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Abstract: In this paper, we present an experimental study on measurement of the thermal conductivity and heat pipe effect in both hydrophilic and hydrophobic (Toray TGP-H60) carbon papers (around 200μm thickness) with/out liquid water. An experimental setup is developed for measuring thermal conductance at different liquid water contents and temperatures without dissembling the testing device for water addition. Theoretical analysis is also performed to evaluate the apparent conductance of heat pipe effect. We found that liquid water presence inside these materials increases the overall thermal conductivity. At high temperature around 80°C, the heat pipe effect is evident for the hydrophilic paper; while for the hydrophobic one, the heat pipe effect is found to be smaller. The distinction is likely due to the different patterns of the capillary liquid flow in the two media. For the hydrophobic paper, liquid water flows back to the evaporation side when the breakthrough pressure is reached and flow is through preferred routes of small flow resistance. As a result, heat pipe effect is active only in part of the medium, therefore smaller than that in the hydrophilic one. The results are important for understanding the heat transfer phenomena occurring in porous media and effects of material surface property. [Copyright &y& Elsevier]

• Numerical model for solution of heat diffusion equation. • Thermographic investigation in optically semi-transparent materials. • Non-destructive detection of graphic features buried in cultural heritage artefacts. • Selective configurations for optical and thermal properties investigation in paper. In this paper, infrared thermography is applied to the quantitative characterization of optically semi-transparent media by using a newly introduced mathematical model. The method is specifically aimed to the detection and evaluations in graphic features buried beneath the surface of some particular kind of cultural heritage artefacts, such as books and documents. A numerical approach based on the finite element method, is proposed for solving the model equations. The computed thermographic signal depends on different inspected material physical properties, which have been selectively evaluated by comparing the theoretical results with the experimental ones in specifically designed test samples. Thereafter, the model is applied to the analysis of the results obtained in the case of graphic elements buried within samples constituted by the same previously characterized materials. [ABSTRACT FROM AUTHOR]

We derive a one-dimensional model that describes pressing of water saturated paper in the press-section of the paper machine. The model involves two nonlinear diffusion equations which are coupled across an internal boundary. Existence and uniqueness as a number of qualitative properties are demonstrated. Further, computational results for a concrete case are discussed. [ABSTRACT FROM AUTHOR]

The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond [Math. Ann. 21 (1883), no. 1, pp. 109–117]). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky [Crelle 80 (1875), pp. 1–32]) that a solution to the heat equation may not be time-analytic at t=0 even if the initial function is real analytic. Recently, it was shown by Dong and Pan [J Math. Fluid Mech. 22 (2020), no. 4, Paper No. 53]; Dong and Zhang [J. Funct. Anal. 279 (2020), no. 4, Paper No. 108563]; Zhang [Proc. Amer. Math. Soc. 148 (2020), no. 4, pp. 1665–1670] that solutions to the heat equation in the whole space, or in the half space with zero boundary value, are analytic in time under an essentially optimal growth condition. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any \delta >0, we find a solution to the heat equation on the whole plane, with exponential growth of order 2+\delta, which is nowhere analytic in time. [ABSTRACT FROM AUTHOR]

In this present study, we reduce the fractional reaction-diffusion equation to a traditional differential equation using the fractional complex transformation and consider the Landau Lifshitz (LLG) equation. Moreover, by using the generalized exponential rational function method and Kudryashov's method respectively we extract new exact and solitary wave solutions for these equations. Some plots of some presented new solutions are represented to exhibit wave characteristics. All results in this paper are essential to understand the physical meaning and behavior of the investigated equation that sheds light on the importance of investigating various nonlinear wave phenomena in mathematical physics. [ABSTRACT FROM AUTHOR]

Traditionally, the ampacity of an overhead transmission line (OHTL) is a static value obtained based on adverse weather conditions, which constrains the transmission capacity. With the continuous growth of power system load, it is increasingly necessary to dynamically adjust the ampacity based on weather conditions. To this end, this paper models the heat balance relationship of the OHTL based on a BP neural network using Bayesian optimization (BO-BP). On this basis, an OHTL ampacity prediction method considering the model error is proposed. First, a two-stage current-stepping ampacity prediction model is established to obtain the initial ampacity prediction results. Then, the risk control strategy of ampacity prediction considering the model error is proposed to correct the ampacity based on the quartile of the model error to reduce the risk of the conductor overheating caused by the model error. Finally, a simulation is carried out based on the operation data of a 220-kV transmission line. The simulation results show that the accuracy of the BO-BP model is improved by more than 20% compared with the traditional heat balance equation. The proposed ampacity prediction method can improve the transmission capacity by more than 150% compared with the original static ampacity. [ABSTRACT FROM AUTHOR]

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations. [ABSTRACT FROM AUTHOR]

In this paper we study the controllability for the 1D-Heat equation with a Dirichlet boundary condition, in the presence of a scale-invariant parameter. First, we construct the scale-invariant solutions for the one-dimensional heat equation. Then we present our problem statement. We finally prove the Dirichlet boundary controllability. [ABSTRACT FROM AUTHOR]

The paper examines stochastic diffusion within an expanding space–time framework motivated by cosmological applications. Contrary to other results in the literature, for the considered general stochastic model, the expansion of space–time leads to a class of stochastic equations with non-constant coefficients that evolve with the expansion factor. The Cauchy problem with random initial conditions is posed and investigated. The exact solution to a stochastic diffusion equation on the expanding sphere is derived. Various probabilistic properties of the solution are studied, including its dependence structure, evolution of the angular power spectrum and local properties of the solution and its approximations by finite truncations. The paper also characterizes the extremal behaviour of the random solution by establishing upper bounds on the probabilities of large deviations. Numerical studies are carried out to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump (possibly degenerate) noise, including stochastic 2D Navier-Stokes equations, stochastic Burgers equations, stochastic singular p-Laplace equations, and stochastic fast diffusion equations. As a further application, we establish the ergodicity of stochastic singular p-Laplace equations and stochastic fast diffusion equations driven by additive pure jump noise, and we remark that the driving noises could be Compound Poisson processes or Lévy processes with heavy tails. [ABSTRACT FROM AUTHOR]

This paper studies optimal control of systems driven by stochastic differential equations, where the control variable has two components, the first being absolutely continuous and the second singular. Our main result is a stochastic maximum principle for relaxed controls, where the first part of the control is a measure valued process. To achieve this result, we establish first order optimality necessary conditions for strict controls by using strong perturbation on the absolutely continuous component of the control and a convex perturbation on the singular one. The proof of the main result is based on the strict maximum principle, Ekeland's variational principle, and some stability properties of the trajectories and adjoint processes with respect to the control variable. [ABSTRACT FROM AUTHOR]

In this paper we establish some exact controllability results for systems of two parabolic equations. First, we prove the existence of insensitizing controls for the L2 norm of the gradient of solutions of linear heat equations. Then, in the worst situation where null controllability for a system of two parabolic equations can hold, we prove this result for some general couplings. [ABSTRACT FROM AUTHOR]

In Physics it is not allowed to add quantities with different units and for that reason the term HT ht is wrong. [Extracted from the article]

In this paper, we develop a new Best Uniform Rational Approximation-Semi-Discrete (BURA-SD) method taking into account the singularities of the solution of fractional diffusion problems in polygonal domains. The complementary capabilities of the exponential convergence rate of BURA-SD and the h p FEM are explored with the aim of maximizing the overall performance. A challenge here is the emerging singularly perturbed diffusion–reaction equations. The main contributions of this paper include asymptotically accurate error estimates, ending with sufficient conditions to balance errors of different origins, thereby guaranteeing the high computational efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we explore the numerical analysis of the microscale heat equation. We present the characteristics of numerical solutions obtained through both semi- and fully-discrete linear finite element methods. We establish a priori estimates and error bounds for both semi-discrete and fully-discrete finite element approximations. Additionally, the existence and uniqueness of the semi-discrete and fully-discrete finite element approximations have been confirmed. The study explores error bounds in various spaces, comparing the semi-discrete to the exact solutions, the semidiscrete against the fully-discrete solutions, and the fully-discrete solutions with the exact ones. A practical algorithm is introduced to address the system emerging from the fully-discrete finite element approximation at every time step. Additionally, the paper presents numerical error calculations to further demonstrate and validate the results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the heat semigroups S ν m - 1 (t) t ≥ 0 related to Δ ν m - 1 given by Δ ν m - 1 = d 2 d x 2 + 2 ν + 1 x + 2 i a b x d dx - a 2 b 2 x 2 - 2 i ν + 1 a b and we study some of its important properties. In the present paper, several uncertainty principles for the canonical Fourier–Bessel transform are given, including the Beurling, Gelfand–Shilov and Cowling–Price uncertainty principles. [ABSTRACT FROM AUTHOR]

In computer vision, moving objects' detection and tracking technology have become hot topics. With the continuous maturity of UAV technology, quadrotor UAV is increasingly widely used in the market, and their maneuverability and concealment are extreme. The application of computer vision technology on drones is a breakthrough and has been widely used in traffic control, crop protection, drone tracking and shooting, and other fields. Therefore, this paper proposed research on the location and tracking environmental pollution sources under the multi-UAV vision based on the target motion model. This article combines the target tracking technology of UAVs and the development and types of UAVs. Firstly, based on the diffusion model of pollutants in the river, a two-dimensional steady-state diffusion equation for pollutants in the river is established. Then, under the improved boundary conditions, the least squares method of the sum of squares of the measured data and theoretical values is used to model the target motion, and due to the increasingly serious pollution, this paper proposes to use UAV sensing technology to locate the pollution source. Finally, by combining pollution source location technology and drone tracking technology, the problem of not being able to quickly identify pollution sources leading to industrial waste discharge or accidental leakage during transportation poses a considerable threat to river safety. Finally, the use of pollution source positioning technology and drone tracking technology to solve the problem of failure to quickly identify pollution sources, resulting in industrial waste discharge or accidental leakage during transportation, poses a considerable threat to river safety. Experiments are carried out on the collected UAV data based on static and dynamic tests of UAV flight platforms, the experimental part, the UAV flight platform was tested, and the actual operation and positioning of the pollution source were carried out. The final experimental results showed that: within 0–360 s, the attitude angle obtained by the gradient descent method had no divergence phenomenon, which could effectively reduce the error caused by integration; the inclination angle deviation of the two groups of experimental equipment was within ± 2.5°, the roll angle deviation was within ± 3°, and the deflection angle was larger at certain moments, but the average deviation was only 0.8°. It also showed that the system could better adapt to the practical application requirements of quadrotor UAVs. [ABSTRACT FROM AUTHOR]

In this paper we consider solvability of an initial boundary value problem for the heat equation with a dynamic type boundary condition. Using some regularity and consistency conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown. This paper also considers an inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition. Conditions for the well-posedness of the formulated problem are found. In contrast to previous works, in this paper the existence and uniqueness of the solution of direct and inverse problems is proved without using orthogonality conditions. [ABSTRACT FROM AUTHOR]

This paper investigates Cauchy problems of nonlinear parabolic equation with a Caputo fractional derivative. When the initial datum is sufficiently small in some appropriate spaces, we demonstrate the existence in global time and uniqueness of a mild solution in fractional Sobolev spaces using some novel techniques. Under some suitable assumptions on the initial datum, we show that the mild solution of the time fractional parabolic equation converges to the mild solution of the classical problem when α → 1 − . Under some appropriate assumptions on the initial datum, we show that the mild solution of the time fractional diffusion equation converges to the mild solution of the classical problem when α → 1 − . Our theoretical results can be widely applied to many different equations such as the Hamilton–Jacobi equation, the Navier–Stokes equation in two cases: the fractional derivative and the classical derivative. Our paper also provides a completely new answer to the related open problem of convergence of solutions to fractional diffusion equations as the order of fractional derivative approaches 1−. [ABSTRACT FROM AUTHOR]

In this paper, a new split iterative compact difference scheme for a class of system is constructed. Then, the conservation properties of the scheme are discussed, and the convergence of the split iterative difference scheme is analyzed by using the discrete energy method on the basis of the prior estimation. Finally, numerical experiments verify these properties of the new scheme. In addition, the numerical results also show the influence of fractional derivative on the variation of the transport equation. [ABSTRACT FROM AUTHOR]

For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem. [ABSTRACT FROM AUTHOR]

The heat kernel associated with a discrete graph Laplacian is the basic solution to the heat diffusion equation of a strict graph or network. In addition, this kernel represents the heat transfer that occurs over time across the network edges. Its computation involves exponentiating the Laplacian eigensystem with respect to time. In this paper, we expand upon this concept by considering a novel network-theoretic approach developed in recent years, which involves defining the k-path Laplacian operator for networks. Prior studies have adopted the notion of integrating long-range interactions (LRI) in the transmission of "information" across the nodes and edges of the network. Various methods have been employed to consider long-range interactions. We explore here the incorporation of long-range interactions in network analysis through the use of Mellin and Laplace transforms applied to the k-path Laplacian matrix. The contribution of this paper is the computation of the heat kernel associated with the k -path Laplacian, called the generalized heat kernel (GHK), and its employment as the basis for extracting stable and useful novel versions of invariants for graph characterization. The results presented in this paper demonstrate that the use of LRI improves the results obtained with classical diffusion methods for networks characterization. [ABSTRACT FROM AUTHOR]

Grain-oriented silicon steel (GO FeSi) laminations are vital components for efficient energy conversion in electromagnetic devices. While traditionally optimized for power frequencies of 50/60 Hz, the pursuit of higher frequency operation (f ≥ 200 Hz) promises enhanced power density. This paper introduces a model for estimating GO FeSi laminations' magnetic behavior under these elevated operational frequencies. The proposed model combines the Maxwell diffusion equation and a material law derived from a fractional differential equation, capturing the viscoelastic characteristics of the magnetization process. Remarkably, the model's dynamical contribution, characterized by only two parameters, achieves a notable 4.8% Euclidean relative distance error across the frequency spectrum from 50 Hz to 1 kHz. The paper's initial section offers an exhaustive description of the model, featuring comprehensive comparisons between simulated and measured data. Subsequently, a methodology is presented for the localized segregation of magnetic losses into three conventional categories: hysteresis, classical, and excess, delineated across various tested frequencies. Further leveraging the model's predictive capabilities, the study extends to investigating the very high-frequency regime, elucidating the spatial distribution of loss contributions. The application of proportional–iterative learning control facilitates the model's adaptation to standard characterization conditions, employing sinusoidal imposed flux density. The paper deliberates on the implications of GO FeSi behavior under extreme operational conditions, offering insights and reflections essential for understanding and optimizing magnetic core performance in high-frequency applications. [ABSTRACT FROM AUTHOR]

This paper focused on the stability analysis of the Dirichlet-to-Neumann (DN) map for the fractional diffusion equation with a reaction coefficient q . The main result provided a Hölder-type stability estimate for the map, which was formulated in terms of the Dirichlet eigenvalues and normal derivatives of eigenfunctions of the operator A q := − Δ + q . This paper focused on the stability analysis of the Dirichlet-to-Neumann (DN) map for the fractional diffusion equation with a reaction coefficient . The main result provided a Hölder-type stability estimate for the map, which was formulated in terms of the Dirichlet eigenvalues and normal derivatives of eigenfunctions of the operator . [ABSTRACT FROM AUTHOR]

In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders α i ∈ (0 , 1) , i = 1 , 2 , ⋯ , n ). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only O (1) storage and O (N T) computational complexity, where N T denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of O ( (Δ t) 2 + N − m ) , where Δ t , N , and m represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions. In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders , ). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only storage and computational complexity, where denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of , where , , and represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions. [ABSTRACT FROM AUTHOR]

This paper tackles efficient methods for Bayesian inverse problems with priors based on Whittle–Matérn Gaussian random fields. The Whittle–Matérn prior is characterized by a mean function and a covariance operator that is taken as a negative power of an elliptic differential operator. This approach is flexible in that it can incorporate a wide range of prior information including nonstationary effects, but it is currently computationally advantageous only for integer values of the exponent. In this paper, we derive an efficient method for handling all admissible noninteger values of the exponent. The method first discretizes the covariance operator using finite elements and quadrature, and uses preconditioned Krylov subspace solvers for shifted linear systems to efficiently apply the resulting covariance matrix to a vector. This approach can be used for generating samples from the distribution in two different ways: by solving a stochastic partial differential equation, and by using a truncated Karhunen–Loève expansion. We show how to incorporate this prior representation into the infinite-dimensional Bayesian formulation, and show how to efficiently compute the maximum a posteriori estimate, and approximate the posterior variance. Although the focus of this paper is on Bayesian inverse problems, the techniques developed here are applicable to solving systems with fractional Laplacians and Gaussian random fields. Numerical experiments demonstrate the performance and scalability of the solvers and their applicability to model and real-data inverse problems in tomography and a time-dependent heat equation. [ABSTRACT FROM AUTHOR]

This paper investigates the longtime behavior of solutions to a temporally discrete diffusion equation with a fixed boundary and a free boundary respectively in one space dimension. Such equation can be equivalent in any case to an integro-difference equation, another important time discrete equation that provides powerful tools for the study of dispersal phenomena. In this paper, we first discuss the global dynamics of the equation in a fixed bounded domain. With a Stefan type free boundary, we then give a new well-posedness proof and the regular spreading-vanishing dichotomy for the corresponding problem. Moreover, a modified comparison principle for the time discrete free boundary problem is proved in an effort to provide the sufficient conditions for dichotomy. It is the first attempt to study the temporally discrete diffusive phenomenon with a free boundary. [ABSTRACT FROM AUTHOR]

The reaction-diffusion equations have been studied in various aspects of nature, such as heat transfer and biological dynamic modelling. In this paper, we study a nonlocal reaction-diffusion model with time delay associated with the density and diffusion behaviour of biological populations. Based on our previous conclusions and numerical strategy of the direct problem, we continue to study the inverse problem about the diffusion coefficient as well as the parameters in the birth function, and also perform numerical simulations with error analysis. In addition, we further generalize the constant diffusion coefficient to the position-dependent diffusion rate, which is intended to improve the generalizability to multiple organisms diffusion in nature. [ABSTRACT FROM AUTHOR]

In contrast with the diffusion equation which smoothens the initial data to C^\infty for t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac–Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods. [ABSTRACT FROM AUTHOR]

This paper derives well-posedness and asymptotic results that provide qualitative information about the behavior, mechanism and strategies used by living organisms to navigate their biological networks. Chemical driven swimming is a captivating phenomenon that is observed in various living organisms like bacteria and protozoa but the problem in weighted networks is more complex, since the equations of parabolic-parabolic Keller-Segel model coupled with incompressible Navier-Stokes equations must be reformulated in a discrete setting. The starting point is to transpose the coupled system from the Euclidean case to the connected networks with certain network-theoretic simplified approaches, which yields fruitful key results. These results not only enable the construction of global solutions but also serve as a foundation for determining information about the stability and large time behavior of the system. Then, decay rates are well established to predict important features, such as how quickly weak solutions at a given point decrease over time due to dissipative processes. Additionally, the L 1 - convergence of cell densities towards the self-similar Gaussian solution of the heat equation is well proved by time dependent scaling, which shows that the solution maintains its shape and only scales in time and space as time evolves. Finally, this paper includes many numerical tests through a recent robust numerical scheme to illustrate the theoretical results and to develop computational control and prediction of living organisms' trajectories around central nodes in networked flows. [ABSTRACT FROM AUTHOR]

In this paper, we define and investigate a system of coupled regular diffusion equations in which each concentration acts as a driving term in the next diffusion equation. Such systems can be understood as a kind of cascade process which appear in different fields of physics like diffusion and reaction processes or turbulence. As a solution, we apply the time-dependent self-similar Ansatz method, the obtained solutions can be expressed as the product of a Gaussian and a Kummer's function. This model physically means that the first diffusion works as a catalyst in the second diffusion system. The coupling of these diffusion systems is only one way. In the second part of the study we investigate mutually coupled diffusion equations which also have the self-similar trial function. The derived solutions show some similarities to the former one. To make our investigation more complete, different kinds of couplings were examined like the linear, the power-law, and the Lorentzian. Finally, a special coupling was investigated which is capable of describing isomerization with temporal decay. [ABSTRACT FROM AUTHOR]

In this paper, high-order numerical algorithms for two classes of time-independent one-sided tempered fractional diffusion equations were studied. The time derivative was discretized by the backward difference formula, the space tempered fractional derivatives were discretized based on tempered weighted and shifted Grünwald difference operators combined with the quasi-compact technique, and the effective second-order numerical approximations of the left and right third-order Riemann-Liouville tempered derivatives were given, thus the detailed fourth-order numerical schemes of these two classes of equations were derived. With the energy method, we proved rigorously that the numerical schemes were stable and convergent with order O (τ + h 4) and were only related to the tempered parameter λ. Finally, some examples were given to verify the validity of the numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of a time-fractional nonlinear diffusion equation. According to Kaplan's first eigenvalue method, we first prove the blow-up of the solutions in finite time under some sufficient conditions. We next provide sufficient conditions for the existence of global solutions by using the results of Zhang and Sun. In conclusion, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity. [ABSTRACT FROM AUTHOR]

This paper is concerned with finding solutions to free-boundary inverse coefficient problems. Mathematically, we handle a one-dimensional non-homogeneous heat equation subject to initial and boundary conditions as well as non-localized integral observations of zeroth and first-order heat momentum. The direct problem is solved for the temperature distribution and the non-localized integral measurements using the Crank–Nicolson finite difference method. The inverse problem is solved by simultaneously finding the temperature distribution, the time-dependent free-boundary function indicating the location of the moving interface, and the time-wise thermal diffusivity or advection velocities. We reformulate the inverse problem as a non-linear optimization problem and use the  l s q n o n l i n  non-linear least-square solver from the MATLAB optimization toolbox. Through examples and discussions, we determine the optimal values of the regulation parameters to ensure accurate, convergent, and stable reconstructions. The direct problem is well-posed, and the Crank–Nicolson method provides accurate solutions with relative errors below  0.006 %  when the discretization elements are  M = N = 80 . The accuracy of the forward solutions helps to obtain sensible solutions for the inverse problem. Although the inverse problem is ill-posed, we determine the optimal regularization parameter values to obtain satisfactory solutions. We also investigate the existence of inverse solutions to the considered problems and verify their uniqueness based on established definitions and theorems. [ABSTRACT FROM AUTHOR]

This paper addresses the null controllability of an abstract boundary control systems in Hilbert spaces where the system operator is of Riesz type. Consequently, this document establishes a criterion for null controllability in such systems based on initial data, utilizing the moment problem. Furthermore, this criterion is formulated by employing a null controllability criterion that is applicable to a corresponding linear system with internal control. Finally, we apply our approach to the heat equation and the Mullins equation, demonstrating the practicality of our methodology. [ABSTRACT FROM AUTHOR]

In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity. [ABSTRACT FROM AUTHOR]

The paper introduces a new deep learning-based high-order spatial approximation for a solution of a high-dimensional Kolmogorov equation where the initial condition is only assumed to be a continuous function and the condition on the vector fields associated with the differential operator is very general, i.e. weaker than Hörmander's hypoelliptic condition. In particular, the deep learning-based method is constructed based on the Kusuoka approximation. Numerical results for high-dimensional partial differential equations up to 500-dimension cases appearing in option pricing problems show the validity of the method. As an application, a computation scheme for the delta is shown using "deep" numerical differentiation. [ABSTRACT FROM AUTHOR]

We study the behavior of solutions of the Cauchy problem for a semilinear heat equation with critical nonlinearity in the sense of Joseph and Lundgren. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its optimal and sharp convergence rate of solutions with a critical exponent and two exponentially approaching initial data. This rate contains a logarithmic term which does not contain in the super critical nonlinearity case. Proofs are given by a comparison method based on matched asymptotic expansion. [ABSTRACT FROM AUTHOR]

We study solutions of the equation u t - Δ u + λ u = f , for initial data that is 'large at infinity' as treated in our previous papers on the unforced heat equation. When f = 0 we characterise those (u 0 , λ) for which solutions converge to 0 as t → ∞ , as not every λ > 0 is able to achieve that for all initial data. When f ≠ 0 we give conditions to guarantee that the solution is given by the usual 'variation of constants formula' u (t) = e - λ t S (t) u 0 + ∫ 0 t e - λ (t - s) S (t - s) f (s) d s , where S (·) is the heat semigroup. We use these results to treat the elliptic problem - Δ u + λ u = f when f is allowed to be 'large at infinity', giving conditions under which a solution exists that is given by convolution with the usual Green's function for the problem. Many of our results are sharp when u 0 , f ≥ 0 . [ABSTRACT FROM AUTHOR]

In this paper, we present a lemma that allows us to characterize a broad class of C0-semigroups on time scales, which can be applied to prove existence and uniqueness of solutions of systems of partial differential equations where the time domain is a time scale. The result obtained is applied to study the controllability of the heat equation on time scales. The lemma presented in this paper can be seen as a unification of the one proved by H. Leiva in [28], for semigroups in [0,∞). [ABSTRACT FROM AUTHOR]

This paper presents a numerical approach to solve fuzzy partial differential equation restrict to fuzzy boundary and initial conditions. The numerical solution to this problem is obtained by the finite difference method considering a particular type of fuzzy arithmetic called J 0 -interactive arithmetic. A study of the computational cost of the method is presented, as well as a comparison with the numerical solution obtained by the standard fuzzy arithmetic, which is associated with the Zadeh's extension principle. The paper focuses on the heat equation in order to illustrate the methodology. [ABSTRACT FROM AUTHOR]

The article presents a study which investigates the benefits of using open boundaries for subduction modelling of lithosphere in two-dimension (2-D). The study adopted the Boussinesq approximation consisting three coupled equations including the Stokes equation, and the heat equations on heat diffusion and heat advection. The results showed that the overall evolution of subduction and mantle flow were not changed with the open sidewalls raising the model aspect ratio.