This paper proposes a new approach for image noise reduction using partial Diffusion equation (PDE). Diffusion coefficient is an important parameter in PDE for image noise reduction. This parameter affects the noise reduction results and quality of edges in the denoised image. The existing PDE-based image denoising techniques experimentally adjust the Diffusion coefficient. This paper proposes a new approach to adaptively adjust the Diffusion coefficient. The proposed approach was applied on a number of standard images to evaluate its performance. The results indicate that the proposed approach outperform the existing PDE-based image denoisng techniques.

We examine the Diffusion equation on the sphere. In this sense, we answer the question of the symmetry classification. We provide the algebra of symmetry and build the optimal system of Lie subalgebras. We prove for one-dimensional optimal systems of Eq. (1. 4), space is expanding Ricci solitons. Reductions of similarities related to subalgebras are classified, and some exact invariant solutions of the Diffusion equation on the sphere are presented.

The purpose of this paper is to analyze in detail a special nonlinear partial differential equation (nPDE) of the second order which is important in physical, chemical and technical applications. The present nPDE describes nonlinear Diffusion and is of interest in several parts of physics, chemistry and engineering problems alike. Since nature is not linear intrinsically the nonlinear case is therefore the general. We determine the classical Lie point symmetries including algebraic properties whereas similarity solutions are given as well as nonlinear transformations could derived. In addition, we discuss the nonclassical case which seems to be not solvable. Moreover we show how one can deduce approximate symmetries modeling the nonlinear part and we deduce new generalized symmetries of lower symmetry. The analysis allows one to deduce wider classes of solutions either of practical and theoretical usage in different domains of science and engineering.

Nowadays, network structures are found in many natural and engineered systems, e.g., river networks, microchannel networks, plant roots, human blood vessels, etc. Therefore, providing efficient methods for modeling phenomena such as Diffusion, advection, etc. is very practical. One of the most common tools for modeling this phenomena is numerical modeling, as mathematics software is well- developed and powerful nowadays. In this research, a new approach called equation-Oriented Modeling has been presented. In this approach each branch of the network has its own differential equation, and these branches are connected or coupled by boundary conditions. In other words, unlike classical modeling, EOM does not solve through the discretization of the partial differential equation in the whole domain of the network, while in this approach, each branch of the network has its own differential equation with its own specific Diffusion coefficient and cross section area, then the problem is solved as a system of PDE. The main point of EOM is to formulate a physical problem in the network into a system of differential equations, which is finally solved by the Method of Lines. MOL is an efficient computational method used to solve partial differential equations or PDE systems. MOL is generally implemented in two steps, in the first step spatial derivatives are replaced by algebraic approximation. In the second step, the ordinary differential equation system is integrated with respect to time using any method, for example, in this research, we use the Runge-Kutta 4th order method. EOM was implemented to solve the Diffusion equation in three types of networks, including tree-shaped and loop network. Then modeling results for 3 networks were presented as spatial concentration profiles in different paths in the networks. The model had reasonable results in the boundaries and branches according to the boundary conditions, loading and concentration functions, as well as the continuity of concentrations and loading by Diffusion in the output results was reasonable. The boundary conditions that apply at the intersections of the branches include the continuity of concentration and the continuity of loading due to the Diffusion phenomenon. The results of test case 3 were compared with another numerical model for validation, and three types of Error Parameters were calculated at different times between these two models. R-Squared (R2) was calculated in path (1-2-3-5-9), and its value was 0.99-1, which was the optimal value. This coefficient shows that the results of the EOM and the other numerical model has the same trend. Then, RMSE and MAE were also calculated and their values were approximately zero for all times. The modeling results for 3 networks were presented as spatial concentration profiles in different paths in the networks. The first advantage of the EOM approach is that the choice of terms in the differential equation is left to the user rather than the software developer, so that a wider range of phenomena can be modeled and the effects of different terms can be seen in the modeling. The second advantage of this approach over classical modeling is that the equations are available to the user as tools and model elements, and modeling complex networks such as tree-shaped, and Loop networks is not as complicated as classical models. The third advantage of EOM is the tools available in mathematical programs for optimization or linking with other programs. Since the heat equation is similar to the Diffusion equation, the results of this research can be used for other important topics, such as solving the heat equation in microchannel networks for cooling systems, modeling pollutant transport in river networks, or Diffusion modeling of solutes in plant roots.

Multigrid methods, such as the Algebraic MultiGrid methods (AMG), are among the most efficient methods for solving discrete linear equations resulting from elliptic PDEs. In this paper, some classic AMG techniques are used for solving a typical two dimensional Diffusion equation. The present work focuses on different coarsening techniques such as the standard and aggressive coarsening based on the RS algorithm. Results indicate that the standard coarsening technique together with direct interpolation minimizes the CPU-time and the computational work. Regarding the memory requirement, using a variant of aggressive coarsening with direct interpolation minimizes the size of memory usage. In addition, the setup phase time (coarsening process) is minimum in this technique. Nevertheless, the increase in convergence factor results in higher overall time for the solution phase. Analysis of the aspect ratio of the computational cells shows that the convergence rate is affected little by this parameter. However, the coarsening process work and the total CPU-time are slightly increased when cells with high aspect ratios are used for the numerical simulations. Overall, the standard-coarsening method with direct interpolation is found to be appropriate for two-dimensional Diffusion problems.

We find a solution of an unknown time-dependent diffusivity a(t) in a linear inverse parabolic problem by a modified genetic algorithm. At first, it is shown that under certain conditions of data, there exists at least one solution for unknown a(t) in (a(t), T (x, t)), which is a solution to the corresponding problem. Then, an optimal estimation for unknown a(t) is found by applying the least-squares method and a modified genetic algo rithm. Results show that an excellent estimation can be obtained by the implementation of a modified real-valued genetic algorithm within an Intel Pentium (R) dual-core CPU with a clock speed of 2.4 GHz.

In this paper, we study the reaction-Diffusion equation $\Delta u(x) = |u(x)|^{\gamma(x)-1} u(x) $ from a regularity point of view. This equation is used for modelling the distribution of a gas in an inhomogeneous porous catalyst. And the power $\gamma(x)$ can be a discontinuous function. In particular, we study the vanishing order of solution near the zero level set $\{u=0\}$.

The Riesz fractional advection-Diffusion is a result of the mechanics of chaotic dynamics. It’, s of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [−, 1, 1] × [−, 1, 1]. Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical solution. Examples show the effectiveness of the method.