One-dimensional Parabolic Partial Differential Equations Research Articles

A novel fitted numerical scheme for singularly perturbed delay parabolic problems with two small parameters

In this paper, fitted operator finite difference methods are presented for two-parameter singularly perturbed one-dimensional parabolic partial differential equations with a delay in the time variable. Such boundary value problems are frequently encountered in the spatial diffusion of reactants and in control systems. To approximate the solution of the problem, we consider the Crank–Nicolson method for time discretization and the non-standard scheme for space discretization on uniform meshes. We prove that the proposed numerical method is uniformly convergent, having convergence of order two in time and one in space. Further, we present numerical results for two test problems in support of our theoretical findings and to demonstrate that the non-standard scheme is more effective than the hybrid scheme comprising central difference, upwind, and mid-point.

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

Optimal investment strategy for a family with a random household expenditure under the CEV model

This paper considers an optimal investment strategy to maximize the expected constant absolute risk averse (CARA) utility of the terminal wealth for a family in the presence of stochastic household expenditure under the constant elasticity of variance (CEV) model. Since the corresponding Hamilton-Jacobi-Bellman (HJB) equation is difficult to solve for the high dimensionality and nonlinearity, previous work only gives an approximate numerical solution for some special model parameters under the slow-fluctuating regime assumption. In this paper, by directly conjecturing the functional form of the value function, we transform the HJB equation into two one-dimensional parabolic partial differential equations (pdes) and further find their explicit solutions via the Feynman-Kac formula. We prove that the exact and explicit solution for the value function as well as the optimal investment strategy can be expressed as integral of confluent hyper-geometric function. Finally, numerical examples are provided to illustrate the effects of parameters on the optimal strategies.

Modified Method for Parabolic Equations in one Dimensional with Nonlocal time Weighting Initial Condition

In this paper, we develop a modified version of explicit scheme based on finite difference method for the one-dimensional parabolic partial differential equations with nonlocal time weighting initial conditions. The dominancy of the Saulyev’s schemes based on finite difference, over the previous explicit FTCS, Duke-Frankel, as well as implicit BTCS, Crandal’s technique and Crank Nicholson’s scheme has already been established, which proved to be unconditionally stable, use less CPU time and computational effort. However, in this paper a modification of Saulyev’s first kind formula is developed. Main focus was to reduce error of the Saulyev’s formula using proposed method. The comparison has been carried out between both methods to observe errors in different conditions and step sizes. The new modified scheme is proved to be satisfactory and unconditionally stable.

Extended BACOLI

BACOLI is a Fortran software package for solving one-dimensional parabolic partial differential equations (PDEs) with separated boundary conditions by B-spline adaptive collocation methods. A distinguishing feature of BACOLI is its ability to estimate and control error and correspondingly adapt meshes in both space and time. Many models of scientific interest, however, can be formulated as multiscale parabolic PDE systems, that is, models that couple a system of parabolic PDEs describing dynamics on a global scale with a system of ordinary differential equations describing dynamics on a local scale. This article describes the Fortran software eBACOLI, the extension of BACOLI to solve such multiscale models. The performance of the extended software is demonstrated to be statistically equivalent to the original for purely parabolic PDE systems. Results from eBACOLI are given for various multiscale models from the extended problem class considered.

Input-to-State Stabilization of Uncertain Parabolic PDEs Using an Observer-Based Fuzzy Control

This paper presents an observer-based fuzzy controller for one-dimensional parabolic partial differential equations with distributed bounded disturbance using discrete point measurements. Sufficient conditions are derived to guarantee the input-to-state stability of the closed-loop system in terms of linear matrix inequalities. A numerical simulation example is given to demonstrate the efficiency of the results.

On the Theoretical and Numerical Control of a One-Dimensional Nonlinear Parabolic Partial Differential Equation

This paper deals with the analysis of the internal and boundary control of a one-dimensional parabolic partial differential equation with nonlinear diffusion. First, we prove a local null controllability result with distributed controls, locally supported in space. The proof relies on local inversion (more precisely, we use Liusternik’s Inverse Function Theorem), together with some appropriate specific estimates. We also establish a similar result with controls on one side of the boundary. Then, we consider an iterative algorithm for the computation of null controls, we prove the convergence of the iterates, and we perform some numerical experiments.

Fitted Numerical Methods for Singularly Perturbed One-Dimensional Parabolic Partial Differential Equations with Small Shifts Arising in the Modelling of Neuronal Variability

In this paper, we presented exponentially fitted finite difference methods for a class of time dependent singularly perturbed one-dimensional convection diffusion problems with small shifts. Similar boundary value problems arise in computational neuroscience in determination of the behavior of a neuron to random synaptic inputs. When the shift parameters are smaller than the perturbation parameter, the shifted terms are expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed. The proposed finite difference scheme is unconditionally stable and is convergent with order $$O(\Delta t+h^{2})$$ where $$\Delta t$$ and h respectively the time and space step-sizes. When the shift parameters are larger than the perturbation parameter a special type of mesh is used for the space variable so that the shifts lie on the nodal points and an exponentially fitted scheme is developed. This scheme is also unconditionally stable. By means of two examples, it is shown that the proposed methods provide uniformly convergent solutions with respect to the perturbation parameter. On the basis of the numerical results, it is concluded that the present methods offer significant advantage for the linear singularly perturbed partial differential difference equations.

Cardinal Hermite interpolant multiscaling functions for solving a parabolic inverse problem

An effective method based upon cardinal Hermite interpolant multiscaling functions is proposed for the solution of the one-dimensional parabolic partial differential equation with given initial condition and known boundary conditions and subject to overspecification at a point in the spatial domain. The properties of multiscaling functions are first presented. These properties together with a collocation method are then utilized to reduce the parabolic inverse problem to the solution of algebraic equations. The scheme described is efficient. The numerical results obtained using the present algorithms for test problems show that this method can solve the model effectively.

ISS In Different Norms For 1-D Parabolic Pdes With Boundary Disturbances

For one-dimensional parabolic partial differential equations with disturbances at both boundaries and distributed disturbances we provide input-to-state stability (ISS) estimates in various norms. Due to the lack of an ISS Lyapunov functional for boundary disturbances, the proof methodology uses (i) an eigenfunction expansion of the solution, and (ii) a finite-difference scheme. The ISS estimate for the sup-norm leads to a refinement of the well-known maximum principle for the heat equation. Finally, the obtained results are applied to quasi-static thermoelasticity models that involve nonlocal boundary conditions. Small-gain conditions that guarantee the global exponential stability of the zero solution for such models are derived.

L-Stable Block Backward Differentiation Formula for Parabolic Partial Differential Equations

Abstract In this paper, an L-stable Second Derivative Block Backward Differentiation Formula (SDBBDF) of order 5 is presented for the solutions of parabolic equations. It applied the use of the classical method of lines for the discretization of the parabolic equations. The method reduces the one-dimensional parabolic partial differential equation which has integral or non-integral boundary conditions to a system of Ordinary Differential Equations (ODEs) with initial conditions. The stability properties of the block method are investigated using the boundary locus plot and the method was found to be L-stable. The derived method is implemented on standard problems of parabolic equations and the results obtained show that the method is accurate and efficient.

Application of Chebyshev collocation method for solving two classes of non-classical parabolic PDEs

This article contributes a numerical scheme for finding approximate solutions of one-dimensional parabolic partial differential equations (PDEs) under non-classical boundary conditions. This scheme is based on the direct Chebyshev collocation method that has been frequently used for problems of ordinary differential equations (ODEs). In fact, the approximate solution of the problem in the truncated Chebyshev series form is obtained by this method. By substituting Chebyshev series solution into the considered problems and by using the matrix operations and the collocation points, the suggested scheme reduces the problems into the associated linear algebraic systems. By solving this system of equations, the unknown Chebyshev coefficients can be determined. To show the accuracy and the efficiency of the method, two numerical examples are implemented and the comparisons are given by a new collocation method.

Transversal Method of Lines for Unsteady Heat Conduction With Uniform Surface Heat Flux

The transversal method of lines (TMOL) is a general hybrid technique for determining approximate, semi-analytic solutions of parabolic partial differential equations. When applied to a one-dimensional (1D) parabolic partial differential equation, TMOL engenders a sequence of adjoint second-order ordinary differential equations, where in the space coordinate is the independent variable and the time appears as an embedded parameter. Essentially, the adjoint second-order ordinary differential equations that result are of quasi-stationary nature, and depending on the coordinate system may have constant or variable coefficients. In this work, TMOL is applied to the unsteady 1D heat equation in simple bodies (large plate, long cylinder, and sphere) with temperature-invariant thermophysical properties, constant initial temperature and uniform heat flux at the surface. In engineering applications, the surface heat flux is customarily provided by electrical heating or radiative heating. Using the first adjoint quasi-stationary heat equation for each simple body with one time jump, it is demonstrated that approximate, semi-analytic TMOL temperature solutions with good quality are easily obtainable, regardless of time. As a consequence, usage of the more involved second adjoint quasi-stationary heat equation accounting for two consecutive time jumps come to be unnecessary.

Simultaneous estimation of nonlinear parameters in parabolic partial differential equation using quantum-behaved particle swarm optimization with Gaussian mutation

In this paper, an improved quantum-behaved particle swarm optimization with Gaussian mutation is proposed to simultaneously estimate nonlinear parameters in a one-dimensional parabolic partial differential equation (PDE). No a priori information about the functional form is available, therefore the problems may be treated as function estimation which is difficult to estimate using traditional gradient-based methods. Measurements on the boundary are used in the least square modelling. Tikhonov regularization technique is used to stabilize the ill-posed problem. The numerical benchmark and experiment results demonstrate the validity and efficiency of the proposed method to solve inverse problems of estimating nonlinear parameters in parabolic PDEs.

ON THE TIME-DISCRETIZATION METHOD FOR THE UNSTEADY HEAT CONDUCTION EQUATION WITH UNIFORM HEAT FLUX BOUNDARY CONDITION

The time-discretization method is a powerful method to provide approximate, semi-analytic solutions of parabolic differential equations in one or more dimensions. In the case of one-dimensional parabolic partial differential equations, the time-discretization method employs the two-point backward approximation for the time-derivative, while leaving the space derivative continuous. This is a simple operation that engenders a sequence of adjoint second-order ordinary differential equations, wherein the space coordinate is the independent variable and a fixed time appears as an embedded parameter. In this work, the time-discretization method is applied to the unsteady 1-D heat equation in a large plate with constant initial temperature and uniform surface heat flux as the boundary condition. Conceptually, the associated sequence of adjoint second-order ordinary differential equations of heat conduction are of quasi-stationary nature. Using the first adjoint quasi-stationary heat equation with one time jump, it is demonstrated that an approximate, semi-analytic temperature solution of good quality is easily obtainable and is valid at all time.

An Efficient Spectral Approximation for Solving Several Types of Parabolic PDEs with Nonlocal Boundary Conditions

The problem of solving several types of one-dimensional parabolic partial differential equations (PDEs) subject to the given initial and nonlocal boundary conditions is considered. The main idea is based on direct collocation and transforming the considered PDEs into their associated algebraic equations. After approximating the solution in the Legendre matrix form, we use Legendre operational matrix of differentiation for representing the mentioned algebraic equations clearly. Three numerical illustrations are provided to show the accuracy of the presented scheme. High accurate results with respect to the Bernstein Tau technique and Sinc collocation method confirm this accuracy.

New algorithm for nonclassical parabolic problems based on the reproducing kernel method

Purpose: In this paper, a new algorithm is presented for solving one-dimensional parabolic partial differential equations subject to integral conditions. Methods: The algorithm is based on the transverse method of lines and the reproducing kernel method. The transverse method of lines can reduce a one-dimensional parabolic partial differential equation subject to integral conditions to a series of ordinary differential equations(ODEs) with integral boundary conditions. The reproducing kernel method is a relative new analytical technique, which can solve successfully ODEs with integral boundary conditions.

Boundary Control of Boundary Coupled Parabolic Distributed Parameter Systems based on Partial State Transformation

This paper considers a boundary stabilization problem of systems described by boundary coupled two one-dimensional parabolic partial differential equations based on the backstepping method. Unlike the conventional backstepping framework, we propose a state transformation that contains the state of the unforced subsystem partially. It is proved that this transformation is well-defined and converts the original system into an exponentially stable system together with an associated state feedback control law under a condition that ensures the stability of the unforced subsystem. The original system with this control law inherits the stability of the transfomed one. The effectiveness of the obtained control law is demonstrated numerically.

Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond

We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in [Berglund and Gentz, J. Phys. A 42:052001 (2009)]. The proofs rely on a spectral Galerkin approximation of the SPDE by a finite-dimensional system, and on a potential-theoretic approach to the computation of transition times in finite dimension.

A Numerical Method of High Accuracy for Linear Parabolic Partial Differential Equations

We report a new numerical algorithm for solving one-dimensional linear parabolic partial differential equations (PDEs). The algorithm employs optimal quadratic spline collocation (QSC) for the space discretization and two-stage Gauss method for the time discretization. The new algorithm results in errors of fourth order at the gridpoints of both the space partition and the time partition, and large time steps are allowed to save computational cost. The stability of the new algorithm is analyzed for a model problem. Numerical experiments are carried out to confirm the theoretical order of accuracy and demonstrate the effectiveness of the new algorithm.