galerkinsā method using cubic Bāsplines as both weight and interpolation functions are set up. It is shown that this method is capable ā¦
Numerical solutions for Burgersā equation based on the Galerkinsā method using cubic Bāsplines as both weight and interpolation functions are set up. It is shown that this method is capable of solving Burgersā equation accurately for values of viscosity ranging from very small to large. Three standard problems are used to validate the proposed algorithm. A linear stability analysis shows that a numerical scheme based on a CranckāNicolson approximation in time is unconditionally stable.
In the present work, a numerical scheme is constructed for approximation of time fractional Black-Scholes model governing European options.The present numerical scheme has the capability to overcome spurious oscillation in ā¦
In the present work, a numerical scheme is constructed for approximation of time fractional Black-Scholes model governing European options.The present numerical scheme has the capability to overcome spurious oscillation in the case of volatility.In the present numerical method, the Laplace transform, radial kernels and quadrature rule are used.The time variable is eliminated by the use of Laplace transform which significantly reduced the computational cost as compared to the time-marching schemes.The spatial operator is discretized using radial kernels in the local setting which results in sparse differentiation matrices.By Laplace transform the solution is represented as integral along a smooth contour in the complex plane which is then evaluated by quadrature.The proposed numerical scheme is used to price several different European options.
We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible ā¦
We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems. We also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials. To construct conservation laws, we develop and apply the most direct method which is effective to use in the case of two independent variables. Admitting possibility of dependence of conserved vectors on a number of potentials, we generalize the iteration procedure proposed by Bluman and Doran-Wu for finding nonlocal (potential) conservation laws. As an example, we completely classify potential conservation laws (including arbitrary order local ones) of diffusion-convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations.
This work applies a novel analytical technique to the fractional view analysis of coupled Burgers equations. The proposed problems have been fractionally analyzed in the Caputo-Fabrizio sense. The Yang transformation ā¦
This work applies a novel analytical technique to the fractional view analysis of coupled Burgers equations. The proposed problems have been fractionally analyzed in the Caputo-Fabrizio sense. The Yang transformation was initially applied to the specified problem in the current approach. The series form solution is then obtained using the Adomian decomposition technique. The desired analytical solution is obtained after performing the inverse transform. Specific examples of fractional Burgers couple systems are used to validate the proposed technique. The current strategy has been found to be a useful methodology with a close match to actual solutions. The proposed method offers a lower computing cost and a faster convergence rate. As a result, the suggested technique can be applied to a variety of fractional order problems.
The present study was suggested by several problems and difficulties that had appeared in previous experimental and theoretical investigations of viscosity effects in compressible fluids. The outstanding problem was the ā¦
The present study was suggested by several problems and difficulties that had appeared in previous experimental and theoretical investigations of viscosity effects in compressible fluids. The outstanding problem was the extension of the classical (Prandtl) boundary-layer theory to high-speed flow, especially supersonic flow. In the boundary-layer theory the equations of motion are simplified by assuming that viscous effects are confined to a narrow region close by the wall through which changes are rapid compared to those in the direction of the wall. Then the resulting non-linear equations are studied with the aim of obtaining the flow field in this narrow region or boundary layer. The pressure is usually obtained from the potential or no-viscous flow about the body. Several authors have studied a boundary-layer theory which has the same basic assumptions but which allows for compressibility and heat conduction. However, in supersonic flow several phenomena are known which show that the basic assumptions of boundary-layer theory do not apply, at least in certain regions.