General Initial Value Research Articles

Direct integrator of block type methods with additional derivative for general third order initial value problems

This paper presents the construction of the two-point and three-point block methods with additional derivatives for directly solving [Formula: see text]. The proposed block methods are formulated using Hermite Interpolating Polynomial and approximate the solution of the problem at two or three-point concurrently. The block methods obtain the numerical solutions directly without reducing the equation into the first order system of initial value problems (IVPs). The order and zero-stability of the proposed methods are also investigated. Numerical results are presented and comparisons with other existing block methods are made. The performance shows that the proposed methods are very efficient in solving the general third order IVPs.

A new variation for the relativistic Euler equations

The Glimm scheme is one of the so famous techniques for getting solutions of the general initial value problem by building a convergent sequence of approximate solutions. The approximation scheme is based on the solution of the Riemann problem. In this paper, we use a new strength function in order to present a new kind of total variation of a solution. Based on this new variation, we use the Glimm scheme to prove the global existence of weak solutions for the nonlinear ultra-relativistic Euler equations for a class of large initial data that involve the interaction of nonlinear waves.

Modified multi-step Nyström methods for oscillatory general second-order initial value problems

Recently, multi-step Nyström methods for the numerical integration of general second-order initial value problems has been proposed and developed. In this paper, we modify this class of methods to make them adapted to solving numerically oscillatory problems. The new methods have the coefficients depending on v = wh and integrate exactly the problem . General order conditions are given, and two explicit methods with orders three and four, respectively, are constructed. The linear stability analysis of the new methods is considered. Numerical results show that our new methods are more efficient in comparison with other well-known high-quality methods proposed in the scientific literature.

On the Temperature-Rate Dependent Two-Temperature Thermoelasticity Theory

Abstract This work aims to formulate the temperature-rate dependent two-temperature (TRDTT) theory of thermoelasticity. The two-temperature thermoelasticity theory and the temperature-rate dependent thermoelasticity theory are two well-established thermoelasticity theories, which are developed from the generalized thermodynamic principles independently. Although the constitutive equations for TRDTT theory have been introduced, the formulation for the theory from the thermodynamical principles is not yet derived. Therefore, this work is an attempt to establish the theory from the generalized laws of thermodynamics and derive all the governing equations and constitutive relations for the theory. We derive a new and more general two-temperature relation that involves the temperature-rate terms of conductive and thermodynamic temperatures. We observe that this relation is different from the two-temperature relation reported in the literature. Further, we prove the uniqueness of solution for a general mixed initial boundary value problem in the context of linear modified TRDTT thermoelasticity theory for anisotropic medium. To investigate the effect of the present modified TRDTT theory, we solve a one-dimensional half-space problem and highlight the significance of the present theory.

An $$L^{p}$$-Approach to the Well-Posedness of Transport Equations Associated with a Regular Field: Part I

Transport equations associated with a Lipschitz field $$\mathscr {F}$$ on some subspace of $${\mathbb {R}}^N$$ endowed with some general measure $$\mu $$ are considered. Our aim is to extend the results obtained in two previous contributions (Arlotti et al. in Mediterr J Math 6:367–402, 2009, Mediterr J Math 8:1–35, 2011) in the $$L^{1}$$-context to $$L^{p}$$-spaces $$1< p <\infty $$. This is the first part of a two-part contribution (see in Arlotti and Lods An $$L^{p}$$-approach to the well-posedness of transport equations associated with a regular field—part II, Mediterr. J. Math. 16:145, 2019, for the second part) and we here establish the general mathematical framework we are dealing with and notably prove trace formula and uniqueness of boundary value transport problems with abstract boundary conditions. The abstract results of this first part will be used in the Part II of this work (Arlotti and Lods in Meditter J Math 16:145, 2019) to deal with general initial and boundary value problems and semigroup generation properties.

An $$L^{p}$$-Approach to the Well-Posedness of Transport Equations Associated to a Regular Field: Part II

Transport equations associated with a Lipschitz field $$\mathscr {F}$$ on some subspace of $${\mathbb {R}}^N$$ endowed with some general measure $$\mu $$ are considered. Our aim is to extend the results obtained in two previous contributions (Arlotti et al. in Mediterr J Math 6:367–402, 2009, Mediterr J Math 8:1–35, 2011) in the $$L^{1}$$-context to $$L^{p}$$-spaces $$1< p <\infty $$. This is the first part of a two-part contribution (see in Arlotti and Lods An $$L^{p}$$-approach to the well-posedness of transport equations associated with a regular field—part II, Mediterr. J. Math. 16:145, 2019, for the second part) and we here establish the general mathematical framework we are dealing with and notably prove trace formula and uniqueness of boundary value transport problems with abstract boundary conditions. The abstract results of this first part will be used in the Part II of this work (Arlotti and Lods in Meditter J Math 16:145, 2019) to deal with general initial and boundary value problems and semigroup generation properties.

An Accurate Continuous Hybrid Block Method for the Direct Solution of General Second Initial Value Problem

In this research, a new hybrid block method of order 5 for solving second order Initial Value Problems (IVPs) is determined. One step with 3 off step points is used to generate the coefficients of the new method. This method approximates the solutions of the second order IVPS at xn+1/6, xn+2/6, xn+3/6, xn+1, concurrently by using interpolation and collocation approach with fixed step size. The developed method don’t need to reduce the IVPs to its equivalent system of first order IVPs. The numerical results obtained show that the new method outperforms the existing methods in terms of error.

Hybrid Third Derivative Block Method for the Solution of General Second Order Initial Value Problems with Generalized One Step Point

This paper deals with two-step hybrid block method with one generalized off-step points for solving second order initial value problem. In derivation of this method, power series of order nine are interpolated at the first two step points while its second and third derivatives are collocated at all point in the selected interval. The new developed method is employed to solve several problems of second order initial value problems. Convergence analysis of the new method alongside numerical procedure is established. The performance of the proposed method is found to be more accurate than existing method available in the literature when solving the same problems.

A posteriori analysis for space-time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain

This paper presents an a posteriori error analysis for the discontinuous in time space–time scheme proposed by Jamet for the heat equation in multi-dimensional, non-cylindrical domains Jamet (SIAM J. Numer. Anal. 15 (1978) 913–928). Using a Clément-type interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of two-dimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with space-time dependent coefficients but posed on a cylindrical domain. We formulate a discontinuous in time space–time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of Evans (American Mathematical Society (1998)) for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso (Comput. Meth. Appl. Mech. Eng. 167 (1998) 223–237), proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.

Functionally Fitted Block Method for Solving the General Oscillatory Second-Order Initial Value Problems and Hyperbolic Partial Differential Equations

We present a block hybrid functionally fitted Runge–Kutta–Nyström method (BHFNM) which is dependent on the stepsize and a fixed frequency. Since the method is implemented in a block-by-block fashion, the method does not require starting values and predictors inherent to other predictor-corrector methods. Upon deriving our method, stability is illustrated, and it is used to numerically solve the general second-order initial value problems as well as hyperbolic partial differential equations. In doing so, we demonstrate the method’s relative accuracy and efficiency.

3- Point Single Hybrid Block Method (3PSHBM) for Direct Solution of General Second Order Initial Value Problem of Ordinary Differential Equations

3- Point Single Hybrid Block Method (3PSHBM) for Direct Solution of General Second Order Initial Value Problem of Ordinary Differential Equations

Well-posedness of general 1D initial boundary value problems for scalar balance laws

We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax–Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kružkov's doubling of variables technique, together with a careful treatment of the boundary terms.

Direct Integrators for the General Third-Order Ordinary Differential Equations with an Application to the Korteweg–de Vries Equation

We construct a new class of implicit continuous linear multistep methods (LMMs) which are used as boundary value methods for the numerical integration of the general third order initial and boundary value problems in ordinary differential equations, including the Korteweg–de Vries equation. The boundary value methods obtained from these continuous LMMs are weighted the same and are used to simultaneously generate approximate solutions to the exact solutions in the entire interval of integration. We established the convergence analysis of the methods and several numerical examples are given to show the performance of the methods.

Trees, Stumps, and Applications

The traditional derivation of Runge–Kutta methods is based on the use of the scalar test problem y′(x)=f(x,y(x)). However, above order 4, this gives less restrictive order conditions than those obtained from a vector test problem using a tree-based theory. In this paper, stumps, or incomplete trees, are introduced to explain the discrepancy between the two alternative theories. Atomic stumps can be combined multiplicatively to generate all trees. For the scalar test problem, these quantities commute, and certain sets of trees form isomeric classes. There is a single order condition for each class, whereas for the general vector-based problem, for which commutation of atomic stumps does not occur, there is exactly one order condition for each tree. In the case of order 5, the only nontrivial isomeric class contains two trees, and the number of order conditions reduces from 17 to 16 for scalar problems. A method is derived that satisfies the 16 conditions for scalar problems but not the complete set based on 17 trees. Hence, as a practical numerical method, it has order 4 for a general initial value problem, but this increases to order 5 for a scalar problem.

General initial value problem for the nonlinear shallow water equations: Runup of long waves on sloping beaches and bays

We formulate a new approach to solving the initial value problem of the shallow water-wave equations utilizing the famous Carrier–Greenspan transformation (Carrier and Greenspan (1957) [9]). We use a Taylor series approximation to deal with the difficulty associated with the initial conditions given on a curve in the transformed space. This extends earlier solutions to waves with near shore initial conditions, large initial velocities, and in more complex U-shaped bathymetries; and allows verification of tsunami wave inundation models in a more realistic 2-D setting.

Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

In some cases, high-order methods are known to provide greater accuracy with larger step-sizes than lower order methods. Hence, in this paper, we present a Block Hybrid Method (BHM) of order 11 for directly solving systems of general second-order initial value problems (IVPs), including Hamiltonian systems and partial differential equations (PDEs), which arise in multiple areas of science and engineering. The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. The properties of the BHM are discussed and the performance of the method is demonstrated on some numerical examples. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.

Multi-step Nyström methods for general second-order initial value problems y″(t) = f(t,y(t), y′(t))

ABSTRACTA class of multi-step Nyström methods for solving numerically general second-order initial value problems are proposed and developed. The new methods inherit the main frameworks of classical Nyström methods and make use of the information of previous steps. Order conditions are deduced by using the theory of Nyström-series based on the set of Nyström-trees, and two explicit methods with convergence orders three and four, respectively, are constructed. The linear stability of the new methods is analysed. Numerical results show that our new methods are more efficient in comparison with Runge–Kutta–Nyström methods and other well-known high quality methods proposed in the scientific literature.

Generation of interfaces for multi-dimensional stochastic Allen–Cahn equation with a noise smooth in space

In this paper, we study the generation of interfaces for a stochastic Allen–Cahn equation with general initial value in the multi-dimensional case that external noise is given by Q-Brownian motion. We prove that interfaces, for d-dimensional stochastic Allen–Cahn equation with scaling parameter , are generated at the time of order . Especially, in one-dimensional case, we give more detailed estimate and shape of the generated interface than that obtained in our previous result. Assuming that the Q-Brownian motion is smooth in space variable, we extend a comparison theorem for PDE to SPDE’s in order to prove the generation. Moreover, we connect the generated interface to the motion of interface in one-dimension. In this case, we consider the white noise only in time multiplied by as the noise term, where a is a smooth function which has a compact support. This is the special case of Q-Brownian motion. We take the time scale of order for studying the motion of interface.

Approximate solutions of general second–order initial value problems using differential evolution

In this paper, it is assumed that the solution of the general second order initial value problem $u^{\prime \prime} = f\left(t, u, u^{\prime} \right);\quad u(t_0) = u_0,\;\;u^{\prime}(t_0) = u^{\prime}_0, \quad t \in \left[t_0,t_n\right]$ can be approximated by a polynomial \emph{u(t)}. To obtain the values of the coefficients of the terms of \emph{u(t)}, the problem is converted to an optimization problem and the simple stochastic function minimizer called differential evolution is used to obtain the optimal values of the coefficients. Numerical examples show the efficiency and accuracy of the proposed technique compared with some existing classical methods. https://doi.org/10.28919/jmcs/3404

Application of the meshless generalised RKPM to the transient advection-diffusion-reaction equation

A variety of physical problems may be expressed using the advection–diffusion-reaction (ADR) equation that encompasses transport processes into a porous or nonporous material. Finding a theoretical solution to the ADR equation is difficult with time-dependent nonlinear coefficients, complicated geometries, general initial value and/or boundary conditions. In this paper a meshless model is developed by adapting the generalised reproducing kernel particle method to the strong and weak integral forms of ADR equation. Moreover, mixed–type boundary conditions are directly enforced via generalising the corrected collocation method. The model is validated using existing analytical solutions and shown to be both accurate and efficient.