Riemann-Liouville Time Derivative Research Articles

Determining of a Space Dependent Coefficient of Fractional Diffusion Equation with the Generalized Riemann–Liouville Time Derivative

Determining of a Space Dependent Coefficient of Fractional Diffusion Equation with the Generalized Riemann–Liouville Time Derivative

Inverse Coefficient Problem for Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative

Inverse Coefficient Problem for Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative

Inverse coefficient problems for a time-fractional wave equation with the generalized Riemann-Liouville time derivative

This paper considers the inverse problem of determining the time-dependent coeffiicient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag-Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved.

Inverse Coefficient Problems for a Time-Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative

Inverse Coefficient Problems for a Time-Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative

Boundary-Value Problem for the Aller–Lykov Nonlocal Moisture Transfer Equation

In this paper, a boundary-value problem for the inhomogeneous Aller–Lykov moisture transfer equation with a fractional Riemann–Liouville time derivative is examined. The equation considered is a generalization of the Aller–Lykov equation obtained by introducing the fractal rate of change of humidity, which explains the appearance of flows directed against the potential of humidity. The existence of a solution to the first boundary-value problem is proved by the Fourier method. Using the method of energy inequalities for solutions of the problem, we obtain an a priori estimate in terms of the fractional Riemann–Liouville derivative, which implies the uniqueness of the solution.

Weak Solvability of the Variable-Order Subdiffusion Equation

Abstract In this work, we study a new type of linear partial differential equations – the variable-order subdiffusion equation. Here the Laplace operator in space acts on the Riemann-Liouville time derivative of space-dependent order. We construct a variable-order Sobolev and prove the weak solvability of the initial-boundary value problem for this equation, which confirms the well-posedness of the problem. Finally, we briefly discuss the application of the developed approach to the more general variable-order reaction-subdiffusion equation.

A higher‐order unconditionally stable scheme for the solution of fractional diffusion equation

In this paper, a higher‐order compact finite difference scheme with multigrid algorithm is applied for solving one‐dimensional time fractional diffusion equation. The second‐order derivative with respect to space is approximated by higher‐order compact difference scheme. Then, Grünwald–Letnikov approximation is used for the Riemann–Liouville time derivative to get an implicit scheme. The scheme is based on a heptadiagonal matrix with eighth‐order accurate local truncation error. Fourier analysis is used to analyze the stability of higher‐order compact finite difference scheme. Matrix analysis is used to show that the scheme is convergent with the accuracy of eighth‐order in space. Numerical experiments confirm our theoretical analysis and demonstrate the performance and accuracy of our proposed scheme.

Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives

We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann–Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge of the state.

Solving k-fractional shallow water wave equations via homotopy analysis transform method

In this paper, homotopy analysis transform method has been applied to obtain the approximate analytical solution of the shallow water equations with time-fractional derivatives. The interesting point is manifested in its emphatic application of k-fractional Riemann-Liouville time derivatives on the classical equations with an achievement of the highly accurate solutions by the known series solutions.

Fractional formalism to DNA chain and impact of the fractional order on breather dynamics

We have investigated the impact of the fractional order derivative on the dynamics of modulated waves of a homogeneous DNA chain that is based on site-dependent finite stacking and pairing enthalpies. We have reformulated the classical Lagrangian of the system by including the coordinates depending on the Riemann-Liouville time derivative of fractional order γ. From the Lagrange equation, we derived the fractional nonlinear equation of motion. We obtained the fractional breather as solutions by means of a fractional perturbation technique. The impact of the fractional order is investigated and we showed that depending on the values of γ, there are three types of waves that propagate in DNA. We have static breathers, breathers of small amplitude and high velocity, and breathers of high amplitude and small velocity.

Subdiffusion in a system with thin membranes

We study both theoretically and experimentally a process of subdiffusion in a system with two thin membranes. The theoretical model uses Green's functions obtained for the membrane system by means of the generalized method of images. These Green's functions are combinations of the fundamental solutions to a fractional subdiffusion equation describing subdiffusion in a homogenous, unbounded system. Using Green's functions we find analytical formulas describing the time evolution of concentration profiles and the time evolution of the amount of substance that remains in the region between the membranes. The concentration profiles fulfill a new boundary condition at the membrane, in which the membrane permeability is assumed to change over time according to the special formula presented in the paper. These concentration profiles fulfill a standard subdiffusion equation with fractional Riemann-Liouville time derivative only approximately, but they coincide very well with the experimental data. Fitting the theoretical functions in with the experimental results, we also estimate the subdiffusion coefficient of polyethylene glycol 2000 in agarose hydrogel.

Compact alternating direction implicit method for two-dimensional time fractional diffusion equation

High-order compact finite difference scheme with operator splitting technique for solving two-dimensional time fractional diffusion equation is considered in this paper. A Grünwald–Letnikov approximation is used for the Riemann–Liouville time derivative, and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme. Alternating direction implicit (ADI) method is used to split the original problem into two separate one-dimensional problems. The local truncation error is analyzed and the stability is discussed by the Fourier method. The proposed scheme is suitable when the order of the time fractional derivative γ lies in the interval 1 2 , 1 . A correction term is added to maintain high accuracy when γ ∈ 0 , 1 2 . Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.