Equivalent Initial Value Problem Research Articles

Global existence of solutions for a free‐boundary tumor model with angiogenesis and a necrotic core

In this paper, we study a free‐boundary problem modeling the growth of spherically symmetric tumors with angiogenesis and a necrotic core, where the Robin boundary condition is imposed for the nutrient concentration. The existence of a global solution is established by first reducing the free‐boundary problem into an equivalent initial boundary value problem for a nonlinear strongly singular parabolic equation on a fixed domain, then proving that an approximation problem admits a unique solution by the Schauder fixed point theorem combined with the estimates for parabolic equations, and finally taking the limit. Compared with the Dirichlet boundary value condition problem, the Robin condition causes some new difficulties in making rigorous analysis of the model, particularly on the uniqueness of solutions to the approximation problem.

Использование Python для численного решения начально-краевых задач уравнения теплопроводности и визуализации результатов

в работе описана реализация численного метода контрольного объема для решения одного класса трехмерных начально-краевых задач теплопроводности в шаре на языке программирования Python. Проведено тестирование на задаче с  аналитическим решением и выполнена визуализация результатов расчета средствами Python. this study uses the control volume method for solving a class of 3D initial-boundary value problems of the heat equation in a ball with Python. The code was tested with an analytically solvable problem, and the results were visualized.

A SHOOTING METHOD SOLUTION FOR A CONVECTIVE-RADIATIVE STRAIGHT LONGITUDINAL FIN WITH VARIABLE THERMAL PROPERTIES AND NONLINEAR BOUNDARY CONDITIONS

We develop in this work a simple and highly efficient shooting approach for solving the fin energy equation with multiple nonlinearities. The present fin problem is characterized by temperature-dependent thermal conductivity, heat transfer coefficient, and surface radiation emissivity, where the fin base is imposed to a constant temperature and the fin tip is subjected to a combination of convective and radiative heat losses. The governing fin boundary value problem is first reduced into an equivalent initial value problem and then integrated using the fourth-order Runge-Kutta method. The temperature gradient at the tip is approximated by a five-point backward finite difference formula, and computed iteratively using the secant method on the base, which is decisive for numerical integration. The fin problem is solved and compared for two cases of tip boundary condition: an adiabatic fin tip and a convective-radiative fin tip. A thermal analysis is performed using Biot number, Stark number, and the geometrical number that stands for the ratio of fin surface area to its cross-sectional area. Solutions are computed and compared to those obtained by the boundary value problem method, Galerkin method, and the Adomian decomposition method under the assumption of adiabatic fin tip. The accuracy of the nonlinear shooting method is checked by evaluating the absolute errors. Comparative results show that the fin temperature distribution and the fin efficiency are significantly impacted not only by Biot number, Stark number, and geometrical number but also by the type of the tip boundary condition, which could dramatically degrade the fin efficiency mainly for small values of geometrical number.

A Predictor-Corrector Method for Nonlinear Fractional Differential Equation Initial Value Problems on Graded Grids

A Predictor-Corrector Method for Nonlinear Fractional Differential Equation Initial Value Problems on Graded Grids

On Euler methods for Caputo fractional differential equations

Numerical methods for fractional differential equations have specific properties with respect to the ones for ordinary differential equations. The paper discusses Euler methods for Caputo differential equation initial value problem. The common properties of the methods are stated and demonstrated by several numerical experiments. Python codes are available to researchers for numerical simulations.

Multirate Exponential Rosenbrock Methods

In this paper we propose a novel class of methods for high order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix $\varphi$-functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called Multirate Exponential Rosenbrock (MERB) methods. They consist of the solution to a sequence of modified "fast" initial-value problems, that may themselves be approximated through subcycling any desired IVP solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders two through six (the highest order ever constructed infinitesimal multirate methods). We then present numerical simulations to confirm these theoretical convergence rates, and to compare the efficiency of MERB methods against other recently-introduced high order multirate methods.

Novel aspects of discrete dynamical type inequalities within fractional operators having generalized ℏ-discrete Mittag-Leffler kernels and application

Discrete fractional calculus (DFC) has had significant advances in the last few decades, being successfully employed in the time scale domain ℏZ. Understanding of DFC has demonstrated a valuable improvement in neural networks and modeling in other terrains. In the context of Riemann form (ABR), we discuss the discrete fractional operator influencing discrete Atangana-Baleanu (AB)-fractional operator having ℏ-discrete generalized Mittag-Leffler kernels. In the approach being presented, some new Pólya-Szegö and Chebyshev type inequalities introduced within discrete AB-fractional operators having ℏ-discrete generalized Mittag-Leffler kernels. By analyzing discrete AB-fractional operators in the time scale domain Z, we can perform a comparison basis for notable outcomes derived from the aforesaid operators. This type of discretization generates novel outcomes for synchronous functions. The specification of this proposed strategy simply demonstrates its efficiency, precision, and accessibility in terms of the methodology of qualitative approach of discrete fractional difference equation solutions, including its stability, consistency, and continual reliance on the initial value for the solutions of many fractional difference equation initial value problems. The repercussions of the discrete AB-fractional operators can depict new presentations for various particular cases. Finally, applications concerning bounding mappings are also illustrated.

On the boundedness of solution of the second order ordinary differential equation with damping term and involution

In the present paper the initial value problem for the second order ordinary differential equation with damping term and involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with damping term and involution is established.

Modeling continuous glucose monitoring with fractional differential equations subject to shocks

Continuous Glucose Monitoring (CGM) produces long time-series of noisy observations of a single variable (tissue glucose concentration), whose evolution may be explained by a dynamical model. In order to represent the unknown mixture of possible control mechanisms of different orders affecting the measured variable, a fractional differential approach seems justified. In any case, variations in food intake and/or physical activity ought to be taken into account if a plausible interpretation of the dynamics is to be obtained. In the present work, the mathematical construction and the numerical implementation of a Fractional Differential Equations (FDE) initial value problem are systematically reviewed, with the intent of offering the reader a concise and mathematically rigorous description of this approach. An FDE model for CGM is formulated: the model includes compartments for stomach and intestinal glucose contents and for blood and tissue (subcutaneous) glucose concentrations, as well as the shock effects of food ingestion and of increased glucose consumption due to physical activity. The model parameters, including the (non-integer) order of differentiation, are estimated from CGM observations on six Type 1 diabetic patients. The best-fit fractional orders for the six subjects range from 1.59 to 2.13. For comparison, best fits have also been computed for all subjects using an average fractional order of 1.9 and integer orders of 1 and 2.The results indicate that in the case of CGM the fractional differential model, which should be physiologically more appropriate, in fact fits the data much better than the first-order model and also better than the 2nd-order model.

Initial value problems for equations with a composition of fractional derivatives

Initial value problems for equations with a composition of fractional derivatives

The fractional derivative type identification for the modelling deformation and strength characteristics of polymer concrete

The article contains the comparative use analysis of fractional derivative three types (Riemann -Liouville, Caputo and model) at the Begley -Torvik equation when modeling the deformation and strength characteristics of polymer concrete. We also consider how to obtain a solution to the Begley -Torvik equation initial value problem for the fractional derivative model using a recurrent kernels sequence. Further, the correctness is substantiated of establishing the type and the fractional derivative order at the Begley -Torvik equation initial value problem.

On the Boundedness of Solution of the Second Order Ordinary Differential Equation with Involution

In the present paper, the initial value problem for the second order ordinary differential equation with involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with involution is established.

A New Series Solution Applicable to a Class of Boundary Layer Equations with Exponential Decay in Solution

A new series solution for the coupled nonlinear boundary value problem (BVP) of the Falkner–Skan-type, resulting from the use of the boundary layer approximation to a mixed convective flow is obtained. The solution domain is transformed from the semi-infinite interval of physical interest to the unit interval by using the transformation $$z=1-e^{-\eta }$$ . The coupled nonlinear BVP is converted into an equivalent initial value problem (IVP) by supplying appropriate initial conditions. Series solution of the equivalent IVP is obtained in powers of $$1-e^{-\eta }$$ with assistance from the Newton–Raphson method. Convergence of the series is assured by the very design of the series as confirmed by the Domb–Sykes plots and the plots of partial sums. The results of the series solution compare well with the numerical results obtained by the shooting method based on the Runge–Kutta–Fehlberg45 and Newton–Raphson methods. The novelty of the methodology lies in the construction of a new series solution for a class of problems with exponential decay in their solution. Such a work has not been reported before.

Solutions of a non‐local aggregation equation: Universal bounds, concavity changes, and efficient numerical solutions

We consider a one‐dimensional aggregation equation for a non‐negative density associated with a quartic potential ( , ). We show that for the case of symmetric initial data [ ], the solution of the aggregation equation can be expressed in terms of an explicit function of , , and , where the functions and are determined by an ordinary differential equation initial value problem, the numerical solution of which is relatively straightforward. The function , which can be interpreted as an upper bound for the radius of the support of , is finite for all , even if the support of is unbounded, while is a potential related to the time integral of the second spatial moment of . We develop various bounds on and and a number of results concerning convexity and monotonicity that require no knowledge of other than its symmetry. We show that many, but not all, of these results persist if the assumption of symmetry is relaxed. Our general results are tested against numerical solutions for two examples of symmetric . We also exhibit some counterintuitive behaviour of the model, by finding conditions under which as , even if we start with and take , which ensures ultimate collapse to a single Dirac component at the origin.

Global optimization of stiff dynamical systems

AbstractWe present a deterministic global optimization method for nonlinear programming formulations constrained by stiff systems of ordinary differential equation (ODE) initial value problems (IVPs). The examples arise from dynamic optimization problems exhibiting both fast and slow transient phenomena commonly encountered in model‐based systems engineering applications. The proposed approach utilizes unconditionally stable implicit integration methods to reformulate the ODE‐constrained problem into a nonconvex nonlinear program (NLP) with implicit functions embedded. This problem is then solved to global optimality in finite time using a spatial branch‐and‐bound framework utilizing convex/concave relaxations of implicit functions constructed by a method which fully exploits problem sparsity. The algorithms were implemented in the Julia programming language within the EAGO.jl package and demonstrated on five illustrative examples with varying complexity relevant in process systems engineering. The developed methods enable the guaranteed global solution of dynamic optimization problems with stiff ODE–IVPs embedded.

Steady incompressible magnetohydrodynamics Casson boundary layer flow past a permeable vertical and exponentially shrinking sheet: A stability analysis

AbstractSteady incompressible magnetohydrodynamic mixed convection boundary layer flow of a Casson fluid on an exponentially vertical shrinking sheet using the non‐Newtonian heating equation is investigated in this paper. There are three main objectives of this study, namely, to develop a new mathematical model, to obtain multiple solutions, and to perform stability analysis. The governing partial differential equations have been changed into nonlinear ordinary differential equations. The resultant equations of boundary value problems are then converted into the equivalent initial value problems using the shooting method before they can be solved using Runge‐Kutta of order four. The numerical results are obtained and found to be in good agreement with the published literature. The results also indicate that the velocity boundary layer becomes thinner as the magnetic, slip, and Casson parameters increase. Dual solutions for temperature and velocity distributions are obtained. Furthermore, the results suggest that the presence of the force of buoyancy (opposing flow case) would cause the occurrence of dual solutions. However, based on the stability analysis, only the first solution is stable.

A Generalized Series Solution for the Boussinesq Equation With Constant Boundary Conditions

AbstractIn this work we consider the Boussinesq equation applied to a one‐dimensional homogeneous aquifer with constant boundary conditions. Such a problem is generally solved (both analytically and numerically) by transforming it from a boundary value problem into an equivalent initial value problem that has to be solved by means of a trial‐and‐error approach. We devise a method that eliminates the need for trial and error that is purely analytical, which can benefit both numerical and analytical solutions. Furthermore, we present a general series solution that solves this equation analytically with arbitrary precision. The series solution generalizes two particular cases that already exist in the literature. With the new proposed series, the need for any numerical integration of differential equations is removed, and fully analytical benchmark solutions are obtained.

A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.

Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova & Rakhlin,2013; Farmer et al., 2013; Donier et al., 2015; Tóth (2016).Mathematically, the Hamilton–Jacobi–Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.

Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013; Farmer et al., 2013; Donier et al., 2015; Toth et al., 2016). Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.