Order Diffusion Equation Research Articles

One-dimensional viscous diffusion equation of higher order with gradient dependent potentials and sources

In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ(s) and sources A(s). We first show the general existence and uniqueness of global classical solutions provided that the first order derivatives of both Φ(s) and A(s) are bounded below. Such a restriction is almost necessary, namely, if one of the derivatives is unbounded from below, then the solution might blow up in a finite time. A more interesting phenomenon is also revealed for potentials or sources being unbounded from below. In fact, if either the source or the potential is dominant, then the solution will blow up definitely in a finite time. Moreover, the viscous coefficient might postpone the blow-up time. Exactly speaking, for any T > 0, the solution will never blow up during the period 0 < t < T, so long as the viscous coefficient is large enough.

On the self-similar, early-time, anomalous diffusion in random networks — Approach by fractional calculus

In a recent work, Zhang and Padrino (2017) derived an equation for diffusion in random networks consisting of junction pockets and connecting channels by applying the ensemble average method to the mass conservation principle. The resulting integro-differential equation was solved numerically using finite volumes for the test case of one-dimensional diffusion in the half-line. For early time, they found that the numerical predictions of pocket mass density depend on the similarity variable xt−1/4, representing sub-diffusion. They argue that the sub-diffusive behavior is a consequence of the time needed to establish a linear concentration profile inside a channel. By theoretical analysis of the diffusion equation for small time, they confirmed this finding. Nevertheless, they did not present an exact solution for the small time limit to compare with. Here, starting with their small-time leading order diffusion equation in (x,t) space, we use elements of fractional calculus to cast it into a form for which an analytical solution has been obtained in the literature for the same boundary and initial conditions in terms of the Fox H-function (Schneider and Wyss, 1989). For ease of computation, we express the solution in terms of the Meijer G-function. We compare the exact solution with Zhang and Padrino's numerical predictions, resulting in excellent agreement, thereby validating their numerical approach.

Methylene blue adsorption from aqueous solution by loofah sponge-based porous carbons

Three environmentally friendly activated carbons with mainly micropores based on loofah sponge (LSC), as the adsorbents, were prepared through a simple carbonization process under argon atmosphere at 600°C, 800°C, and 1000°C. The adsorption properties of methylene blue on the LSCs were studied through varying the experimental parameters such as the temperature, pH, dosage and contact time. The adsorption equilibrium data was described by the Freundlich and Langmuir isotherms. The maximum adsorption capacity calculated based on Langmuir isotherm equation was 210.97mg/g for the LSC sample obtained at 800°C. The kinetic data of LSCs was evaluated by using the pseudo-first order, pseudo-second order, and intraparticle diffusion equation. The data of thermodynamic analysis indicated that the adsorption reaction of methylene blue on to LSCs was endothermic. The relationship between the structure and adsorption properties of LSCs had been discussed and analyzed based on the experimental and theoretical data.

Retracted: New two step Laplace Adam‐Bashforth method for integer a noninteger order partial differential equations

This article presents a novel method that allows to generalize the use of the Adam‐Bashforth to Partial Differential Equations with local and nonlocal operator. The Method derives a two step Adam‐Bashforth numerical scheme in Laplace space and the solution is taken back into the real space via inverse Laplace transform. The method yields a powerful numerical algorithm for fractional order derivative where the usually very difficult to manage summation in the numerical scheme disappears. Error Analysis of the method is also presented. Applications of the method and numerical simulations are presented on a wave–equation‐like, and on a fractional order diffusion equation.

Preparation and characterization of a novel magnetic biosorbent functionalized with biomass of Bacillus Subtilis: Kinetic and isotherm studies of biosorption processes in the removal of Methylene Blue

A novel magnetic biosorbent (M-BSub) was successfully prepared by the immobilization of Bacillus subtilis (B. subtilis) with nano-sized magnetic silica which was confirmed by Fourier transform infrared spectroscopy (FTIR), thermogravimetric analysis (TGA), vibrating sample magnetometry (VSM), scanning electron microscope (SEM). Optimum biosorption conditions were determined as a function of pH, contact time and initial concentration of Methylene Blue (MB). The maximum biosorption efficiency of MB was obtained at pH 6.8 and 30°C. The uptake of dye was very fast initially, and achieved equilibrium after 3h. The isotherm models like Langmuir, Freundlich, Dubinin–Radushkevich (D–R) were used to analyze the equilibrium data. The equilibrium data fit better to the Freundlich model compared to the Langmuir model in concentration range studied (50–300mg/L). The maximum biosorption capacity of biosorbent reached up to 59±0.6mgg−1 at pH value of 6.8 and 30°C. The kinetics data were analyzed using adsorption kinetic models like the pseudo-first order, pseudo-second order and intraparticle diffusion equation. Kinetic data showed good agreement with the pseudo-second order kinetic model. In addition, biosorbent can be easily regenerated and reused for five cycles with high biosorption capacity. This study indicated that M-BSub is a reusable biosorbent for rapid, convenient, and efficient removal of MB from contaminated aquatic systems.

Spatial-fractional order diffusion filtering

In order to reduce noise of a spectrum while preserving the peak, a smoothing method based on the spatial-fractional order diffusion equation was proposed. Three methods including explicit difference scheme, implicit difference scheme and Crank–Nicolson scheme were presented to implement the spatial-fractional order diffusion filtering. Some simulated signals and a NMR spectrum have been used to verify the effectiveness of the proposed method. The performance of the proposed method was compared with other smoothing methods such as regularization method, Savitzky–Golay method and wavelet method. Results show that the spatial-fractional order diffusion filtering can improve the signal-to-noise ratio while preserving the peak.

Adsorption and Kinetic studies of dyeingClitoria ternatea L. natural dye onto bamboo yarn

In the present study, the bamboo yarn is dyed with a natural dye extracted from Clitoria ternatea L.flower. The effect of adsorption mechanism on the dyeing Clitoria ternatea L. natural dyes onto bamboo yarn were investigated. It has been observed that the adsorption process of time in the range of 10-90 minutes, with maximum adsorption capacity at 60th minutes.Assessment of the effect of time on dye adsorption showed an increase in adsorption capacity with increasing time, with maximum adsorption at 60 minutes. Pseudo-first order, Pseudo-second order, and Intraparticle diffusion equations were employed to investigate adsorption rates. Pseudo-first order model provided the best fit indicating physisorption. The equilibrium adsorption data were fitted to Langmuir, Freundlich, Temkin, and Dubinin Radushkevish isotherm models. Langmuir isotherm model agreed well with the experimental data having high regression coefficients for dyeing bamboo yarn with dye concentration at 0.12 g/mL. These findings suggest that the application of natural dye from Clitoria ternatea L.flower onto bamboo yarn which is a new resource of the natural fiber in the industry is a good approach to get the desired dyeing, adsorption, and kinetic property.

A new fractional operator of variable order: Application in the description of anomalous diffusion

In this paper, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type. The properties in term of the Laplace and Fourier transforms are analyzed and the results for the anomalous diffusion equations of variable order are discussed. The new formulation is efficient in modeling a class of concentrations in the complex transport process.

Null-Controllability of A Fractional Order Diffusion Equation

AbstractThe article considers the controllability of a fractional order diffusion equation. We show that the resulting fractional order diffusion equation is null-controllable. Our method reduces essentially to the study of a moment problem related to the Mittag-Leffler functions. Paley-Wiener type theorems are applied to construct biorthogonal sequence to a family of complex Mittag-Leffler functions.

Filtration and adsorption properties of porous calcium alginate membrane for methylene blue removal from water

Porous sodium alginate membrane was first prepared by vacuum freeze-dry method, then, sodium ions were substituted by calcium ions through cross-linking reaction to obtain the porous calcium alginate membrane via natural drying. The filtration and the adsorption properties of methylene blue on the porous calcium alginate membrane were studied through investigating the experimental parameters such as the pH, contact time, temperature and dosage. The data of the adsorption equilibrium were described by the Langmuir and Freundlich isotherms. The pseudo-first order, pseudo-second order, and intra-particle diffusion equation were used to evaluate the kinetic data. Thermodynamic analysis indicated that the adsorption of methylene blue on to the porous calcium alginate membrane was an exothermic reaction.

Subdiffusion model with time-dependent diffusion coefficient: Integral-balance solution and analysis

The paper addresses approximate integral-balance approach to a time-fractional diffusion equation of order 0 &lt; ? &lt; 1 with a time-dependent diffusion coefficient of power-law type D(t)=D0t? where 0 &lt; ? &lt; 1. The form of the solution spreading in a semi-infinite medium through an analysis of the second moment of the approximate solution reveals that depending on the sum ?+? the solution can model subdiffusive (?+?&lt;1), superdiffusive (?+?&gt;1) or Gaussian (?+?=1) process of transport. The optimal exponents of the approximate parabolic profiles have been determined by minimization the mean squared error of approximation over the penetration depth.

Comparison of Numerical Approximations of One-Dimensional Space Fractional Diffusion Equation Using Different Types of Collocation Points in Spectral Method Based on Lagrange’s Basis Polynomials

Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order diffusion equations. Spectral collocation method based on Lagrange’s basis polynomials to approximate numerical solutions of one-dimensional (1D) space fractional diffusion equations are introduced in this research paper. The proposed form of approximate solution satisfies non-zero Dirichlet’s boundary conditions on both boundaries. Collocation scheme produce a system of first order Ordinary Differential Equations (ODE) from the fractional diffusion equation. We applied this method with four different sets of collocation points to compare their performance.

Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods

Gradient schemes is a framework which enables the unified convergence analysis of many different methods–such as finite elements (conforming, non-conforming and mixed) and finite volumes methods–for 2nd order diffusion equations. We show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed Dirichlet, Neumann and Signorini boundary conditions. This extension allows us to provide error estimates for numerical approximations of such models, recovering known convergence rates for some methods, and establishing new convergence rates for schemes not previously studied for variational inequalities. The general framework we develop also enables us to design a new numerical method for the obstacle and Signorini problems, based on hybrid mimetic mixed schemes. We provide numerical results that demonstrate the accuracy of these schemes, and confirm our theoretical rates of convergence.

Analytical Solutions of Fractional Order Diffusion Equations by Natural Transform Method

In this article, we develop an analytical method for solving fractional order partial differential equations. Our method is the generalizations of homotopy perturbations Laplace transform method and homotopy perturbations Sumudu transform method. The solutions obtained using the proposed method implies that the method is highly accurate and easy to implement. The proposed method can be easily applied to a large variety of problems, which are modeled in terms of fractional order partial differential equations. Some test problems are solved to show the accuracy of the proposed method. Some results are shown graphically also.

An inverse source problem in a semilinear time-fractional diffusion equation

We study an inverse source problem for a semilinear time-fractional diffusion equation of second order in a bounded domain in Rd. The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is addressed. We design a numerical algorithm based on Rothe’s method, derive a priori estimates and prove convergence of iterates towards the exact solution. Theoretical results are supported by a numerical experiment.

Generalized distributed order diffusion equations with composite time fractional derivative

In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier–Laplace transform method. We represent solutions in terms of infinite series in Fox H-functions. The fractional and second moments are derived by using Mittag-Leffler functions. We observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives. Generalized uniformly distributed order diffusion equation, as a model for strong anomalous behavior, is analyzed by using Tauberian theorem. Some previously obtained results are special cases of those presented in this paper.

Maximum principles for a time–space fractional diffusion equation

In this paper, we focus on maximum principles of a time–space fractional diffusion equation. Maximum principles for classical solution and weak solution are all obtained by using properties of the time fractional derivative operator and the fractional Laplace operator. We deduce maximum principles for a full fractional diffusion equation, other than time-fractional and spatial-integer order diffusion equations.

Long-time behavior of a finite volume discretization for a fourth order diffusion equation

We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary . The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker–Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.

Existence and regularity of solutions to time-fractional diffusion equations

We investigate some initial–boundary value problems for time-fractional diffusion equations of order α∈(0,1). Such equations model anomalous diffusion on fractals. The existence of solution irrelevant to α is established only if the external force function f is weighted Hölder continuous, which is weaker than Hölder continuous. Some interesting versions of maximal and spatial regularity criteria depending on the fractional exponent α are also discussed.

Exact solution of certain time fractional nonlinear partial differential equations

Given a time fractional nonlinear partial differential equation, we show how to derive its exact solution using invariant subspace method. This has been illustrated through time fractional diffusion convection equation, time fractional nonlinear dispersive Boussinesq equation, time fractional reaction diffusion equation of second order, time fractional thin-film equation, and time fractional quadratic wave equation. Also, we explicitly shown that time fractional nonlinear partial differential equations admit more than one invariant subspaces which in turn helps to derive more than one exact solution.