Picard Iteration Technique Research Articles

Stochastic Burgers type equations with two reflecting walls: Existence, uniqueness and exponential ergodicity

In this paper, we study stochastic Burgers type equations with two reflecting walls driven by multiplicative noise. We first establish the existence and uniqueness of the solution by Picard iteration technique, the main trouble is to handle the complicated nonlinear term ∂g(u(x,t))∂x which involves the derivative of the solution u(x, t) and the singularities due to reflection. Furthermore, we prove the exponential ergodicity for the equations driven by possibly degenerate, multiplicative noise through asymptotic coupling. More generally, we also give a simplified criterion for exponential ergodicity of parabolic stochastic partial differential equations with reflections driven by degenerate multiplicative noise.

Stability analysis of random fractional-order nonlinear systems and its application

The research on stability analysis and control design for random nonlinear systems have been greatly popularized in recent ten years, but almost no literature focuses on the fractional-order case. This paper explores the stability problem for a class of random Caputo fractional-order nonlinear systems. As a prerequisite, under the globally and the locally Lipschitz conditions, it is shown that such systems have a global unique solution with the aid of the generalized Gronwall inequality and a Picard iterative technique. By resorting to Laplace transformation and Lyapunov stability theory, some feasible conditions are established such that the considered fractional-order nonlinear systems are respectively Mittag-Leffler noise-to-state stable, Mittag-Leffler globally asymptotically stable. Then, a tracking control strategy is established for a class of random Caputo fractional-order strict-feedback systems. The feasibility analysis is addressed according to the established stability criteria. Finally, a power system and a mass–spring-damper system modeled by the random fractional-order method are employed to demonstrate the efficiency of the established analysis approach. More critically, the deficiency in the existing literatures is covered up by the current work and a set of new theories and methods in studying random Caputo fractional-order nonlinear systems is built up.

The Existence and Averaging Principle for a Class of Fractional Hadamard Itô–Doob Stochastic Integral Equations

In this paper, we investigate the existence and uniqueness properties pertaining to a class of fractional Hadamard Itô–Doob stochastic integral equations (FHIDSIE). Our study centers around the utilization of the Picard iteration technique (PIT), which not only establishes these fundamental properties but also unveils the remarkable averaging principle within FHIDSIE. To accomplish this, we harness powerful mathematical tools, including the Hölder and Gronwall inequalities.

A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results.

A new result on averaging principle for Caputo-type fractional delay stochastic differential equations with Brownian motion

In this paper, we mainly explore the averaging principle of Caputo-type fractional delay stochastic differential equations with Brownian motion. Firstly, the solutions of this considered system are derived with the aid of the Picard iteration technique along with the Laplace transformation and its inverse. Secondly, we obtain the unique result by using the contradiction method. In addition, the averaging principle is discussed by means of the Burkholder-Davis-Gundy inequality, Jensen inequality, Hölder inequality and Grönwall-Bellman inequality under some hypotheses. Finally, an example with numerical simulations is carried out to prove the relevant theories.

Fractional Itô–Doob Stochastic Differential Equations Driven by Countably Many Brownian Motions

This article is devoted to showing the existence and uniqueness (EU) of a solution with non-Lipschitz coefficients (NLC) of fractional Itô-Doob stochastic differential equations driven by countably many Brownian motions (FIDSDECBMs) of order ϰ∈(0,1) by using the Picard iteration technique (PIT) and the semimartingale local time (SLT).

Numerical solution of two-point nonlinear boundary value problems via Legendre–Picard iteration method

The aim of the present work is to introduce an effective numerical method for solving two-point nonlinear boundary value problems. The proposed iterative scheme, called the Legendre–Picard iteration method is based on the Picard iteration technique, shifted Legendre polynomials and Legendre–Gauss quadrature formula. In the Legendre–Picard iteration method, the boundary value problem is reduced to an iterative formula for updating the coefficients of the approximate solution in each step and with a straightforward manner, the integrals of the shifted Legendre polynomials are calculated. In addition, to reduce the CPU time, a vector–matrix scheme of the Legendre–Picard iteration method is constructed. The convergence analysis of the method is studied. Five nonlinear boundary value problems are given to illustrate the validity of the Legendre–Picard iteration method. Numerical results indicate the good performance and the precision of the proposed procedure.

A dimensionally-reduced fracture flow model for poroelastic media with fluid entry resistance and fluid slip

We develop a model which couples the flow in a discrete fracture to a deformable porous medium. To account for the discrete representation of the fracture, a dimensionally-reduced fluid flow model is proposed. The fluid flow model incorporates both a reduced permeability of the fracture walls due to the skin effect, and a slip of fluid flowing along the permeable fracture walls. Biot's model for poroelastic media is coupled to a fracture flow model based on a thin-film approximation of the compressible Navier-Stokes equations. The fracture flow model incorporates a fluid entry resistance parameter to relate the leak-off through the fracture walls to a pressure jump across the fracture walls, and the Beavers-Joseph-Saffman slip rate coefficient to represent the fluid slip along the fracture walls. The numerical model is based on a thermodynamic framework in which all energy storage and dissipative mechanisms in the problem are identified, including the mechanisms related to the interface effects. The thermodynamic framework is employed to solve the nonlinear coupled problem up to a specified energy range through a Picard iteration technique and to study the model and its results. Studies are presented for a range of fluid entry resistance parameters and Beavers-Joseph-Saffman slip rate coefficients, showing the capability of the model to simulate skin and slip effects in a dimensionally-reduced fracture setting.

Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems

In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi‐Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi‐Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation. Also, a vector‐matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method.

Modified Chebyshev Wavelet-Picard Technique for Thin Film Flow of Non-Newtonian Fluid Down an Inclined Plane

For the past few decades, solutions of the nonlinear differential equations are of great importance and interest for researchers and scientists around the globe. Most of the physical phenomena’s around us can easily be modeled into nonlinear differential equations. In this article, a new method is proposed by using shifted Chebyshev wavelets and Picard iteration technique to tackle with the nonlinearity of these physical problems. The proposed scheme is very user friendly and extremely accurate. The accuracy of the proposed scheme is verified by the help of a nonlinear physical model representing the thin film flow of a third grade fluid down an inclined plane. Numerical solution is also sought using Runge–Kutta order 4 method. Results are obtained by shifted Chebyshev wavelets at different iterations of Picard technique for different values of parameters are described in table and graphs which confirms the accuracy and stability of the proposed technique.

Boundary value problems for impulsive Bagley–Torvik models involving the Riemann–Liouville fractional derivatives

By using the Picard iterative technique, we get the exact expression of solutions of linear fractional Bagley–Torvik model. Then the piecewise continuous solutions of this kind of models are obtained. By using these results, we convert the boundary value problems for impulsive fractional differential equations to integral equations technically. By using the weighted function Banach spaces, the completely continuous operators and the Schauder’s fixed point theorem, some existence results for solutions of the boundary value problems for impulsive Bagley–Torvik fractional differential equations are established. Examples are given to illustrate the main results.

Some Methods of Numerical Solutions of Singular System of Transistor Circuits

In this paper, observer design in generalized state space also known as singular system of transistor circuits is solved using the Differential transform method (DTM) and the Picard iterative technique (PIT). The numerical results obtained via these methods converge rapidly to their associated exact solutions upon comparison. It is obvious that these results are in excellent agreement with those already in literature via other numerical methods. However, the DTM reveals the ease of application and fewer computations compared to other numerical methods. Whereas, the PIT requires the Lipschitzian continuity condition to be satisfied.

Picard Iterations for Diffusions on Symmetric Matrices

Matrix-valued stochastic processes have been of significant importance in areas such as physics, engineering and mathematical finance. One of the first models studied has been the so-called Wishart process, which is described as the solution of a stochastic differential equation in the space of matrices. In this paper, we analyze natural extensions of this model and prove the existence and uniqueness of the solution. We do this by carrying out a Picard iteration technique in the space of symmetric matrices. This approach takes into account the operator character of the matrices, which helps to corroborate how the Lipchitz conditions also arise naturally in this context.

Picard iterative processes for initial value problems of singular fractional differential equations

In this paper, the initial value problems of singular fractional differential equations are discussed. New criteria on the existence and uniqueness of solutions are obtained. The well-known Picard iterative technique is then extended for fractional differential equations which provides computable sequences that converge uniformly to the solution of the problems discussed. We obtain not only the existence and uniqueness of solutions for the problems, but we also establish iterative schemes for uniformly approximating the solutions. Two examples are given to illustrate the main theorems.

Second‐order two‐dimensional solution for the drainage of recharge based on Picard's iteration technique: A generalized Dupuit‐Forchheimer equation

Aquifer recharge is one of the most important problems in hydrology from both theoretical and practical points of view. One of the most widely accepted methods to deal with this problem is the use of the Dupuit‐Forchheimer theory. This theory assumes that the water table is almost horizontal, the vertical velocity is zero, and the horizontal velocity is uniform with depth. Surfaces of seepage are not considered. Despite these strong limitations the theory is applied, and success is frequently found in many cases despite its fundamental assumptions being violated. In this work an approximate 2‐D solution to the problem is sought on the basis of Picard's iteration technique, from which a second‐order differential equation for recharge problems is found. On the basis of this solution, a modified, analytical Dupuit‐Forchheimer (DF) ellipse is found which compares favorably with the full 2‐D solution of the problem. The analytical developments of this theory provide a generalized DF theory which permits as an outcome the analytical determination of the surface of seepage.

Acceleration techniques for the iterative resolution of the Richards equation by the finite volume method

Groundwater flow in variably saturated soils is described by the nonlinear Richards' equation. The mass-conservative finite volume discretization recently proposed in (Adv. Water Resour. 2004; 27:1199–1215) produces a nonlinear algebraic problem, whose resolution demands for the application of an appropriate iterative strategy, such as the Picard or the Newton scheme. The Picard iterative technique results in a robust fixed-point method, which is globally convergent at a linear or sub-linear rate. On the other hand, the Newton iterative technique shows better convergence rates, but the computational cost of the full calculation is still comparable to the one of the Picard schemes, due to the additional calculation of the Jacobian matrix and its formal inversion. In this work, we investigate two acceleration techniques to reduce the cost of Newton iterations. These techniques are respectively based on a linearization of the nonlinear Jacobian matrix in accordance with the quasi-Newton approach and on a Broyden-type rank-one correction for a fixed number of sub-iterations. Numerical experiments on a set of two-dimensional test case modeling the uptake of plant roots in a highly heterogeneous soil show the performance and the effectiveness of both acceleration schemes in comparison with the original Picard and Newton methods. Copyright © 2009 John Wiley & Sons, Ltd.

Soil Moisture Dynamics Modeling Considering the Root Compensation Mechanism for Water Uptake by Plants

A numerical model is developed in this study for simulating soil moisture flow in layered soil profile with plant growth. A dynamic root compensation mechanism (RCM) is used for a nonuniform root distribution pattern to compute water uptake by plants in a moisture scarce environment. The governing soil moisture flow equation integrated with the roots water uptake function is solved numerically by the implicit finite difference method coupled with the Picard iteration technique. The model is first tested for a barren layered soil profile using numerical simulation data available in the literature. A nonlinear function for water uptake by roots is then incorporated in the flow equation and the rate of water removal is simulated with and without considering the RCM for a characteristic example under optimal and water scare conditions. The model is finally applied to a rain-fed wheat (Triticum aestivum) field using a dynamic root growth model. The simulation considering the RCM shows better agreement with the observed data compared to the model results obtained without considering the RCM. Results show that under favorable soil moisture conditions, plants extract water at the maximum rate according to the root distribution pattern and when the moisture stress is developed in the upper soil profile, the diminished water uptake rate in the water scarce region is compensated for by an enhanced water uptake from the lower wetter layers. The model can be used for sound irrigation management practices especially in the water scarce arid and semiarid regions and can also be integrated with a transport equation to predict the solute uptake by plant root biomass.

Modeling rhizofiltration: heavy-metal uptake by plant roots

The discovery of phytoaccumulation potential of plant species has led to its application for remediation of heavy-metal-contaminated soil and wastewater, which is termed as phytoextraction/rhizofiltration. For prediction, analysis, planning and cost-effective design of such systems, mathematical models not only are used as a screening tool but also provide optimal parameters like harvesting time, irrigation schedule, etc. Several laboratory and field scale studies have been carried out in the past, and mathematical expressions have been developed by various researchers for different phenomena like metal adsorption in soil, plant root growth with time, moisture and metal uptake by plant root, moisture movement in unsaturated zone, soil moisture relationship, etc. The complete design of any such phytoremediation program would require the knowledge of behavior of heavy-metal movement in soil, water and plant root system. In this paper, a model for simulating heavy-metal dynamics in soil, water and plant root system is developed and discussed. The governing non-linear partial differential equation is solved numerically by implicit finite difference method using Picard's iterative technique, and the formulation has been illustrated using a characteristic example. The source code is written in MATLAB.

SIMULATING WATER MOVEMENT AND ITS UPTAKE BY PLANT ROOTS IN UNSATURATED ZONES

Existing mathematical models that simulate water movement and contaminant transport in unsaturated zone do not take into account the water uptake by plant crop roots resulting in an error in the prediction of water flux and, thereby, contaminant concentration. Moreover the application of these models is often limited due to the lack of easily accessible and representative soil hydraulic properties, like moisture retention characteristic and unsaturated hydraulic conductivity. In this study, a mathematical model is developed for simulating water movement in unsaturated zones by integrating the one‐dimensional transient unsaturated water flow equation (Richard's equation) with a root water extraction term (sink), and also incorporates pedo‐transfer functions (PTF) for estimating soil hydraulic properties. The governing non‐linear partial differential equation is solved numerically by the implicit finite difference method using Picard's iterative technique, the formulation has been illustrated by a characteristic example.

The ac-dc difference of single-junction thermal converters

A nonlinear parabolic partial differential equation of heat conduction subject to the Robin boundary conditions is considered. The equation describes energy conservation in the heater wire of a single-junction thermal converter for both ac and dc currents. In the audio-frequency range the temperature distribution along the heater is calculated and the ac-dc difference deduced by means of the Picard iterative technique. The previously neglected effects of radiation and the thermal properties of the heater wire are included. An expression for the ac-dc difference is also derived in the low-frequency regime. The thermal conductance of the thermocouple, which has previously been neglected, is taken into account. The calculated increase in the ac-dc difference is consistent with recent measurements. The solution in this limit is found by both an eigenfunction-expansion method and the Laplace-transform method. Comparison of the solutions obtained by the two methods gives some useful formulae for the summation of numerical series.