Solution Of Diffusion Equation Research Articles

Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications

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.

An RBF-FD Method for Numerical Solutions of 2D Diffusion-Wave and Diffusion Equations of Distributed Fractional Order

AbstractThe subject of this paper is to propose a numerical algorithm for solving 2D diffusion and diffusion-wave equations of distributed order fractional derivatives. Such equations arise in modelling complex systems and have many important applications. Existence of integral term over the order of fractional derivative causes the high complexity of these equations and so their numerical solutions needs special cares. Using Gauss quadrature approach for discretizing the integral term of fractional derivative converts the distributed equation into a multi-term fractional differential equation. Then, the time variable is discretized with a suitable finite difference approach. The resultant semi-discretized equations are fully discretized by a radial basis function-generated finite difference based method. Convergence of the method are studied numerically. Various kind of test problems are considered for a comprehensive numerical study and the results confirm the efficiency of the method.

On the analysis and application of a spectral collocation scheme for the nonlinear two-dimensional fractional diffusion equation

In this paper, we propose and analyze a novel spectral scheme for the numerical solution of a two-dimensional time-fractional diffusion equation. The proposed scheme approximates the unknown function and its derivatives in space by finite Lucas and Fibonacci polynomial expansion, and time derivative/variable by L1 formula with θ-weighted difference scheme. Error estimates are derived and convergence is proved. Six test problems are then solved to verify the theory numerically. The obtained results confirm the low computational cost and better accuracy than known results in the literature.

EXtended Hybridizable Discontinuous Galerkin (X-HDG) method for linear convection-diffusion equations on unfitted domains

In this work, we propose a novel strategy for the numerical solution of linear convection diffusion equation (CDE) over unfitted domains. In the proposed numerical scheme, strategies from high order Hybridized Discontinuous Galerkin method and eXtended Finite Element method are combined with the level set definition of the boundaries. The proposed scheme and hence, is named as eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted domains; i.e., the computational mesh does not need to fit to the domain boundary; instead, the boundary is defined by a level set function and cuts through the background mesh arbitrarily. The original unknown structure of HDG and its hybrid nature ensuring the local conservation of fluxes is kept, while developing a modified bilinear form for the elements cut by the boundary. At every cut element, an auxiliary nodal trace variable on the boundary is introduced, which is eliminated afterwards while imposing the boundary conditions. Both stationary and time dependent CDEs are studied over a range of flow regimes from diffusion to convection dominated; using high order (p≤4) XHDG through benchmark numerical examples over arbitrary unfitted domains. Results proved that XHDG inherits optimal (p+1) and super (p+2) convergence properties of HDG while removing the fitting mesh restriction.

Accurate numerical simulations for fractional diffusion equations using spectral deferred correction methods

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.

Modeling the impregnation of roselle antioxidants into papaya cubes

The increasing demand for healthy foods has recently encouraged research focused on incorporating antioxidant compounds from bioactive-rich sources into other foodstuffs; however, the studies aimed at modeling the mass transfer occurring during the enrichment of food matrices with bioactive substances remain scarce. This work investigated the modeling of water loss, solids gain, impregnation of bioactive compounds - total phenolics (TPC) and total anthocyanins (TAN) - and changes in total antioxidant capacity (TAC) of papaya cubes infused with a roselle extract/sucrose solution (RESS). The RESS was used to impregnate 15 mm papaya cubes in the mass ratio of 15:1 at 45, 55 and 65 °C for different immersion times (0, 10, 15, 30, 60, 90, 120, 180, 240, 300, and 360 min). A novel semi-analytical solution for the unsteady-state diffusion equation considering the product shrinkage was proposed for describing experimental mass transfer data and for obtaining equilibrium points and effective diffusivities of migrating species. After 360 min, RESS-impregnated papaya cubes increased their TPC, TAN, and TAC in about 670–990 mg GAE/100 g fresh weight (FW), 48.8–65.0 mg C3G/100 g FW, and 805–1004 mg TE/100 g FW from 402 ± 53 mg GAE/100 g FW, 0 mg C3G/100 g FW, and 679 ± 62 mg TE/100 g FW. Diffusivities for water, total solids, TPC, TAN, and TAC were in the ranges of 1.76–12.6, 0.90–2.54, 0.65–10.8, 0.36–1.32, and 3.31–11.1 ( × 10−10) m2/s, respectively. These values were overestimated in about 23 ± 6% when papaya shrinkage, accounting up to a 50% volume loss, was not considered in the calculations.

Convolution kernel determination problem for the time-fractional diffusion equation

In this work, we study the inverse problem of determining the convolution kernel in the Cauchy problem for the time-fractional diffusion equation by a single observation at the point x=0 of the diffusion process. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation is constructed. In particular, using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel function which will be used for the study inverse problem. The inverse problem is reduced to the equivalent integral equation of the Volterra type. By the contracted mapping principle, the local existence and global uniqueness results are proven. Also the stability estimate is obtained.

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems.

The solution of the time-space fractional diffusion equation based on the Chebyshev collocation method

The solution of the time-space fractional diffusion equation based on the Chebyshev collocation method

Production Forecasting of Unruly Geoenergy Extraction Wells Using Gaussian Decline Curve Analysis

Fast and rigorous well performance evaluation is made possible by new solutions of the pressure diffusion equation. The derived Gaussian pressure transient (GPT) solutions can be practically formulated as a decline curve analysis (DCA) equation for history matching of historic well rates to then forecast the future well performance and estimate the remaining reserves. Application in rate transient analysis (RTA) mode is also possible to estimate fracture half-lengths. Because GPT solutions are physics-based, these can be used for production forecasting as well as in reservoir simulation mode (by computing the spatial and temporal pressure gradients everywhere in the reservoir section drained by either an existing or a planned well). The present paper focuses on the physics-based production forecasting of so-called “unruly” wells, which at first seem to have production behavior noncompliant with any DCA curve. Four shale wells (one from the Utica, Ohio; one from the Eagle Ford Formation, East Texas; and two from the Wolfcamp Formation, West Texas) are analyzed in detail. Physics-based adjustments are made to the Gaussian DCA history matching process, showing how the production rate of these wells is fully compliant with the rate implied by the hydraulic diffusivity of the reservoir sections where these wells drain from.

A solution for the neutron diffusion equation in the spherical and hemispherical reactors using the residual power series

A novel analytical solution to the neutron diffusion equation is proposed in this study using the residual power series approach for both spherical and hemispherical fissile material reactors. Various boundary conditions are investigated, including zero flux on the boundary, zero flux on the extrapolated boundary distance, and the radiation boundary condition (RBC). The study also shows how two hemispheres with opposing flat faces interact. We give numerical results for the same energy neutrons diffused in pure P239u. By qualitative comparison with the homotopy perturbation method and Bessel function-based solutions, the residual power series method (RPSM) presents accurate series solutions that converge to the exact solutions, as shown in this study. Moreover, numerical results were shown to be improved by the computer implementation of the analytic solutions.

Recovering solution of the Reverse nonlinear time Fractional diffusion equations with fluctuations data


 In this study, our focus is on obtaining an estimated solution for the nonlinear fractional time diffusion equation. Specifically, we have utilized the Riemann Liouville fractional derivative. Additionally, we have concerned Gaussian white noise in the input data. As we are aware, this problem is considered ill-posed according to Hadamard's definition. To tackle this problem, we have proposed a regularized solution and demonstrated the convergence between the mild solution and the regularized solution.

Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows

We present a new exact solution of the thermal diffusion equations for steady-state shear flows of a binary fluid. Shear fluid flows are used in modeling and simulating large-scale currents of the world ocean, motions in thin layers of fluid, fluid flows in processes, and apparatuses of chemical technology. To describe the steady shear flows of an incompressible fluid, the system of Navier–Stokes equations in the Boussinesq approximation is redefined, so the construction of exact and numerical solutions to the equations of hydrodynamics is a very difficult and urgent task. A non-trivial exact solution is constructed in the Lin-Sidorov-Aristov class. For this class of exact solutions, the hydrodynamic fields (velocity field, pressure field, temperature field, and solute concentration field) were considered as linear forms in the x and y coordinates. The coefficients of linear forms depend on the third coordinate z. Thus, when considering a shear flow, the two-dimensional velocity field depends on three coordinates. It is worth noting that the solvability condition given in the article imposes a condition (relation) only between the velocity gradients. A theorem on the uniqueness of the exact solution in the Lin–Sidorov–Aristov class is formulated. The remaining coefficients of linear forms for hydrodynamic fields have functional arbitrariness. To illustrate the exact solution of the overdetermined system of Oberbeck–Boussinesq equations, a boundary value problem was solved to describe the complex convection of a vertical swirling fluid without its preliminary rotation. It was shown that the velocity field is highly stratified. Complex countercurrents are recorded in the fluid.

Numerical Solution of Two-Dimensional Fractional Order Diffusion Equation by using Elzaki Transform with Residual Power Series Method

The Elzaki transform with RPS method is an efficient and reliable approach for solution of non-integer order linear as well as non-linear differential equations. The major endeavourof current work is to find the solution of two dimensional non-integer order diffusion equation by Elzaki transform with RPS method. Firstly, Elzaki is applied on this equation. Secondly, inverse Elzaki is taken on this equation for finding the expressionof solution. Thirdly, assumed approximate solution is substituted on this equation. Then unidentified coefficient functions are obtained by using residual function is equal to zero as well as combining its initial circumstances. At last, coefficient functions are substituted in power series form for finding finite approximate logical solutions. The judgment between exact solution and approximate analytic solutions with different number of terms of this equation are determined and compared for reliability. This method reduces the size of computational works of solution of non-integer order non-linear diffusion equation

Asymptotic behaviour of blow-up solutions of the fast diffusion equation

Asymptotic behaviour of blow-up solutions of the fast diffusion equation

Global existence and convergence results for a class of nonlinear time fractional diffusion equation

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 . 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 . 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−.

Rotating Disk Electrodes beyond the Levich Approximation: Physics-Informed Neural Networks Reveal and Quantify Edge Effects.

Physics-informed neural networks are used to characterize the mass transport to the rotating disk electrode (RDE), the most widely employed hydrodynamic electrode in electroanalysis. The PINN approach was first quantitatively verified via 1D simulations under the Levich approximation for cyclic voltammetry and chronoamperometry, allowing comparison of the results with finite difference simulations and analytical equations. However, the Levich approximation is only accurate for high Schmidt numbers (Sc > 1000). The PINN approach allowed consideration of smaller Sc, achieving an analytical level of accuracy (error <0.1%) comparable with independent numerical evaluation and confirming that the errors in the Levich equation can be as high as 3% when Sc = 1000 for rapidly diffusing species in aqueous solution. Entirely novel, the PINNs permit the solution of the 2D diffusion equation under cylindrical geometry incorporating radial diffusion and reveal the rotating disk electrode edge effect as a consequence of the nonuniform accessibility of the disc with greater currents flowing near the extremities. The contribution to the total current is quantified as a function of the rotation speed, disk radius, and analyte diffusion coefficient. The success in extending the theory for the rotating disk electrode beyond the Levich equation shows that PINNs can be an easier and more powerful substitute for conventional methods, both analytical and simulation based.

Continuation of Radial Positive Definite Functions and Their Characterization

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.

Chloride penetration in concrete with trilinear binding isotherm

Chloride binding in concrete is an important factor and it can significantly affect the penetration speed of chlorides in concrete and thus the service life of concrete structures. In this paper a new chloride binding isotherm is proposed to be included in the diffusion model of chlorides in concrete. Instead of the use of the traditional Freundlich or Langmuir isotherm, the present chloride binding model uses a trilinear function to describe the chloride binding in concrete. The trilinear binding isotherm not only reflects the nonlinear binding feature of chlorides in concrete when it is exposed to a chloride environment, but also makes it possible to achieve an analytical solution of the chloride diffusion equation. In the present paper, the superior of using trilinear binding isotherm to other binding isotherms is demonstrated. The detailed derivation of the analytical solution of the chloride diffusion equation with nonlinear binding feature is described and illustrated. To demonstrate the rationality and accuracy of the analytical solution derived, comparisons between the analytical solution and the numerical solution and experimental results are also provided. It is shown that the chloride diffusion profiles obtained from the chloride diffusion equation with the trilinear binding isotherm are much closer the experimental results than that obtained from the chloride diffusion equation with linear or bilinear binding isotherm.

Measurement of Mass Gain Behavior due to Hydration and Reduction of Uncertainty Induced by Time Lag of Weighing on Pristine Kapton Film after Heating in Vacuum

To simulate material degradations in space environments, materials are exposed to various space environment factors in vacuum. In vacuum, low-molecular-weight component, moisture, and etc. desorb from the materials, and it follows the mass loss. The vacuum-desorbed materials, however, adsorb the moisture due to retrieval from vacuum chamber into the atmosphere condition. The mass gain due to hydration disturb the accuracy of an atomic oxygen fluence measurement and an outgassing test. In this study, we have evaluated the mass gain behaviour of vacuum-desorbed material due to hydration. To desorb the moisture, the test specimens were heated in vacuum. After dehydration in vacuum, the mass of test specimen was measured by using an ultra-micro balance inside the low-humidity controlled globe box to evaluate the mass equivalent to that in vacuum condition and then the mass was measured under controlled condition of 50% relative humidity to evaluate the mass gain in atmosphere. From the experimental results, we have evaluated the uncertainties due to hydration occurring during the time lag between sample retrieval from a vacuum chamber to sample weighing, quantitatively. The back extrapolation using the regression curve based on the solution of diffusion equation is effective on the reduction of uncertainties.