Second-order Parabolic Partial Differential Equations Research Articles

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

Linear-quadratic optimal control of second-order parabolic systems

This paper considers the linear-quadratic (LQ) optimal control problem for systems governed by a class of second-order parabolic partial differential equations (PDEs). This problem is significant in many areas of mathematics, control, engineering and so on, and thus has received great attention in the past decades. Different from the previous articles where the operator is applied to present the controller, the main contribution of this paper is to propose the discretisation-then-continuousization method, which is explicit and implementable. The solvability condition of the LQ optimal control problems is given based on a set of differential Riccati equations, and an explicit numerical calculation way of these equations and the design of the optimal controller are provided.

MATHEMATICAL MODELING OF DIFFUSION BOUNDARY PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS USING MATLAB AND C++ FOR NUMERICAL AND ANALYTICAL SOLUTIONS

The article examines a second-order parabolic partial differential equation of a three-dimensional (3D) non-stationary boundary problem with constant diffusion coefficients and periodic boundary conditions in the x and y directions. The method for reducing the (3D) non-stationary boundary problem to the corresponding one-dimensional (1D) non-stationary boundary problem using periodic boundary conditions in the x and y directions is discussed. The stationary (analytical) solution of the obtained (1D) stationary boundary problem is also obtained. The numerical solutions of the 1D boundary problem are obtained using the Matlab package "pdepe" and the C++ programming language. As a practical application of the developed mathematical model, the article discusses calculating the concentration of heavy metal Ca in a peat layer based on the obtained experimental data (measurements).

Numerical Simulation of Hydrogen Diffusion in Cement Sheath of Wells Used for Underground Hydrogen Storage

The negative environmental impact of carbon emissions from fossil fuels has promoted hydrogen utilization and storage in underground structures. Hydrogen leakage from storage structures through wells is a major concern due to the small hydrogen molecules that diffuse fast in the porous well cement sheath. The second-order parabolic partial differential equation describing the hydrogen diffusion in well cement was solved numerically using the finite difference method (FDM). The numerical model was verified with an analytical solution for an ideal case where the matrix and fluid have invariant properties. Sensitivity analyses with the model revealed several possibilities. Based on simulation studies and underlying assumptions such as non-dissolvable hydrogen gas in water present in the cement pore spaces, constant hydrogen diffusion coefficient, cement properties such as porosity and saturation, etc., hydrogen should take about 7.5 days to fully penetrate a 35 cm cement sheath under expected well conditions. The relatively short duration for hydrogen breakthrough in the cement sheath is mainly due to the small molecule size and high hydrogen diffusivity. If the hydrogen reaches a vertical channel behind the casing, a hydrogen leak from the well is soon expected. Also, the simulation result reveals that hydrogen migration along the axial direction of the cement column from a storage reservoir to the top of a 50 m caprock is likely to occur in 500 years. Hydrogen diffusion into cement sheaths increases with increased cement porosity and diffusion coefficient and decreases with water saturation (and increases with hydrogen saturation). Hence, cement with a low water-to-cement ratio to reduce water content and low cement porosity is desirable for completing hydrogen storage wells.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

Applications of a space-time FOSLS formulation for parabolic PDEs

Abstract In this work, we show that the space-time first-order system least-squares formulation (Führer, T. & Karkulik, M. (2021) Space–time least-squares finite elements for parabolic equations. Comput. Math. Appl.92, 27–36) for the heat equation and its recent generalization (Gantner, G. & Stevenson, R. (2021) Further results on a space-time FOSLS formulation of parabolic PDEs. ESAIM Math. Model. Numer. Anal.55, 283–299) to arbitrary second-order parabolic partial differential equations can be used to efficiently solve parameter-dependent problems, optimal control problems and problems on time-dependent spatial domains.

О корректности начальных задач для уравнения дробной диффузии

The paper studies a second-order parabolic partial differential equation with fractional differentiation with respect to a time variable. The fractional differentiation operator is a linear combination of the Riemann-Liouville and Gerasimov-Caputo fractional derivatives. It is shown that the distribution of orders of fractional derivatives, included in the equation affects the correctness of the initial problems for the equation under consideration.

Parabolic differential equations with bounded delay

We show the continuous dependence of solutions of linear nonautonomous second-order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.

Time-dependent finite-difference model for transient and steady-state analysis of thermoelectric bulk materials

A mathematical model of coupled thermoelectricity is presented to investigate the transient and steady-state behaviour of thermoelectric bulk material. Governing partial differential equations (PDEs) for the coupled thermal and electrical behaviour of the thermoelectric model are discretised using the explicit finite-difference method. Differencing schemes like Upwind and Lax–Wendroff methods are employed to obtain solutions for the first-order hyperbolic PDEs, whereas FTCS (Forward Time, Centred Space) scheme is employed to solve second-order parabolic PDEs. Courant-Friedrichs-Lewy and Von Neumann stability analyses are done to ensure the stability and convergence of the model. The model considers the temperature dependency of thermal conductivity, electrical conductivity, and Seebeck coefficient of the P/N materials separately. and accounts for the Seebeck, Peltier, and Joule-Thomson effects in thermoelectric materials. The new model is practically useful to predict the transient and steady-state behaviours of a thermoelectric device with multiple P-N elements. The results of the presented finite-difference model are proven to agree well with experimental values as well as 3D simulations with ANSYS®.

Digital Simulations for Three-dimensional Nonlinear Advection-diffusion Equations Using Quasi-variable Meshes High-resolution Implicit Compact Scheme

A two-level implicit compact formulation with quasi-variable meshes is reported for solving three-dimensions second-order nonlinear parabolic partial differential equations. The new nineteen-point compact scheme exhibit fourth and second-order accuracy in space and time on a variable mesh steps and uniformly spaced mesh points. We have also developed an operator-splitting technique to implement the alternating direction implicit (ADI) scheme for computing the 3D advection-diffusion equation. Thomas algorithm computes each tri-diagonal matrix that arises from ADI steps in minimal computing time. The operator-splitting form is unconditionally stable. The improved accuracy is achieved at a lower cost of computation and storage because the spatial mesh parameters tune the mesh location according to solution values' behavior. The new method is successfully applied to the Navier-Stokes equation, advection-diffusion equation, and Burger's equation for the computational illustrations that corroborate the order, accuracies, and robustness of the new high-order implicit compact scheme. The main highlight of the present work lies in obtaining a fourth-order scheme on a quasi-variable mesh network, and its superiority over the comparable uniform meshes high-order compact scheme.

A Numerical Solution of Heat Equation for Several Thermal Diffusivity Using Finite Difference Scheme with Stability Conditions

The heat equation is a second-order parabolic partial differential equation, which can be solved in many ways using numerical methods. This paper provides a numerical solution that uses the finite difference method like the explicit center difference method. The forward time and centered space (FTCS) is used to a problem containing the one-dimensional heat equation and the stability condition of the scheme is reported with different thermal conductivity of different materials. In this study, results obtained for different thermal conductivity of distinct materials are compared. Also, the results reveal the well-behavior properties of the materials in good agreement.

A highly accurate algorithm for retrieving the predicted behavior of problems with piecewise-smooth initial data

A numerical scheme is constructed for the second-order parabolic partial differential equation with piecewise smooth initial data. The scheme comprises an orthogonal spline collocation strategy with the Rannacher time-marching. The irregular behavior of the underlying initial conditions of such differential equations results in inaccurate approximations due to the quantization error. For such problems, even the A-stable Crank-Nicolson scheme yields only first-order convergence in the temporal direction, with oscillations near the discontinuity. Applying mathematical perspective to dampen these oscillations, we present a highly accurate orthogonal spline collocation method with a smooth but straightforward time-marching scheme that significantly improves the convergence order. Through rigorous analysis, the present conjunctive scheme's convergence in the spatial direction is shown fourth-order (in L2 and L∞-norms) and third-order (in H1-norm), and it is shown second-order in the temporal direction. The performance and robustness of the contributed scheme are conclusively demonstrated with two test examples.

Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise

We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener process. The equation is spatially discretized by Galerkin methods and temporally discretized by drift-implicit Euler and Milstein schemes. By the monotone and Lyapunov assumptions, we use both the variational and semigroup approaches to derive a spatial Sobolev regularity under the $L_\omega^p L_t^\infty \dot H^{1+\gamma}$-norm and a temporal H\"older regularity under the $L_\omega^p L_x^2$-norm for the solution of the proposed equation with an $\dot H^{1+\gamma}$-valued initial datum for $\gamma\in [0,1]$. Then we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates $O(h^{1+\gamma}+\tau^{1/2})$ and $O(h^{1+\gamma}+\tau^{(1+\gamma)/2})$ for the Galerkin-based Euler and Milstein schemes, respectively.

On Black–Scholes option pricing model with stochastic volatility: an information theoretic approach

In this article, we derive the risk-neutral measures of the stock options price and volatility by incorporating a simple constrained minimization of the Kullback measure of relative information. We obtain a second-order parabolic partial differential equation, the generalized Black–Scholes equation based on the theoretical analysis when the underlying financial asset is estimated using a stochastic volatility model. Also to investigate the analytical solution of this generalized Black–Scholes equation we have used Laplace transform homotopy perturbation method.

The Regularity of the Solutions to the Cauchy Problem for the Quasilinear Second-Order Parabolic Partial Differential Equations

This article is dedicated to expanding our comprehension of the regularity of the solutions to the Cauchy problem for the quasilinear second-order parabolic partial differential equations under fair general conditions on the nonlinear perturbations. In this paper have been obtained that the sequence of the weak solutions uz ∈ V1,02, z = 1,2,..... to the Cauchy problems for the Equations (15) under the initial conditions uz (0,x) = φ0z converges to the weak solution to the Cauchy problem for the Equation (1) under the initial condition u(0, x) = u0 in V1,02.

Second-Order Parabolic Equation to Model, Analyze, and Forecast Thermal-Stress Distribution in Aircraft Plate Attack Wing–Fuselage

During a flight, the steel plate attack wing–fuselage of an aircraft is subjected to cyclical thermal stress caused by flight altitude variation that could compromise the functionality of the plate. Thus, it is compulsory after a sequence of flights to evaluate the state of plate health. In this work, we propose a new dynamic model on the basis of the physical transmission of heat by conduction governed by a second-order parabolic partial differential equation with suitable initial and boundary conditions to analyze and forecast thermal stresses in the plate of a P64 OSCAR B airplane. Developing this model in the COMSOL Multi-Physics® environment, a finite-element technique was applied to achieve the thermal-stress map on the plate. The achieved results, equivalent to those obtained by a campaign of infrared thermographic experiment measurements (not yet used in the aeronautical industry), highlight the evolution of the thermal load of the steel plate attack wing–fuselage, adding evidence of possible incoming fatigue phenomena to identify in advance if the steel plate must be replaced.

The phase-change heat conduction analysis during solidification processes by a hybrid generalized FDM

In this paper, the single-domain enthalpy model is adopted for heat transfer analysis of phase change during solidification processes. The resulting second-order parabolic partial differential equations (PDEs) with varying thermophysical coefficients is numerically solved by a hybrid generalized finite difference method (GFDM) under mixed boundary conditions. The spatial derivatives in the PDEs are approximated by the Taylor series expansions combining with the moving-least squares technique. The temporal derivative is evaluated with a six-point symmetric difference by the classical Crank-Nicholson technique. The Newton-Raphson iteration method is used to solve the resulting nonlinear algebraic equations. Finally, the transient temperature field and the moving phase-change interface are obtained by analysing the nodal temperature distribution. Several examples are presented for verify the stability and effectiveness of this meshless method.

Strong approximation of monotone stochastic partial differential equations driven by white noise

Abstract We establish an optimal strong convergence rate of a fully discrete numerical scheme for second-order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen–Cahn equation, driven by an additive space-time white noise. Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one. Then we use the backward Euler in time and spectral Galerkin in space to fully discretise this random equation. By the monotonicity assumption, in combination with the factorisation method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a Hölder-type regularity for both stochastic and random equations. Finally, the strong convergence rate of the proposed fully discrete scheme is obtained. Several numerical experiments are carried out to verify the theoretical result.

Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance

AbstractThe invariant approach is employed to solve the Cauchy problem for the bond-pricing partial differential equation (PDE) of mathematical finance. We first briefly review the invariant criteria for a scalar second-order parabolic PDE in two independent variables and then utilize it to reduce the bond-pricing equation to different Lie canonical forms. We show that the invariant approach aids in transforming the bond-pricing equation to the second Lie canonical form and that with a proper parametric selection, the bond-pricing PDE can be converted to the first Lie canonical form which is the classical heat equation. Different cases are deduced for which the original equation reduces to the first and second Lie canonical forms. For each of the cases, we work out the transformations which map the bond-pricing equation into the heat equation and also to the second Lie canonical form. We construct the fundamental solutions for the bond-pricing model via these transformations by utilizing the fundamental solutions of the classical heat equation as well as solution to the second Lie canonical form. Finally, the closed-form analytical solutions of the Cauchy initial value problems for the bond-pricing model with proper choice of terminal conditions are obtained.

Stochastic and partial differential equations on non-smooth time-dependent domains

In this article, we consider non-smooth time-dependent domains whose boundary is W1,p in time and single-valued, smoothly varying directions of reflection at the boundary. In this setting, we first prove existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection. Secondly, we prove, using the theory of viscosity solutions, a comparison principle for fully nonlinear second-order parabolic partial differential equations with oblique derivative boundary conditions. As a consequence, we obtain uniqueness, and, by barrier construction and Perron’s method, we also conclude existence of viscosity solutions. Our results generalize two articles by Dupuis and Ishii to time-dependent domains.