This last part is really the core of the work, where all the tools developed in Parts I and II are combined to prove the main results. The natural setting is provided by a Polish topological space (X, τ ) endowed with a σ-finite reference Borel measure m and a strongly local symmetric Dirichlet form E in L 2 (X, m) enjoying a Carr´e du Champ Γ : D( E ) × D( E ) → L 1 (X, m) and a Γ-calculus. All the estimates about the Bakry- ´ Emery condition discussed in Section 10 and the action estimates for nonlinear **diffusion** **equations** provided in Section 11 do not really need an underlying compatible metric structure. In any case, in Section 12, they will be applied to the case of the Cheeger energy (thus assumed to be quadratic) of the metric measure space (X, d , m) in order to prove the main results of the paper. Let us now discuss in more detail the content of Part III.

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**diffusion** equation in term of the Caputo-Fabrizio time fractional derivative. In [10], Hristov proposes new **equations** related to the fractional **diffusion** **equations** using the Atangana-Baleanu fractional derivative, see others models in [23]. In [1], Alkahtani and Atangana discuss the numerical solution of the Cattaneo-Hristov **diffusion** equation. In [26] Koca et al. propose the numerical solution of the second term of the Cattaneo- Hristov **diffusion** equation. In [27], Li et al. have studied the Cauchy problem for nonlinear fractional time-space generalized Keller-Segel equation using the Caputo fractional derivative. In [29], Yranli et al. devoted to comparing the smoothing performance between the time fractional **diffusion** equation and the classical **diffusion** equation using the regulation method, Savitzky-Golay, and coverer method. In [37], Ruan et al. study a simultaneous identification problem of piecewise source term and the fractional order for time-fractional **diffusion** equation. In [45], Zhang and al. propose a discrete form for solving time fractional convection-**diffusion** equation. In [31], Ma et al. study asymptotic of the solutions to the fractional anomalous **diffusion** **equations**. Several works related to the fractional **diffusion** **equations** exist in the literature. The papers [5] [33] [39] [44] treat on fractional **diffusion** **equations**.

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Unfortunately, neither of these two approaches possesses all the properties given in the background introduction of this paper. Both of them are not consistent on non-uniform meshes, and neither of them can satisfy discrete maximum principle (DMP) without imposing severe restrictions on both meshes and problem coefficients, as shown in chapter 4. A detailed accuracy and consistency analysis of these methods is presented in chapter 4. Indeed, a second order cell-centered grid-transparent finite volume scheme on non-uniform and highly distorted grids for **diffusion** **equations** is still at large.

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Control and preventing inflation is a focus goal in the world. It is also an important task of macro-control in China in 2008 and 2011. Now, the price problem is widely concerned in China. However, the solutions of the problem are more on practice monitoring and control (investigation, statistics, monitoring, etc.), rather than on theoretical study. As the theoretical research, focuses are more on qualitative analysis, or on statistic probability rather than on quantitative analysis. The core of solutions is on policy making, price reform, price system, and price forming, so far no paper on price **diffusion** is found in literature. This paper extends the author’s paper [1], in which the **diffusion** of price changing is quantitatively analyzed by a partial differential equation based on ana- logy of heat **diffusion** equation explained by Newton’s second law via “time-space exchanging”, to a fractal and fractional **diffusion** **equations**.

Hurst exponents H≠1/2 are perfectly consistent with Markov processes and the EMH. A Hurst exponent, taken alone, tells us nothing about autocorrelations. Scaling solutions with arbitrary Hurst exponents H can be reduced for Markov processes to a single integration. A truly nonlinear **diffusion** equation has no underlying Langevin description. Any nonlinear **diffusion** equation with a Langevin description is a linear Fokker-Planck equation in disguised form. The Tsallis model is Markovian, does not describe fractional Brownian motion. A Hurst exponent H≠1/2 in a Markov process x(t) describes nonstationary increments, not autocorrelations in x(t).

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[4] Gantulga Tsedendorjand Hiroshi Isshiki (2017) Numerical Solution of Two-Dimensional Advection–**Diffusion** Equation Using Generalized Integral Representation Method. International Journal of Computational MethodsVol. 14, No. 01. [5] Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied, Ayman Marrouf (2018) Analytical Solution of Time Dependent **Diffusion** Equation in Stable Case. American Journal of Environmental Science and Engineering; 2 (2): 32-36. [6] Chatterjee, A. and Singh, M. K. (2018) Two-dimensional

This ODE has a unique solution. Roughly speaking this means that the solution to the original reaction **diffusion** equation (1.2) behaves for large t as a wave U (x − t √ 2ˆ cD) and it can be characterized by its shape U (x) and the speed √ 2ˆ cD. Freidlin ([25], [24], [26], [27]) considered generalized KPP equation where, in the reaction-**diffusion** (1.2), he considered more general nonlinear term instead of ˆ cu(1 − u). He defined the asymptotic wave speed α to be the number such that: for any h > 0, lim t→∞ sup x>(α+h)t u(t, x) = 0

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impossibility of simultaneously measuring canonical variables) distribution functions have proven [14,18] very useful in quantum mechanical systems as they provide insights into the connection between classical and quantum mechanics allowing one to express quantum mechanical averages in a form which is very similar to that of classical averages. Thus they are ideally suited to the study of the quantum-classical correspondence as is well illustrated by the following remarks of Baker [17] and Stratonovich [19]; Baker: “The large-scale experimental validity of classical mechanics tells us that quantum theory must, in some sense, correspond closely to classical mechanics. We have altered the classical concept of a moving point in phase space to that of a quasi-probability distribution which changes in time. This distribution is imagined to be concentrated about the classical point, so that a crude measurement will be unable to differentiate between the two theories. To ensure this correspondence, we use the statement which actually seems to be given by experiments – on the average, Hamilton’s canonical **equations** hold”; Stratonovich: “The statistical nature of quantum theory is manifested in the process of physical “measurement.” From this it follows that the “pre-observation” state of a quantum system, which exists before the “measurement” and is independent of it (one can speak of such a state, to be understood in a definite sense), is a statistical state.

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In this paper, we consider the dynamics of a reaction–diﬀusion equation with fading memory and nonlinearity satisfying arbitrary polynomial growth condition. Firstly, we prove a criterion in a general setting as an alternative method (or technique) to the existence of the bi-spaces attractors for the nonlinear evolutionary **equations** (see Theorem 2.14). Secondly, we prove the asymptotic compactness of the semigroup on L 2 ( Ω ) × L 2 μ (R; H 1 0 ( Ω )) by using the contractive function, and the global attractor is

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In recent years, the evolution **equations** on noncylindrical domains, that is, spatial domains which vary in time so their Cartesian products with the time variable are noncylindrical sets, have been investigated extensively see, e.g., 1–3. Much of the progress has been made for nested spatial domains which expand in time. However, the results focus mainly on formulation of the problems and existence and uniqueness theory, while the existence of attractors of such systems has been less considered, except some recent works for the reaction-diﬀusion equation or the heat equation 4, 5. This is not really surprising since such systems are intrinsically nonautonomous even if the **equations** themselves contain no time-dependent terms and require the concept of a nonautonomous attractor, which has only been introduced in recent years.

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Let us start our discussion by presenting the approach used here to investigate the electrical response of the ionic solutions obtained from the salts previously mentioned. It is based on the fractional **diffusion** equation and its connection with equivalent circuits with CPE elements. In this regard, it is interesting to mention that, the presence of these elements depend on the boundary conditions requeired by the system, i.e., the surface effects.

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Fractional reaction–diﬀusion **equations** have been broadly examined as of lately. These **equations** emerge normally as description models of numerous evolution processes in var- ious branches of science [12, 21, 33]. Furthermore by continuation to the above literature, we demonstrate the utility of the C–F operator on one- and two-dimensional reaction– diﬀusion **equations**, namely the Fitzhugh–Nagumo (FN) equation and the Fisher equation, respectively given by

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We introduce mesoscopic and macroscopic model **equations** of chemotaxis with anomalous sub- **diffusion** for modelling chemically directed transport of biological organisms in changing chemical environments with **diffusion** hindered by traps or macro-molecular crowding. The mesoscopic mod- els are formulated using Continuous Time Random Walk master **equations** and the macroscopic models are formulated with fractional order differential **equations**. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdif- fusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model **equations**. The model **equations** developed here could be used to replace Keller-Segel type **equations** in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.

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Content extraction from images typically relies on segmentation, i.e., extraction of the borders of target structures. Automated segmentation by computer algorithms has been a focus of decades of research [1–3] and remains an active problem in the computer vision literature [4–6]. In practice, the accuracy of segmentation algorithms can be hampered by noise in the image acquisition and the complexity of the arrangement of target objects with respect to their surroundings within the image. In order to achieve robustness to such hindrances, many algorithms demand an increase in computational cost. However, practically useful segmentation techniques should be accurate and computationally efficient for clinical interpretation and so that extensive quantitative analysis can be automated. In this study, highly efficient and mathematically principled techniques are presented to segment the boundaries of closed structures. The techniques are based on ideas of anisotropic information propagation apparent in certain types of partial differential **equations** (PDEs). This work is motivated by anatomical structures such as lymph nodes, as shown in Fig. 1, whose extraction from medical images, such as magnetic resonance (MR) images, is an important task for subsequent quantitative analysis.

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Our work is partially motivated by numerical simulations of periodic solutions of advection-diﬀusion **equations** on moving two-dimensional surfaces for f = 0 performed in [11] using an evolving surface ﬁnite element method. The periodic solutions were obtained by computing the initial value problem for arbitrary chosen initial conditions. Indeed, the numerical solutions for diﬀerent initial conditions with same initial mass appear to converge very quickly to the same time-periodic solution. We would like to emphasize that in this numerical work the formulation (1.1), which entirely avoids the use of local coordinates and surface parametrisations, is very suitable for the evolving surface ﬁnite element method.

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In this paper, a class of second-order tempered diﬀerence operators for the left and right Riemann–Liouville tempered fractional derivatives is constructed. And a class of second-order numerical methods is presented for solving the space tempered fractional diﬀusion **equations**, where the space tempered fractional derivatives are evaluated by the proposed tempered diﬀerence operators, and in the time direction is discreted by the Crank–Nicolson method. Numerical schemes are proved to be unconditionally stable and convergent with order O(h 2 + τ 2 ). Numerical experiments

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vated by the work of Kiselev and Xu [24] where the authors prove a similar statement for the two- and three-dimensional parabolic-elliptic Keller-Segel model. We would like to stress that to the authors’ knowledge in the arguably more realistic setting of a coupled chemotaxis-fluid system, where the flow is not explicitly prescribed but gov- erned by the laws of fluid dynamics, there is no result in the literature proving the existence of global-in-time solutions for a model for which it is known that without the flow there exist solutions exploding in finite time. The class of flows we focus on is a generalisation of weakly mixing flows in the ergodic sense, and a natural adaptation of the class of relaxation enhancing flows considered in [24] to the case of fractional dissipation. The notion of relaxation enhancing flows was introduced in the work [14] by Constantin, Kiselev, Ryzhik and Zlatoˇ s, which constitutes a core reference for our approach. For more background on fluid mixing and its possibly regularising effects in the context of reaction-**diffusion** **equations**, we refer to [24] and references therein. We conclude by pointing out another interesting work [3], which demonstrates that chemo- tactic singularity formation can also be prevented by mixing due to a fast shear flow. The underlying mechanism is, however, rather different from the one considered here and is not able to suppress more than one dimension (of the Keller-Segel model which is L 1 -critical for d = 2 and L 1 -supercritical in higher dimensions). Our second main result

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approximation, accurate. Also Monte Carlo simulations of the transformations of grains which grow according to a **diffusion** controlled mechanism (i.e. growth rate is proportional to the inverse of the particle radius) indicate that JMAK kinetics is, at least, a good approximation for **diffusion** controlled transformations [ 44 ]. A similar conclusion was reached by Uebele and Hermann [38] who approximated **diffusion** controlled growth in a mathematical model which considers parabolic growth and hard impingement. Their work shows that although strictly speaking the Eq. 1 is not valid, deviations from this equation are very small and may in practice be neglected. From these works we may conclude that Eq. 1 is at least a very good approximation of the transformed fraction in a **diffusion** controlled reaction. As it will be shown in this paper that for precipitation reactions the JMAK assumptions are generally not valid, the discussion on whether JMAK kinetics is exact or just an accurate approximation, will not be further pursued.

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Au atoms in Si occupy interstitial and substitutional sites, and the substitutional Au exists in three states depending on the heat treatment history [1]: high-temperature sub- stitutional Au, low-temperature substitutional Au, and ag- glomerations of substitutional Au. During the heat treat- ment, high-temperature substitutional Au diffuses very slowly itself and the change in its concentration is domi- nated by an interchange mechanism with interstitial Au and substitutional Au associated with vacancies [2] and self-interstitials [3]. In this case, the concentrations of substitutional Au, interstitial Au, vacancies and self-in- terstitials can be obtained from four partial differential **equations** [4,5]. The concentration of substitutional Au exhibits a **diffusion** profile depending on the relative contributions of vacancies and interstitial Si atoms [6]. The author has previously investigated Au **diffusion** us- ing Java programming by obtaining a numerical solution to a single partial differential equation obtained from the above four differential **equations** under several approxi- mations. This approach was adopted because of the dif- ficulty of directly numerically solving the above four partial differential **equations** due to the limitation in the capacity of personal computers [7]. However, recently the capacity of personal computers has been progressing rapidly, and the author has been able to directly solve the

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II. D EVELOPMENT OF HPM FOR SOLVING FPDE The HPM was applied to derive an approximate- analytical solution of linear and nonlinear time dependent partial differential **equations** [17,18], and these works motivated us to develop our proposed method. The methodology for the development of HPM for solving PDEs in fuzzy environment is given as follows. Let the succeeding FPDE,