The explicit solution of a diffusion equation with singularity

We give the explicit solution of the Cauchy problem for the diffusion equation with a singular term:

where . We construct the solution on the basis of a generalization of the Fourier transform. We next show that the solution is expressed by an analytic semigroup, and examine smoothness of and continuity of .

     

The uniqueness of the solution to the diffusion equation with a damping term

Consider a non-linear diffusion equation with a damping term. If the diffusion coefficient is positive, then the solutions are not unique generally. However, if the diffusion coefficient degenerates, the situation may change. In this paper, not only the existence of the weak solution is established, but also the uniqueness of the weak solutions is proved, even the boundary value condition is not imposed. The conclusions imply that, on the boundary, the degeneracy of diffusion coefficient can eliminate the action from the damping term.     

Uniqueness of solutions to a viscous diffusion equation

We prove the uniqueness of solutions to the initial boundary value problem of a viscous diffusion equation by means of a regularizing technique based on elliptic operators.     

Stochastic process model for solute transport and the associated transport equation

A mathematical model of Lagrangian motions of a particle in turbulent flows is developed on the basis of a stochastic differential equation. The model expresses uncertainties involved in turbulence by standard Brownian motion. Because the model does not guarantee smoothness of the path of the particle, local velocity is newly defined so as to be suitable for observation of a velocity time series at a fixed point. Then, it is shown that the newly defined local velocity is governed by a Gaussian distribution. In addition, an estimation method of the turbulent diffusion coefficient involved in the model is proposed by using the local velocity. The estimation method does not require tracer experiments. In order to assess the validity of the proposed local velocity, velocity measurements with three-dimensional acoustic Doppler velocimeters were conducted in agricultural drainage canals. Also, the turbulent diffusion coefficient was estimated by the derived time series of the observed local velocity. Finally, a transport equation of conservative solute is derived by using the linearity of the Kolmogorov forward equation without using gradient-type lows.     

On the Cauchy problem for the transport equation with random noise

We establish the existence and uniqueness of a solution to the Cauchy problem for the transport equation with random noise in Rd. When the vector field is time-periodic and the noise is multiplicative and nondegenerate, we show the existence of a time-periodic measure, which includes an invariant measure as a special case.     

Closed form solution of a periodically forced logistic model

We give the exact closed form solution of the following ordinary differential equation:

Analysis of the time fractional nonlinear diffusion equation from diffusion process

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.     

Global well-posedness for the diffusion equation of population genetics

In this paper, we study the forward diffusion equation of population genetics. We prove the global existence of smooth solutions if the initial value is smooth. We also show that if the initial value is singular on the boundary, in a weighted Sobolev space, the diffusion equation exists a unique weak solution which is a probability density function. Moreover, we investigate the asymptotic behavior of the weak solution by the entropy method.     

Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate

A coefficient inverse problem for the non-stationary single-speed transport equation for t∈(0,T) with the lateral boundary data and initial condition at t=0 is considered. Global uniqueness result is obtained by the method of Carleman estimates.     

The equimaterial approach for the numerical solution of the one dimensional transient diffusion equation with zero flux boundary conditions

A new approach is proposed for the grid motion for the numerical solution of a general transient diffusion equation in one spatial dimension with zero flux boundary conditions. The new criterion for grid motion is that the solute amount contained in each discretization section should be a pre-described fraction of the total solute amount at each time step. This requirement is not explicitly enforced to the solution technique but it is implicitly included in the equation through the appropriate variable transformation. The results showed that although the technique leads to the required grid motion the numerical results are of pure quality due to the appearance of singularities during the variable transformation procedure. Nevertheless, it is shown that by appropriate numerical handling of the solution at the singularity region the technique can lead to accurate results and potentially can replace the existing moving grid algorithms at least for the particular problem at hand.     

Optimal control of the multidimensional diffusion equation

The existence and numerical estimation of a boundary control for then-dimensional linear diffusion equation is considered. The problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures. The existence of an optimal measure corresponding to the above problem is shown, and the optimal measure is approximated by a finite convex combination of atomic measures. This construction gives rise to a finite-dimensional linear programming problem, whose solution can be used to construct the combination of atomic measures, and thus a piecewise-constant control function which approximates the action of the optimal measure, so that the final state corresponding to the above control function is close to the desired final state, and the value it assigns to the performance criterion is close to the corresponding infimum. A numerical procedure is developed for the estimation of these controls, entailing the solution of large, finite-dimensional linear programming problems. This procedure is illustrated by several examples.     

Numerical solution of a nonlinear reaction diffusion equation

In this paper, the authors propose a numerical method to compute the solution of a Cauchy problem with blow-up of the solution. The problem is split in two parts: a hyperbolic problem which is solved by using Hopf and Lax formula and a parabolic problem solved by a backward linearized Euler method in time and a finite element method in space. It is proved that the numerical solution blows up in a finite time as the exact solution and the support of the approximation of a self-similar solution remains bounded. The convergence of the scheme is obtained.     

A non-separable solution of the diffusion equation based on the Galerkin’s method using cubic splines

The two dimensional diffusion equation of the form is considered in this paper. We try a bi-cubic spline function of the form as its solution. The initial coefficients C_i,j(0) are computed simply by applying a collocation method; C_i,j = f(x_i, y_j) where f(x, y) = u(x, y, 0) is the given initial condition. Then the coefficients C_i,j(t) are computed by X(t) = e^tQX(0) where X(t) = (C_0,1, C_0,1, C_0,2, … , C_0,N, C_1,0, … , C_N,N) is a one dimensional array and the square matrix Q is derived from applying the Galerkin’s method to the diffusion equation. Note that this expression provides a solution that is not necessarily separable in space coordinates x, y. The results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included.     

Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique

Much of the literatures are directed toward the development of a mathematical formalism for a rigorous estimation of the ensemble average of the solution process of a stochastic differential equation (SDE). The Random Variable Transformation technique (RVT) is a powerful technique to get the complete solution for the SDE represented by the probability-density function of the solution process. In this paper, the RVT technique together with a simple integral transformation to the input stochastic process are implemented to get the complete solution of the one-speed transport equation for neutral particles in a semi-infinite stochastic medium with linear anisotropic scattering. The extinction function of the medium (input stochastic process) is assumed to be a continuous random function of position. The probability-density function and hence, the higher order statistical moments of the solution process are presented. Numerical results are given for different distributions of the input stochastic process.     

Hypercyclicity in the scattering theory for linear transport equation

We show how the hypercyclicity of the transport semigroup can intervene in the scattering theory to characterize the density property of the Lax and Phillips representation theorem and conversely, how the existence of the wave operators of the scattering theory can be used for recovering the hypercyclicity of the absorbing transport group in some weighted spaces.

     

A free boundary problem of diffusion equation with integral condition

We consider a free boundary problem of heat equation with integral condition on the unknown free boundary. Results of solution regularity and problem well-posedness are presented.     

Symmetry analysis of the two-dimensional diffusion equation with a source term

A symmetry analysis is performed on a (2+1)-dimensional linear diffusion equation with a nonlinear source term involving the dependent variable and its spatial derivatives. In the first part of the paper, we use the classical method to classify source terms where the original equation admits a nontrivial symmetry. In the second part of the paper, we use the nonclassical method and show that we simply recover the classical symmetries.     

非定常BOLTZMANN方程的谱有限元方法

In this paper,a new numerical method,the coupling method of spherical harmonic function spectral and finite elements,for a unsteady transport equation is dlscussed,and the error analysis of this scheme is proved.     

Solving diffusion equation using wavelet method

A wavelet-based numerical method based on wavelets of Hermite cubic splines is presented for computing singularly perturbed convection-dominated diffusion equation. The advantages of the method are explained. To improve the accuracy of singular areas, wavelets are configured hierarchically for solving algebraic equations. Numerical examples show that the proposed method has good efficiency and precision.