Order Diffusion Equation Research Articles

Existence and Uniqueness of Almost Periodic Solution for a Mathematical Model of Tumor Growth

This article is concerned with a mathematical model of tumor growth governed by 2nd order diffusion equation . The source of mitotic inhibitor is almost periodic and time-dependent within the tissue. The system is set up with the initial condition C(r, 0) = C0(r) and Robin type inhomogeneous boundary condition . Under certain conditions we show that there exists a unique solution for this model which is almost periodic.

On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

An inverse source problem for a non-automonous time fractional diffusion equation of order \((0<\beta <1)\) is considered in a bounded Lipschitz domain in \(\mathbb {R}^d\). The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is studied. We design two numerical algorithms based on Rothe’s method over uniform and graded grids, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the fractional subdiffusion problem is that the solution lacks the smoothness near the initial time, although it would be smooth away from \(t = 0\). Rothe’s method on a uniform grid addresses the existence of a such a solution (non-smooth with \(t^\gamma \) term where \(1>\gamma > \beta \)) under low regularity assumptions, whilst Rothe’s method over graded grids has the advantage to cope better with the behaviour at \(t=0\) (also here \(t^\beta \) is included in the class of admissible solutions) for the considered problems. The theoretical obtained results are supported by numerical experiments and stay valid in case of smooth solutions to the problem.

Shock filter coupled with a high-order PDE for additive noise removal and image quality enhancement

We are interested in repairing blurry images degraded by additive noise. We present a new filter for image enhancement based on the combination of a shock filter with a fourth order diffusion equation. The algorithm that we propose eliminates not only the blurry but also the noise. The control of the diffusion weight and that of the enhancement of the model is done by a parameter. After experimentation on some test images and on some real images of degraded old documents, the results obtained are satisfactory compared to certain models in the literature.

Investigation of dye removal ability and reusability of green and sustainable silica and carbon-silica hybrid aerogels prepared from paddy waste ash

In this study, the paddy waste ash (PA) was utilized for producing carbon-silica hybrid (PCS) and silica (PS) aerogels. X-ray fluorescence (XRF) spectrometer analysis revealed a high silica content in PA. Crystalline SiO2-cristobalite structure was observed in the X-ray diffraction (XRD) patterns of PA. As expected, the PCS and PS samples exhibited an amorphous SiO2 phase. The pore enlargement and structural improvement after aerogel synthesis were determined using a N2 adsorption-desorption analysis. The specific surface area of the PCS and PS samples were found to be 519 and 652 m2/g, respectively. From the zeta potential results, the points of zero charge (pHPZC) of the samples were found below “2” except for the PCS sample (pHPZC = 2.08). In the first stage of the Remazol Turquoise GN (RTGN) adsorption experiments, the effect of initial pH (2–8) was examined for PA, PCS, and PS sorbents. The highest dye removal value of the PCS sorbent was 61% at pH = 2, while this value was 17.9% and 21.4% for PS and PA sorbents, respectively. The removal percentage curve exhibited a similar trend to the zeta potential curve of the samples. In the experiments conducted with different amounts of sorbents, the dye removal percentage increased with increasing amounts of sorbents. The experimental results and thermodynamic calculations showed that the RTGN adsorption onto PCS was an endothermic process. Pseudo-first order, pseudo-second order and intra-particle diffusion equations were applied to determine the adsorption kinetics of RTGN onto PCS, and it was found that the experimental data was well-fitted to the pseudo-second order kinetic model. The Langmuir, Freundlich, Temkin and Dubinin-Radushkevich (D-R) adsorption isotherm models were used to define the adsorption mechanism of RTGN onto PCS. The Freundlich and Langmuir isotherm models were in agreement with our experimental data and it was concluded that RTGN adsorption on PCS surface occurred both as single layer and multi-layer, as well as took place on heterogeneous and homogeneous surfaces. Characterization analyses were conducted for the PCS sorbent after use to understand the structural changes after dye removal. The regeneration and reusability of the PCS sorbent were investigated and it was found that it preserved its initial removal ability by about 95% after four cycles. The dye removal percentage of PCS decreased by 28% after five cycles.

A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions

An inverse source problem for non-smooth multiterm time Caputo fractional diffusion with fractional order designed as β0 < β1 < ⋯ < βM < 1 is the case of study in a bounded Lipschitz domain in $\mathbb {R}^{d}$ . The missing solely time-dependent source function is reconstructed from an additional integral measurement. The existence, uniqueness and regularity of a weak solution for the inverse source problem is investigated. We design a numerical algorithm based on Rothe’s method over graded meshes, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the multiterm time Caputo fractional subdiffusion problem is that the solution possibly lacks the smoothness near the initial time, although it would be smooth away from t = 0. In this contribution, we will establish an extension of Grönwall’s inequalities for multiterm fractional operators. This extension will be crucial for showing the existence of a unique solution to the inverse problem. The theoretical obtained results are supported by some numerical experiments.

NUMERICAL SIMULATIONS FOR THE VARIABLE ORDER TWO-DIMENSIONAL REACTION SUB-DIFFUSION EQUATION: LINEAR AND NONLINEAR

The applications and the fields that use the anomalous sub-diffusion equations cannot be easily listed due to their wide area. Sure, one of the main physical reasons for using and researching fractional order diffusion equations is to explain anomalous diffusion that occurs in transport processes through complex and/or disordered structures, such as fractal media. One of the important applications is their use in chemical reactions, where a single material continues to shift from a high concentration area to a low concentration area until the concentration across the space is equal. The mathematical model that describes these physical-chemical phenomena is called the reaction sub-diffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) by using an accurate numerical technique which is the variable weighted average finite difference method (WAFDM). We will analyze the stability of the resulting scheme by using a modified suitable version of the John Von Neumann procedure. Specific stability conditions that occur or some parameters in the resulting schemes are derived and checked. At the end of the study, numerical examples are simulated to check the stability and the accuracy of the proposed technique.

Some new applications of β-Laplace integral transform

The β-Laplace integral transform(β-LIT) has been introduced recently by Gaur et. al. which is a generalization of Laplace transform. This new β-LIT shows some special properties over Laplace integral transform. In the current paper, we investigate the applicability of the β-LIT technique to solve partial differential equation such as initial value problem of first order and second order PDE, wave equation, Diffusion equation, Potential and current equation, Heat conduction equation, Axisymmetric Heat conduction equation, Inhomogeneous partial differential equation, inhomogeneous wave equation, Stroke and Rayleigh problem in fluid dynamics and illustrate by some examples.

Loading and sustained release of sodium alginate membranes on pyridirubicin chloride

To develop diversified drug carriers for local treatment of cancer, three types of alginate membranes were fabricated by spin-coating, freeze drying and electrospinning. The uniaxial tensile test is shown that spin-coated membranes possess the maximum tensile strength. All the three types of membranes present excellent adsorption performance toward pyridirubicin chloride (THP) and the electrospinning membranes exhibit the maximum adsorption mass of 144.13 mg g−1. The pseudo-first order, pseudo-second order, and intra-particle diffusion equation were used to evaluate the kinetic data and the drug adsorption process followed the pseudo-first-order model well. The drug releasing mechanism was studied by using different release medium, and it shows that the release process was undertaken by ions exchange reaction. The result shows that the three types of membranes are promising materials for the local treatment of cancer such as local cancers of the pancreas and trachea.

Numerical Solution of Fractional Order Advection Reaction Diffusion Equation with Fibonacci Neural Network

The authors have developed an efficient method with the Fibonacci neural network’s help to solve the fractional-order reaction-diffusion equation in the present article. The Fibonacci neural network consists of an input layer with one perceptron, a hidden layer with $$n\times m$$ perceptions, and an output layer with one perceptron. The authors have used various degrees of the Fibonacci polynomial as an activation function to the input in the hidden layer. The authors then convert the fractional order diffusion equation with initial and boundary conditions into a non-constrained optimization problem, which is then called the cost function. Marquardt’s method is used to update the values of weights to minimize the cost function. After that, the authors used the discussed method on four examples with an exact solution and showed that our method works more accurately than previously existing methods through comparison. In the last, authors have used the discussed method to solve the unsolved diffusion equation and observe the change in solute concentration for different fractional-order at different times.

A numerical solution of variable order diffusion and wave equations

In this work, we consider variable order difusion and wave equations. The derivative is described in the Caputo sence of variable order. We use the Genocchi polynomials as basic functions and obtain operational matrices via these polynomials. These matrices and collocation method help us to convert variable order diffusion and wave equations to an algebraic system. Some examples are given to show the validity of the presented method.

A subordination principle for subdiffusion equations with memory

We study the existence of mild solutions to subdiffusion equations with memory ( ∗ ) ∂ t α u ( t ) = A u ( t ) + ∫ 0 t κ ( t − s ) A u ( s ) d s , t ≥ 0 , with the initial condition u(0)=x, where 0<α<1, A is a closed linear operator defined on a Banach space X, the initial value x belongs to X and κ is a suitable kernel in Lloc1(ℝ+). First, we find a subordination formula for the solution operator of (∗) and then we study its connection with the existence of mild solution to the first order diffusion equation with memory.

Kernel functions‐based approach for distributed order diffusion equations

AbstractIn this work, we solve distributed order diffusion equations (DODEs) by applying the theory on reproducing kernel functions (RKFs). The classical numerical quadrature formulae is used to approximate the DODE to a multi‐term Caputo fractional order diffusion equation (FDE). The Mittag‐Leffler RKF is introduced to estimate fractional derivatives of Caputo. And a space–time RKFs collocation scheme is derived for the multi‐term Caputo time FDEs. The accuracy of the present numerical technique is indicated by employing several experiments.

A backward problem for distributed order diffusion equation: uniqueness and numerical solution

In this paper we consider the identification of the initial condition for a distributed order diffusion equation. We first prove the unique existence and the regularity properties of the strong solution on the bounded temporal-spacial domain based on the eigenfunction expansions. The ill-posedness of the backward problem is interpreted by the compactness of the observation operator. Next the Laplace transformation technique and analytic continuation method are adopted to prove the uniqueness of the backward problem. Then for stabilizing the ill-posed problem, the backward problem is formulated as a Tikhonov type optimization problems, and the conjugate gradient method is adopted to solve the optimization problem with the help of the variational adjoint technique. Finally four numerical examples are given to show the efficiency and stability of the proposed method.

Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme

This paper develops a numerical scheme for finding the approximate solution of space fractional order of the diffusion equation (SFODE). Firstly, the compact finite difference (CFD) with convergence order \begin{document}$ \mathcal{O}(\delta \tau ^{2}) $\end{document} is used for discretizing time derivative. Afterwards, the spatial fractional derivative is approximated by the Chebyshev collocation method of the fourth kind. Furthermore, time-discrete stability and convergence analysis are presented. Finally, two examples are numerically investigated by the proposed method. The examples illustrate the performance and accuracy of our method compared to existing methods presented in the literature.

Are waves with negative spatial damping unstable?

Conventional plane harmonic waves decay in direction of propagation, but unconventional harmonic waves grow in the direction of propagation. While a single unconventional wave cannot be a solution to a physically meaningful boundary value problem, these waves may have an essential contribution to the overall solution of a problem as long as this is a superposition of unconventional and conventional waves. A fourth order diffusion equation with proper thermodynamic structure, and the Burnett equations of rarefied gas dynamics exhibit conventional and unconventional waves. Steady state oscillating boundary value problems are considered to discuss the interplay of conventional and unconventional waves. Results show that as long as the second law of thermodynamics is valid, unconventional waves may contribute to the overall solution, which, however is dominated by conventional waves, and behaves as these.

Computation of solution to fractional order partial reaction diffusion equations

In this article, the considered problem of Cauchy reaction diffusion equation of fractional order is solved by using integral transform of Laplace coupled with decomposition technique due to Adomian scheme. This combination led us to a hybrid method which has been properly used to handle nonlinear and linear problems. The considered problem is used in modeling spatial effects in engineering, biology and ecology. The fractional derivative is considered in Caputo sense. The results are obtained in series form corresponding to the proposed problem of fractional order. To present the analytical procedure of the proposed method, some test examples are provided. An approximate solution of a fractional order diffusion equation were obtained. This solution was rapidly convergent to the exact solution with less computational cost. For the computation purposes, we used MATLAB.

A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model

Multi-term time fractional diffusion model is not only an important physical subject, but also a practical problem commonly involved in engineering. In this paper, we apply the alternating segment technique to combine the classical explicit and implicit schemes, and propose a parallel nature difference method alternating segment pure explicit–implicit (PASE-I) and alternating segment pure implicit–explicit (PASI-E) difference schemes for multi-term time fractional order diffusion equations. The existence and uniqueness of the solutions are proved, and stability and convergence analysis of the two schemes are also given. Theoretical analyses and numerical experiments show that the PASE-I and PASI-E schemes are unconditionally stable and satisfy second-order accuracy in spatial precision and 2 − α order in time precision. When the computational accuracy is equivalent, the CPU time of the two schemes are reduced by up to 2 / 3 compared with the classical implicit difference method. It indicates that the PASE-I and PASI-E parallel difference methods are efficient and feasible for solving multi-term time fractional diffusion equations.

Numerical Solution of a Class of Time-Fractional Order Diffusion Equations in a New Reproducing Kernel Space

In this paper, we structure some new reproducing kernel spaces based on Jacobi polynomial and give a numerical solution of a class of time fractional order diffusion equations using piecewise reproducing kernel method (RKM). Compared with other methods, numerical results show the reliability of the present method.

The capture of particles on absorbing traps in the medium with anomalous diffusion: The effective fractional order diffusion equation and the slow temporal asymptotic of survival probability

The new asymptotic behavior of the survival probability of particles, which move in the anomalous sub-diffusion way, in the medium with absolutely absorbing traps was established. The new power temporal behavior of the survival probability of diffusing particles at long times was obtained. This result is caused by the anomalous character of diffusion.

Solution of Fourth Order Diffusion Equations and Analysis Using the Second Moment

The classical concept of diffusion characterized by Fick’s law is well suited for describing a wide class of practical problems of interest. Nevertheless, it has been observed that it is not enough to properly represent other relevant applications of practical interest. When in a system of particles their spreading is slower or faster than predicted by the classical diffusion model, such a phenomenon is referred to as anomalous diffusion. Time fractional, space fractional and even space-time fractional equations are widely used to model phenomena such as solute transport in porous media, financial modelling and cancer tumor behavior. Considering the effects of partial and temporary retention in dispersion processes a new analytical formulation was derived to simulate anomalous diffusion. The new approach leads to a fourth-order partial differential equation (PDE) and assumes the existence of two concomitant fluxes. This work investigates the behavior of the bi-flux approach in one dimensional (1D) medium evaluating the mean square displacement for different cases in order to classify the diffusion process in normal, sub-diffusive or super-diffusive.