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This paper is concerned with the backward problem of recovering initial value for a homogeneous time-space fractional pseudo parabolic differential equation. Since the problem is ill-posed, a version of modified quasi boundary value method is used as the method of regularization for obtaining stable approximations. Error analysis and parameter choice strategies are done for both a priori and a posteriori cases. It is shown that, the order of the convergence rate can exceed 2 3. Hence, the obtained rate improves upon the previously known rates of order 2 3 for the a priori case, and 1 2 for the a posteriori case for the considered problem. [ABSTRACT FROM AUTHOR]

In this paper, the Local Fractional Laplace Decomposition Method (LFLDM) is used for solving a type of Two-Dimensional Fractional Diffusion Equation (TDFDE). In this method, first we apply the Laplace transform and its inverse to the main equation, and then the Adomian decomposition is used to obtain approximate/analytical solution. The accuracy and applicability of the LFLDM is shown through two examples. The LFLDM results are in good agreement with the exact solution of the problems. [ABSTRACT FROM AUTHOR]

An efficient numerical method with high accuracy both in time and in space is proposed for solving d$$ d $$‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an s$$ s $$‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order s+1$$ s+1 $$ in time, when the implicit Runge‐Kutta method with classical order p$$ p $$ (p≥s+1$$ p\ge s+1 $$) is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter α$$ \alpha $$. This improves the previous result which depends on the fractional parameter α$$ \alpha $$. Numerical experiments verify and complement our theoretical results. [ABSTRACT FROM AUTHOR]

In a lot of engineering applications, one has to recover the temperature of a heat body having a surface that cannot be measured directly. This raises an inverse problem of determining the temperature on the surface of a body using interior measurements, which is called the sideways problem. Many papers investigated the problem (with or without fractional derivatives) by using the Cauchy data u(x0,t),ux(x0,t)$$ u\left({x}_0,t\right),{u}_x\left({x}_0,t\right) $$ measured at one interior point x=x0∈(0,L)$$ x={x}_0\in \left(0,L\right) $$ or using an interior data u(x0,t)$$ u\left({x}_0,t\right) $$ and the assumption limx→∞u(x,t)=0$$ {\lim}_{x\to \infty }u\left(x,t\right)=0 $$. However, the flux ux(x0,t)$$ {u}_x\left({x}_0,t\right) $$ is not easy to measure, and the temperature assumption at infinity is inappropriate for a bounded body. Hence, in the present paper, we consider a fractional sideways problem, in which the interior measurements at two interior point, namely, x=1$$ x=1 $$ and x=2$$ x=2 $$, are given by continuous data with deterministic noises and by discrete data contaminated with random noises. We show that the problem is severely ill‐posed and further constructs an a posteriori optimal truncation regularization method for the deterministic data, and we also construct a nonparametric regularization for the discrete random data. Numerical examples show that the proposed method works well. [ABSTRACT FROM AUTHOR]

The paper considers an inverse problem for a one-dimensional time-fractional subdiffusion equation with a generalized impedance boundary condition. This boundary condition is given by a second-order spatial differential operator imposed on the boundary. The inverse problem is the problem of determining the time dependent source parameter from the energy measurement. The well-posedness of the inverse problem is established by applying the Fourier expansion in terms of eigenfunctions of a spectral problem which has the spectral parameter also in the boundary condition, Volterra type integral equation with the kernel may have a diagonal singularity and fractional type Gronwall inequality. [ABSTRACT FROM AUTHOR]

In this paper we employ Sinc functions as the basis functions and call the resulting method Sinc wavelet collocation method. The main distinguishing feature of the present method, in contrast to the standard Sinc collocation method, is that we do not use a mapping to transform the Sinc functions to a finite domain on which the BVPs are defined to an infinite domain in order to apply them. This results in a much simplified algorithm which is easy to implement while maintaining the high accuracy of the traditional Sinc collocation method. We apply the Sinc collocation method to BVPs and the fractional diffusion equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse problem of recovering the time dependent source term in a time fractional diffusion equation, in the case of nonlocal boundary condition and integral overdetermination condition. The boundary conditions of this problem are regular but not strongly regular. The existence and uniqueness of the solution are established by applying generalized Fourier method based on the expansion in terms of root functions of a spectral problem, weakly singular Volterra integral equation and fractional type Gronwall's inequality. Moreover, we show its continuous dependence on the data. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method based on an operational matrix of Legendre polynomials for resolving the class of time fractional diffusion (TFD) equations. The operational matrix of fractional order derivatives of the Legendre polynomials is derived as a product of matrices. The collocation method together with the operational matrix of Legendre polynomials are employed to transform the TFD equations into a set of algebraic equations. The perturbation method is applied to show the stability of the discussed method. The accuracy of the suggested method is validated using numerical experiments. The solution obtained by this method is in excellent agreement with the exact solution for the integer order of derivatives and is more precise than the solution obtained by the existing method in which Bernstein polynomials are taken as the basis polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study a Cauchy problem of fractional diffusion equation concerning fractional Laplacian. We prove the existence and uniqueness of solution to the problem under investigation in Hölder space. Then we apply the Monte Carlo method to solving this Cauchy problem. For the problem with free force term, we derive an unbiased scheme which only produces the statistic error. For the problem with inhomogeneous force term, we establish the simple and high-order jump-adapted schemes by approximating the trajectory of symmetric stable process. The error estimates of these two jump-adapted schemes are given based on the Hölder regularity of the solution. Numerical experiments including even one hundred dimensional case confirm the theoretical analysis and show the numerical efficiency. [ABSTRACT FROM AUTHOR]

In this work, a new type of the unique continuation property for time-fractional diffusion equations is studied. The proof is mainly based on the Laplace transform and the properties of Bessel functions. As an application, the uniqueness of the inverse problem in the simultaneous determination of spatially dependent source terms and fractional order from sparse boundary observation data is established. [ABSTRACT FROM AUTHOR]

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme. [ABSTRACT FROM AUTHOR]

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with the tempered fractional derivative of order α . Although some of its variants are considered in many recent numerical analysis works, there are still some significant differences. Here we first provide the regularity estimates of the solution. Then a modified L1 scheme inspired by the middle rectangle quadrature formula on graded meshes is employed to compensate for the singularity of the solution at t → 0+, while the five-point difference scheme is used in space. Stability and convergence are proved in the sense of L∞ norm, getting a sharp error estimate σ (τmin{2−α,rα}) on graded meshes. Furthermore, the constant multipliers in the analysis do not blow up as the order of Caputo fractional derivative α approaches the classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and convergence orders of the presented schemes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the time-fractional non-linear diffusion equations for describing the pollution caused by oil in the water. The hybrid computational technique, the Natural Transform Homotopy Perturbation Technique is applied to get a numerical solution to the time-fractional non-linear diffusion equation. The existence and uniqueness are analyzed with the help of the fixed point theorem, also the stability analysis is discussed by using the Lyapunov function. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying the fractional diffusion equation with Riemann–Liouville as follows: D 0 + α y − y x x = 0 , 0 < x < π , with nonlocal in time condition. We are going to study the well-posedness of the above problem with some assumptions of the input data. On the other hand, in Hilbert scale and L p spaces, we provide several estimates of regularity results of the mild solution. We also establish the evaluation for gradient term of the mild solution. We also show that the nonlocal problem is ill-posed in the sense of Hadamard. We also derive the regularity result by applying Fourier truncation method. The main tool of the paper is to use some estimates of Wright functions and Sobolev embeddings. In addition, we also obtain a lower bound of the solution according to the input data. [ABSTRACT FROM AUTHOR]

Calcium signaling in nerve cells is a crucial activity for the human brain to execute a diversity of its functions. An alteration in the signaling process leads to cell death. To date, several attempts registered to study the calcium distribution in nerve cells like neurons, astrocytes, etc. in the form of the integer-order model. In this paper, a fractional-order mathematical model to study the spatiotemporal profile of calcium in nerve cells is assembled and analyzed. The proposed model is solved by the finite element method for space derivative and finite difference method for time derivative. The classical case of the calcium dynamics model is recovered by setting the fractional parameter and that validates the model for classical sense. The numerical computations have systematically presented the impact of a fractional parameter on nerve cells. It is observed that calbindin-D28k provides a significant effect on the spatiotemporal variation of calcium profile due to the amalgamation of the memory of nerve cells. The presence of excess amounts of calbindin-D28k controls the intracellular calcium level and prevents the nerve cell from toxicity. [ABSTRACT FROM AUTHOR]

This paper delves into the extension and characterization of radial positive definite functions into non-integer dimensions. We provide a thorough investigation by employing the Riemann–Liouville fractional integral and fractional Caputo derivatives, enabling a comprehensive understanding of these functions. Additionally, we introduce a secondary characterization based on the Bernstein characterization of completely monotone functions. The practical significance of our study is showcased through an examination of the positivity of the fundamental solution of the space-fractional Bessel diffusion equation, highlighting the real-world applicability of the developed concepts. Through this work, we contribute to the broader understanding of radial positive definite functions and their utility in diverse mathematical and applied contexts. [ABSTRACT FROM AUTHOR]

The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type time-fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we discuss the uniqueness and existence of solutions to these initial-boundary value problems. Under some suitable conditions on the problem data, we then prove positivity of the solutions. Based on these results, several comparison principles for the solutions to the initial-boundary value problems for the linear time-fractional diffusion equations are derived. [ABSTRACT FROM AUTHOR]

In this paper, we try to recover an unknown source in a time-fractional diffusion equation. In order to overcome the influence of boundary conditions on source conditions, we introduce the Jacobi polynomials to construct the approximation and a modified Tikhonov regularization method is proposed to deal with the illposedness. Error estimates are obtained under a discrepancy principle as the parameter choice rule. Numerical results are also presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we consider a boundary value problem for a loaded fractional diffusion equation. The loaded term has the form of the Riemann-Liouville fractional derivative or integral. The BVP is considered in the open right upper quadrant. The problem is reduced to an integral equation that, in some cases, belongs to the pseudo-Volterra type, and its solvability depends on the order of differentiation in the loaded term and the behavior of the support line of the load in a neighborhood of the origin. All these cases are considered. In particular, we establish sufficient conditions for the unique solvability of the problem. Moreover, we give an example showing that violation of these conditions can lead to nonuniqueness of the solution. [ABSTRACT FROM AUTHOR]

This article is concerned with the study of unique solvability of an inverse coefficient problem of determining the coefficient at the lower term of a fractional diffusion equation. The direct problem is the initial-boundary problem for this equation with usual initial and homogeneous Dirichlet conditions. To determine the unknown coefficient, an overdetermination condition is given as the Neumann condition at the left end of the spatial interval. The theorems of existence and uniqueness of inverse problem solution are obtained. Furthermore, we propose a numerical algorithm based on a finite-difference scheme to accurately compute the inverse problem of simultaneously determining a time-dependent coefficient in a fractional diffusion equation, together with its solution. A test example using the developed numerical algorithm is presented herein. [ABSTRACT FROM AUTHOR]

The anomalous transport of particles in the presence of a time-dependent field is considered in the framework of a comb model. This turbulent-like dynamics consists of inhomogeneous time-dependent advection along the x-backbone and Brownian motion along the y-side branches. This geometrically constrained transport leads to anomalous diffusion along the backbone, which is described by a fractional diffusion equation with time-dependent coefficients. The time periodic process leads to localization of the transport and a particular form of relaxation. The analytical approach is considered in the framework of the Floquet theory, which is developed for the fractional diffusion equation with periodic in time coefficients. This physical situation is considered in detail and analytical expressions for both the probability density function and the mean squared displacement are obtained. The new analytical approach is developed in the framework of the fractional Floquet theory that makes it possible to investigate a new class of anomalous diffusion in the presence of time periodic fields. [ABSTRACT FROM AUTHOR]

In this article, we have studied solutions of a generalised multiorder fractional partial differential equations involving the Caputo time‐fractional derivative and the Riemann–Liouville space fractional derivatives using Laplace–Fourier transform technique. Proposed generalised multiorder fractional partial differential equation is reducible to Schrödinger equation, wave equation and diffusion equation in a more general sense, and hence, solutions of these equations are specifically noted. Not only this, solutions of equation proposed in the stochastic resetting theory in the context of Brownian motion can also be found in a general regime. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study a problem of recovering a parabolic equation with fractional derivative from its time averaging. This problem can be established as a new boundary value problem where a Cauchy condition is replaced by a prescribed time average of the solution. By applying some properties of the Mittag–Leffler function, we set some of the results above existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space. Moreover, we also show the ill‐posedness of our problem in the sense of Hadamard. The regularized solution is given, and convergence rate between the regularized solution and the exact solution in Lp space is also derived. [ABSTRACT FROM AUTHOR]

In this paper, approximation of space fractional order diffusion equation are considered using compact finite difference technique to discretize the time derivative, which was then approximated via shifted Gegenbauer polynomials using zeros of (N − 1) degree shifted Gegenbauer polynomial as collocation points. The important feature in this approach is that it reduces the problems to algebraic linear system of equations together with the boundary conditions gives (N + 1) linear equations. Some theorems are given to establish the convergence and the stability of the proposed method. To validate the efficiency and the accuracy of the method, obtained results are compared with the existing results in the literature. The graphical representation are also displayed for various values of β− Gegenbauer polynomials. It can be observe in the tables of the results and figures that the proposed method performs better than the existing one in the literature. [ABSTRACT FROM AUTHOR]

The source term identification of the time‐fractional diffusion equation under Robin boundary condition is studied. This problem is ill‐posed. Therefore, we apply Landweber iterative regularization method, fractional Landweber iterative regularization method, TSVD method, combining TSVD method, and fractional Landweber iterative regularization method, respectively. The comparisons of these four methods are given, which can help us select the most effective method. The error estimates between the regularized approximate solutions and the exact solution are given under the a priori and a posteriori regularization parameter choice rules. Finally, numerical examples verify the effectiveness of the methods. [ABSTRACT FROM AUTHOR]

This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle < π / 2 , that is the resolvent set of this operator contains { z ∈ C ∖ { 0 } : | Arg z | < θ } for some π / 2 < θ < π. The relationship between the time fractional order α ∈ (0 , 1) and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as α approaches 1. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study g-fractional diffusion on bounded domains in R d with absorbing boundary conditions. A new general and explicit representation of the solution is obtained. We study the first-passage time distribution, showing the dependence on the particular choice of the function g. Then, we specialize the analysis to the interesting case of a rectangular domain. Finally, we briefly discuss the connection of this general theory with the physical application to the so-called fractional Dodson diffusion model, recently discussed in the literature. [ABSTRACT FROM AUTHOR]

Our main goal in this paper is to investigate the regularity of the mild solution fractional diffusion equation which can be used in the modelling of heat transfer with memory effects. Under some various assumptions of the input data, we obtain two main results. We also provide the upper bound and lower bound of the source function. The main tool is to use complex evaluations involving the Wright function. [ABSTRACT FROM AUTHOR]

We consider initial boundary value problems for one-dimensional diffusion equation with time-fractional derivative of order α ∈ (0 , 1) which are subject to non-zero Neumann boundary conditions. We prove the uniqueness for an inverse coefficient problem of determining a spatially varying potential and the order of the time-fractional derivative by Dirichlet data at one end point of the spatial interval. The imposed Neumann conditions are required to be within the correct Sobolev space of order α. Our proof is based on a representation formula of solution to an initial boundary value problem with non-zero boundary data. Moreover, we apply such a formula and prove the uniqueness in the determination of boundary value at another end point by Cauchy data at one end point. [ABSTRACT FROM AUTHOR]

In this paper, we study the variable-order generalized time fractional Oldroyd-B fluid model, use the reduced order method and the L2-1σ method to establish the differential format with second-order accuracy, prove the stability and convergence of the format, and give numerical examples to illustrate the effectiveness of the differential format. [ABSTRACT FROM AUTHOR]

In this paper, we study a τ-preconditioner for non-symmetric linear system arising from a steady-state multi-dimensional Riemann-Liouville (RL) fractional diffusion equation. The generalized minimal residual (GMRES) method is applied to solve the preconditioned linear system. Theoretically, we show that the GMRES solver for the preconditioned linear system has a convergence rate independent of discretization stepsizes. To the best of our knowledge, this is the first iterative solver with stepsize-independent convergence rate for the non-symmetric linear system. The proposed τ-preconditioner is diagonalizable by the sine transform matrix, thanks to which the matrix-vector multiplication in each iteration step can be fast implemented by the fast sine transform (FST). Hence, the total operation cost of the proposed solver for the non-symmetric problem is linearithmic. Numerical results are reported to show the efficiency of the proposed preconditioner. [ABSTRACT FROM AUTHOR]

This paper is devoted to recovering simultaneously the fractional order and the space-dependent source term from partial Cauchy's boundary data in a multidimensional time-fractional diffusion equation. The uniqueness of the inverse problem is obtained by employing analytic continuation and the Laplace transform. Then a modified non-stationary iterative Tikhonov regularization method with a regularization parameter chosen by a sigmoid-type function is used to find a stable approximate solution for the source term and the fractional order. Numerical examples in one-dimensional and two-dimensional cases are provided to illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Fractional diffusion equations have applications in various fields and in this paper we consider a fractional diffusion equation with a Riemann–Liouville derivative. The main objective is to investigate the convergence of solutions of the problem when the fractional order tends to 1 − . Under some suitable conditions on the Cauchy data, we prove the convergence results in a reasonable sense. [ABSTRACT FROM AUTHOR]

The source term identification for the time-fractional diffusion equation with Robin boundary conditions was studied. Since the ill-posedness of this problem, an iterative regularization method was constructed to calculate the regularized approximate solution of the source term. The error estimates between the regularized approximate solution and the exact solution were given under the priori and the posteriori regularization parameter choice rules. Numerical examples verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In the current research work we have introduced the Variational Iteration Method to obtain the Analytical Approximate Solution of the Two-Dimensional Diffusion Equation. We have used the initial value to determine the solution for various mathematical problems. This in turn has helped in accelerating the rapid convergence of the solution for the series. We have used the Variational Iteration method to solve the fractional-order diffusion equations for two dimensions. The series form solution for the fractional -order diffusion problem is obtained by the proposed method. We have effectively identified solutions for a wide range of physical and analytical problems during our research. Since we have demonstrated its effectiveness in finding the solutions for the fractional diffusion equations of two dimension, we can now say that VIM can be extremely helpful in tackling a wide range of engineering complexities. The Variation Iteration method has time and again proved to give exemplary result in terms of efficacy and simplicity. It has been proved superior and advantageous over the previously used analytical techniques Adomian Method, New Homotopy Perturbation Method, Homotopy [7.11,12] etc. The primary benefit of this strategy is that it allows us to achieve the desired result in a short amount of time. The obtained results are incredibly accurate and effective. We can also adapt this method for various other non-linear mathematical and scientific problems. We have tried to project mathematical results for various problems in the current research paper. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our current work is to show the existence and uniqueness of mild solutions. Our desired goal is achieved using the Picard iteration method, and our analysis is based on properties of Mittag–Leffler functions and embeddings between Hilbert scales spaces and Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this work, we focus on an initial value problem for a class of 2D time-fractional diffusion evolution equations with Riemann–Liouville fractional derivative. There are three new results in this paper. First of all, the existence and ill-posedness result (in the sense of Hadamard) in three cases which are consisting of homogeneous, inhomogeneous, and nonlinear problems are considered. Next, by using the Fourier truncation method, we show that regularized problems are well-posed. Finally, some demonstrated examples are presented to test the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an equation of nonlinear fractional diffusion with the derivative of Riemann–Liouville. Firstly, we determine the global existence and uniqueness of the mild solution. Next, under some assumptions on the input data, we discuss continuity with regard to the fractional derivative order for the time. Our key idea is to combine the theories Mittag–Leffler functions and Banach fixed‐point theorem. Finally, we present some examples to test the proposed theory. [ABSTRACT FROM AUTHOR]

Recent decades have witnessed the emergence of interesting models of fractional partial differential equations. In the current work, a class of parabolic equations with regularized Hyper-Bessel derivative and the exponential source is investigated. More specifically, we examine the existence and uniqueness of mild solutions in Hilbert scale-spaces which are constructed by a uniformly elliptic symmetry operator on a smooth bounded domain. Our main argument is based on the Banach principle argument. In order to achieve the necessary and sufficient requirements of this argument, we have smoothly combined the application of the Fourier series supportively represented by Mittag-Leffler functions, with Hilbert spaces and Sobolev embeddings. Because of the presence of the fractional operator, we face many challenges in handling proper integrals which appear in the representation of mild solutions. Besides, the source term of an exponential type also causes trouble for us when deriving the desired results. Therefore, powerful embeddings are used to limit the growth of nonlinearity. [ABSTRACT FROM AUTHOR]

We consider the boundary output feedback Mittag-Leffler (M-L) stabilization of a spatial multi-dimensional time fractional distributed parameter systems (DPSs), where the control input is actuated by Neumann boundary control input with the external disturbance. By employing the active disturbance rejection control (ADRC) approach, we design a disturbance observer consisting of two subsystems in light of the boundary output signal. The first one is utilized to channel the unknown disturbance to a stable system so that the second can observe the unknown disturbance. By employing the fractional Lyapunov method, the bounded disturbance observer can well estimate the external disturbance which means the observation error decays to zero in H 1 / 2 (Γ 1). Then a disturbance-observer-based stabilizing controller is proposed to ensure the M-L stability of the original system part in the closed-loop system. Finally, some simulations are included to demonstrate the theoretical results. • The boundary disturbance rejection of multi-dimensional fractional DPSs is studied. • A bounded disturbance observer is designed to estimate the disturbance in H 1 / 2 (Γ 1). • The designed controller doesn't involve high gain and achieve more rapid convergence. [ABSTRACT FROM AUTHOR]

• Two efficient numerical schemes are proposed for Riesz fractional diffusion equation. • A series of estimates are constructed to prove the properties of the schemes. • Numerical experiments are performed to confirm the efficiency of our schemes. This paper is concerned with the study of second order scheme for finding the approximate solutions of space-fractional diffusion equations with fractional Neumann boundary conditions, which are based on the Crank-Nicholson method in time and the shifted Grünwald-Letnikov operator and Taylor expansion method in space. The fractional differential operators are taken in sense of the Riesz fractional derivative. The convergence, solvability and stability of the proposed scheme are proved. In addition, a new numerical scheme is proposed by using different approaches to deal with the boundary conditions. The accuracy and efficiency of the presented method is shown by conducting two numerical examples. [ABSTRACT FROM AUTHOR]

A novel technique is presented to analytically solve the fractional diffusion equation for non-reactive air pollutants emitted from an elevated continuous source into the air. A generalized methodical solution combining first-order Wentzel–Kramers–Brillouin (WKB) approximation theory and the Sturm–Liouville problem is used to solve the air pollutants' fractional dispersion equation. Drawing insight from previous analysis, we expanded the initial issue assuming the turbulent flow characteristics appearing in the diffusion process in a non-integer dimensional space. We solved the transformed problem and compared the solutions against data from real experiment. Physical consequences connecting to the conventional generalized diffusion equations are presented. The results indicate that the present solutions are in accordance with those obtained in literature. This report demonstrates that fractional equations can be applied in a practical prediction of pollutant dissemination in a turbulent atmosphere. [ABSTRACT FROM AUTHOR]

We consider one-dimensional infinite chains of harmonic oscillators with random exchanges of momenta and long-range interaction potentials which have polynomial decay rate | x |âˆ' θ , x â†' ∞, θ > 1 where x ∈ Z is the interaction range. The dynamics conserve total momentum, total length and total energy. We prove that the systems evolve macroscopically on superballistic space-time scale (y ε âˆ' 1 , t ε âˆ' θ âˆ' 1 2 ) when 1 < θ < 3, (y ε âˆ' 1 , t ε âˆ' 1 âˆ' log ( ε âˆ' 1 ) âˆ' 1 ) when θ = 3, and hyperbolic space-time scale (yÉ› âˆ'1, tÉ› âˆ'1) when θ > 3. Combining our results and the results in (Suda 2021 Ann. Inst. Henri Poincare B 57 2268â€"2314), we show the existence of two different space-time scales on which the systems evolve. In addition, we prove the fluctuations of the normal modes of the superballistic wave equation, which are analogues of the Riemann invariants and capture fluctuations along the characteristics. For the normal modes, the space-time scale is superdiffusive when 2 < θ â©˝ 4 and diffusive when θ > 4. [ABSTRACT FROM AUTHOR]

We consider an initial-boundary value problem for the time-fractional diffusion equation. We prove the equivalence of two notions of weak solutions, viscosity solutions and distributional solutions. [ABSTRACT FROM AUTHOR]

Numerical solution and parameter estimation for a type of fractional diffusion equation are considered. Firstly, the symmetrical compact difference scheme is applied to solve the forward problem of the fractional diffusion equation. The stability and convergence of the symmetrical difference scheme are presented. Then, the Bayesian method is considered to estimate the unknown fractional order α of the fractional diffusion equation model. To validate the efficiency of the symmetrical numerical scheme and the estimation method, some simulation tests are considered. The simulation results demonstrate the accuracy of the compact difference scheme and show that the proposed estimation algorithm can provide effective statistical characteristics of the parameter. [ABSTRACT FROM AUTHOR]

In this paper, we study a boundary-value problem with discontinuous conjugation conditions on the line of type changing for a model equation of mixed hyperbolic-parabolic type with degeneration of order in the hyperbolicity domain. In the parabolic domain, the equation is the fractional diffusion equation, whereas in the hyperbolic domain it is the loaded one-speed transfer equation. We prove the uniqueness and existence theorem and propose an explicit solution of the problem in the parabolic and hyperbolic domains. [ABSTRACT FROM AUTHOR]