Diffusion Equation Research Articles

Exponential Convergence and Computational Efficiency of BURA-SD Method for Fractional Diffusion Equations in Polygons

In this paper, we develop a new Best Uniform Rational Approximation-Semi-Discrete (BURA-SD) method taking into account the singularities of the solution of fractional diffusion problems in polygonal domains. The complementary capabilities of the exponential convergence rate of BURA-SD and the hp FEM are explored with the aim of maximizing the overall performance. A challenge here is the emerging singularly perturbed diffusion–reaction equations. The main contributions of this paper include asymptotically accurate error estimates, ending with sufficient conditions to balance errors of different origins, thereby guaranteeing the high computational efficiency of the method.

Two quantum algorithms for solving the one-dimensional advection–diffusion equation

Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.

Thermal Spreading Resistance of Near Surface Localized Heating-Induced Size Effects in Semiconductors

Abstract Localized heating is encountered in various scenarios, including the operation of transistors, light-emitting diodes, and some thermal spectroscopy techniques. When localized heating occurs on a scale comparable to the mean free path of the dominant energy carriers, additional thermal resistance is observed due to ballistic effects. The main objective of this study is to find a relation between this resistance, problem geometry, and material thermal properties in situations involving localized heating. Models based on the solution of the Fourier heat diffusion equation and the gray phonon Boltzmann Transport Equation are solved simultaneously to calculate the additional thermal resistances that arise from localized heating. Subsequently, the results are analyzed to derive the desired relationship. It is noted that in the context of localized heating resistance, the effects of geometrical variables are non-linear and substantial, particularly when the Knudsen numbers for the boundary and heat source exceed certain thresholds. Specifically, when the Knudsen number for the heat source becomes comparable to 1 localized heating resistance is observed. However, when the Knudsen number based on heat source height and width surpasses 8 and 20, respectively the heat source behaves akin to a point source, no longer significantly affecting the localized heating resistance. At this juncture, the maximum resistance limit is reached.

Effect of Radial Neutron Reflector on the Characteristics of Nuclear Fuel Burn-Up Wave in a Fast Neutron Energy Spectrum Multiplying Medium: A Consistent Parametric Approach

Abstract The influence of radial neutron reflector on the build-up and propagation of a nuclear fuel burn-up wave in a fast multiplying medium is investigated using a consistent parametric approach. Coupled multigroup neutron diffusion equations with a burn-up evolution model are simulated on the two-dimensional cylindrical reactor geometry with azimuthal symmetry. Uranium–plutonium transmutation model is considered, and the simulation is performed by using the finite element multiphysics software package comsol. Transient characteristics of the burn-up wave are represented by two new parameters, namely, transient time (TT) and transient length (TL). TT and TL are defined as the time and distance required for the burn-up wave to attain its steady-state nature. Steady-state phases are characterized in terms of wave velocity, full width half maximum (FWHM), and full width 10% of maximum (FW10M). A sensitivity study of steady-state and transient parameters is conducted for the different values of radial reflector thickness. The potential relevance of these characterization parameters on the development of optimal geometrical configuration of radial neutron reflector in breed and burn (B&B)-based reactor design is addressed based on the sensitivity study.

Building up a model family for inflammations

The paper presents an approach for overcoming modeling problems of typical life science applications with partly unknown mechanisms and lacking quantitative data: A model family of reaction–diffusion equations is built up on a mesoscopic scale and uses classes of feasible functions for reaction and taxis terms. The classes are found by translating biological knowledge into mathematical conditions and the analysis of the models further constrains the classes. Numerical simulations allow comparing single models out of the model family with available qualitative information on the solutions from observations. The method provides insight into a hierarchical order of the mechanisms. The method is applied to the clinics for liver inflammation such as metabolic dysfunction-associated steatohepatitis or viral hepatitis where reasons for the chronification of disease are still unclear and time- and space-dependent data is unavailable.

Renormalization group model of turbulent bioconvection of gyrotactic microorganisms

The problems connected with processes of turbulent bioconvection are considered. These processes are typical for technologies of microbiological and other industries. The concept of using renormalization groups in the theory of turbulence is shown. A step-by-step algorithm for applying the renormalization methodology to deriving the equation for mass transfer in bioconvective processes, which includes renormalization of the turbulent diffusion coefficient, is presented. The dependence for the turbulent Schmidt number was obtained. It has been shown that turbulent Schmidt number is function of six parameters: molecular Schmidt number, turbulent viscosity, molecular viscosity, parameter of geometric shape of microorganisms, average velocity of motion of microorganisms relative to the fluid, orientation parameter. The turbulent Schmidt number enables closing the diffusion equation. The effect of the geometric shape of microorganisms on the development of turbulence in a bioconvective flow in the presence of gyrotactic microorganisms is analyzed. Analysis shows that microorganisms with spherical symmetry make a small contribution to the formation of turbulent mass transfer processes. The results of the analysis are in a good agreement with data from studies of the nonlinear hydrodynamic stability of bioconvective processes based on the Lorentz model.

Transition fronts of combustion reaction–diffusion equations around an obstacle

Transition fronts of combustion reaction–diffusion equations around an obstacle

Enhancing Charge Generation in Nonfullerene Interdigitated Heterojunction Organic Solar Cells

Interdigitated heterojunction (IHJ) structures have recently been investigated as an alternatives to bulk heterojunction (BHJ) structures due to their better film morphology stability, reproducibility, and contact selectivity. Herein, the electrical and optical properties of a nonfullerene IHJ structure are investigated using drift diffusion and Maxwell equations and compared with other conventional structures. Based on simulation results, the IHJ structure demonstrates electrical advantages such as enhanced charge transport pathways and reduced nonradiative recombination and exhibits superior optical absorption profiles compared to the BHJ structure, owing to its photonic crystal‐like structure. A method is also proposed to further enhance the optical absorption of this structure by introducing a third organic material with absorption capabilities in the near‐infrared range, increasing the solar cell power conversion efficiency from 18.42% to over 19.5%.

Semilinear approximations of quasilinear parabolic equations with applications

An approach to solvability of certain quasilinear parabolic equations is presented by approximating the quasilinear equation under consideration with a parameter family of semilinear problems with stronger linear fractional diffusion term. Defined on arbitrarily long time intervals, solutions to the original problem are found as a suitable limit of global solutions to those semilinear approximations. The method is applied to nonlinear parabolic Kirchhoff equation, quasilinear reaction–diffusion equation, and critical 2D surface quasi‐geostrophic equation associated with Dirichlet boundary conditions.

Fractional analysis of unsteady magnetohydrodynamics Jeffrey flow over an infinite vertical plate in the presence of Hall current

The impact of Hall current on the multiphase thermal transfer of an incompressible electrically conductive Jeffrey flow over an infinitely vertical plate when heat absorption and chemical reaction are present has been examined. Partial differential equations have been used to describe the process, accounting for heat and mass transfer effects. This study uses extended Fourier's and Fick's laws together with the recently announced constant proportional Caputo (CPC) fractional operator. The fractional model is converted into a nondimensional form by applying some appropriate quantities. The nondimensional produced fractional model for momentum, heat, and diffusion equations based on the CPC fractional operator has been calculated semi‐analytically by applying the Laplace method. The Mathcad 15 software to sketch the graphs for several factors, like the Grashof number, mass Grashof number, Schmidt number, Prandtl number, Hall, and magnetic field parameters, is used to describe the velocity profile. Additionally, a graphical explanation is provided for the influence of the appeared parameters, particularly the effect of the fractional parameters. It is concluded that the result of the fluid model developed by the generalized constitutive relations is more accurate and generalized than the results of the artificially contracted fractional model. A fractional derivative is therefore the ideal option to achieve controlled concentration, temperature, and velocity. The current study is immediately relevant to geophysical, cosmically fluid dynamics, medical, biological, and any other processes that are significantly enhanced by a low gas density and a high magnetic field.

Superconvergence analysis of the conforming discontinuous Galerkin method on a Bakhvalov-type mesh for singularly perturbed reaction–diffusion equation

The conforming discontinuous Galerkin (CDG) method maximizes the utilization of all degrees of freedom of the discontinuous Pk polynomial to achieve a convergence rate two orders higher than its counterpart conforming finite element method employing continuous Pk element. Despite this superiority, there is little theory of the CDG methods for singular perturbation problems. In this paper, superconvergence of the CDG method is studied on a Bakhvalov-type mesh for a singularly perturbed reaction–diffusion problem. For this goal, a pre-existing least squares method has been utilized to ensure better approximation properties of the projection. On the basis of that, we derive superconvergence results for the CDG finite element solution in the energy norm and L2-norm and obtain uniform convergence of the CDG method for the first time.

Solitons for a generalized reaction–diffusion equation with the higher‐order power‐law nonlinearity in (1+1)‐ and (2+1)‐dimensional systems

Solitons for a generalized reaction–diffusion equation with the higher‐order power‐law nonlinearity in (1+1)‐ and (2+1)‐dimensional systems

The Theory and Computation of the Semi-Linear Reaction–Diffusion Equation with Dirichlet Boundaries

The Theory and Computation of the Semi-Linear Reaction–Diffusion Equation with Dirichlet Boundaries

Regularization methods for the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition ★ ★ The project is supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX23_2549) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190578).

In the article, we focus on the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition on a cylindrical symmetric field. The ill-posedness of this problem is proved. We introduce the modified Landweber iteration method(MLIM), the Truncated singular value decomposition(TSVD) method for solving it and propose a new regularization method, named as the TSVD-modified Landweber iteration method(TMLIM). The error estimates between the exact solution and the regularized approximate solution are presented by using two regularization parameter selection rules. Finally, numerical examples are provided to demonstrate the effectiveness and feasibility of the regularization methods.

Fuzzy discrete fractional calculus and fuzzy fractional discrete equations

This paper aims to highlight certain limitations in the study of fuzzy fractional discrete equations (FFDEs) based on the generalized Hukuhara difference (gH-difference) in the previous papers. In general, the equivalence between FFDEs and the associated fuzzy discrete fractional sum equations (FDFSEs) is not achieved, requiring the introduction of an appropriate hypothesis to establish this equivalence. Furthermore, this paper introduces the fundamental theory of fuzzy fractional discrete calculus through granular arithmetic operations between fuzzy intervals to address restrictions in the formerly mentioned approaches involving the generalized Hukuhara difference. These operations are constructed based on the concept of the horizontal membership function (HMF) utilized in multidimensional fuzzy arithmetic (MFA). Additionally, the paper proposes the application of fractional discrete calculus to two types of time-discretization diffusion equations with non-zero right-hand sides. Finally, several numerical examples are provided to validate the main results.

Mathematical analysis of batch reactor performance for the enzymatic synthesis of cephalexin: Laplace Homotopy perturbation method and Adomian decomposition method

In this article, a mathematical model of diffusion reaction in kinetically controlled cephalexin synthesis in the batch reactor with penicillin acylase immobilized in glyoxyl-agarose is analyzed. The kinetic model is a non-linear non-stead-state reaction–diffusion equation with non-linear terms related to the Fick’s law. We have presented the approximate analytical expression for the non-steady-state non-linear reaction–diffusion equation by utilizing the Laplace homotopy perturbation method (LHPM) and for steady-state by using LHPM and Adomian decomposition method (ADM). The approximate analytical solution obtained from these methods proved that they are fit for every values of the reaction–diffusion and kinetic parameters. We also present the numerical solution of the considered reaction–diffusion equation by using pdepe tool in MATLAB software. When comparing the semi-analytical solution with the numerical solution, a satisfactory result is noted for all the possible values of the parameters. The closed-form analytical expressions for the effectiveness factor in non-steady and steady-state conditions are obtained and analyzed using LHPM and ADM.

A Generalized Iterated Tikhonov Method in the Fourier Domain for Determining the Unknown Source of the Time-Fractional Diffusion Equation

In this paper, an inverse problem of determining a source in a time-fractional diffusion equation is investigated. A Fourier extension scheme is used to approximate the solution to avoid the impact on smoothness caused by directly using singular system eigenfunctions for approximation. A modified implicit iteration method is proposed as a regularization technique to stabilize the solution process. The convergence rates are derived when a discrepancy principle serves as the principle for choosing the regularization parameters. Numerical tests are provided to further verify the efficacy of the proposed method.

A Reduced-Dimension Weighted Explicit Finite Difference Method Based on the Proper Orthogonal Decomposition Technique for the Space-Fractional Diffusion Equation

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure a sufficient accuracy order but also reduce the degrees of freedom, decrease storage requirements, and accelerate the computation rate. Uniqueness, stabilization, and error estimation are demonstrated by matrix analysis. The procedural steps of the POD algorithm, which reduces dimensionality, are outlined. Numerical simulations to assess the viability and effectiveness of the reduced-dimension weighted explicit finite difference method are given. A comparison between the reduced-dimension method and the classical weighted explicit finite difference scheme is presented, including the error in the L2 norm, the accuracy order, and the CPU time.

First transition dynamics of reaction–diffusion equations with higher order nonlinearity

AbstractIn this study, the dynamics of a reaction–diffusion equation on a bounded interval at the first dynamic transition point is investigated. The main difference of this study from the existing ones in the dynamic transition literature is that in this work, no specific form of the nonlinear operator is assumed except that it is an analytic function of and . The linear operator is also assumed to be a general operator with sinusoidal eigenvectors. Moreover, we assume that there is a first transition as a real simple eigenvalue changes sign. With this general framework, we make a rigorous analysis of the dependence of the first transition dynamics on the coefficients of the Taylor expansion of the nonlinear operator. The main tool we use in this study is the center manifold reduction combined with the classification of the dynamic transition theory. This study is a generalization of a recent work where the nonlinear operator contains only low‐order (quadratic and cubic) nonlinearities. The current generalization requires certain technical difficulties such as the validity of the genericity conditions on the Taylor coefficients of the nonlinear operator and a bootstrapping argument using these genericity conditions. The results of this work can be generalized in various directions, which are discussed in the conclusions section.

Control-oriented multiphysics model of a lithium-ion battery for thermal runaway estimation under operational and abuse conditions

This study proposes a control-oriented multiphysics model for lithium-ion batteries (LIBs) that can estimate electrochemical-thermal responses in real time under normal operation and abuse conditions. The proposed model integrates the simplified electrochemical model, the thermal resistance network, and the adaptive time-stepping method to ensure computational efficiency without sacrificing the accuracy. Specifically, the diffusion equation of the electrochemical model is simplified by addressing Padé approximation. The thermal resistance network estimates 3D temperature distribution through simple matrix multiplication to account for the entropic, ohmic, and chemical reaction during thermal runaway. The adaptive time-stepping method further secures accurate yet fast explicit calculation. Quantitative experimental validation reveals the high accuracy and robustness, and fast inference time of the proposed model to estimate the electrochemical-thermal responses with the average inference time of 0.0047 s per step. The application of the proposed model on the 26650 LFP cell also demonstrates the versatility on various shapes and types of LIBs. The systematic analysis on the 3D temperature distribution in the LIB of interest not only confirms the effectiveness of the internal temperature monitoring but also ensures the virtual sensing capability. The versatility of the proposed model underscores both design- and control- enabling solutions for battery thermal management.