Initial Value Ordinary Differential Equations Research Articles

Significance of Cattaneo-Christov double diffusion and induced magnetic field on Maxwell ternary nanofluid flow with magnetic response boundary

The present study delves into the significance of Cattaneo-Christov double diffusion and the induced magnetic field (IMF) on a stagnation-point flow of Maxwell ternary nanofluids. A new boundary condition with the magnetic response is designed to study superior heat and mass transfer of Maxwell ternary nanofluid combined with double diffusion and IMF. The governing equations are formulated by mathematically modeling of the current flow in a Cartesian coordinate system and solved using an improved shooting method and the Runge-Kutta method. Through the application of similarity transformations, the governing equations are transformed into a system of initial boundary value ordinary differential equations. These equations are further transformed into linear equations with initial value problems using the shooting method and subsequently solved using the Runge-Kutta method and Newton's iterative techniques. The numerical results and correlation analysis vividly demonstrate the profound influence of double diffusion and IMF on thermal and concentration patterns via graphical representations. It is found that the double diffusion and the induced magnetic field always helps to achieve lower temperature and concentration. The magnetic response boundary leads to higher heat and mass transfer efficiency for the suction case. The magnetically responsive boundary is verified to be effective for regulating the heat and mass transfer of ternary nanofluid.

Efficient Runge–Kutta solver for stochastic crack growth analysis

Stochastic crack growth analysis by simulation may easily require a significant amount of computational effort. The Initial Value ordinary differential equations appearing in crack propagation problems may be solved by the Runge–Kutta (RK) family of solvers. However, crack propagation considering uncertainties imposes additional challenges for RK solvers: possibility of vectorized and/or parallelized computation of sample realizations; and adaptation of time discretization for each sample, to handle changes and discontinuities in the derivatives of crack growth curves, caused by changes in loads and in crack propagation rates. Existing adaptive RK methods, with change in time step length during crack growth computation, struggle with these challenges. In this setting, the present paper proposes a modified adaptive RK method which allows for simultaneous crack growth computations with different time-step discretizations, and aims at efficiently dealing with cases where discontinuities are present in the derivative of the crack growth curve. Application of the proposed method to three crack propagation examples, including one related to weld flaws in pipes, indicates that it is as accurate as other adaptive methods from the literature, while requiring only a fraction of the computational time.

SOLUTION OF ORDINARY DIFFERENTIAL EQUATION vvi (u)=f(u,v,v',v'',v''') USING EIGHTH AND NINTH ORDER RUNGE-KUTTA TYPE METHOD

The present paper presents the numerical conclusion to solve sixth order initial value ordinary differential equation (ODE). The concept of order conditions for three stage eighth order (RKSD8) & four stage ninth order Runge-Kutta methods (RKSD9) has been derived for finding global truncation error of differential equation The global and local truncated errors norms, zero stability of extended Runge-Kutta method (RK) is well defined and demonstrated with the help of an example.

The stability of the parametric Cauchy problem of initial-value ordinary differential equations revisited

The stability of the parametric Cauchy problem of initial-value ordinary differential equations revisited

A simple model for the reflection by a vertical barrier of a dambreak flow over a dry or pre-wetted bottom

We revisit the theoretical solutions of the problem of the reflected flow from a vertical obstacle encountered by a current generated by dambreak of a Cartesian long reservoir over a dry bottom and with tailwaters. Previous investigations derived accurate solutions by quite complex manipulation of the balances along the characteristics, supported by numerical solutions, of the shallow-water equations (a significant extension is the recent paper Hogg & Skevington, Q. J. Mech. Appl. Maths, vol. 74, no. 4, 2021, pp. 441–465). Here we developed a simplified model, based on the major assumption that the fluid between the obstacle (wall) and the reflected jump is stagnant. This allows the solution of the problem by a straightforward numerical integration of one initial-value ordinary differential equation. The model provides the position, height and speed of the jump as functions of time. For long times a simple analytical approximation is also available. The model points out clear-cut effects of the presence of the tailwaters. The model has been validated by comparisons with exact solutions of the shallow-water equations of the recent study of Hogg & Skevington (Q. J. Mech. Appl. Maths, vol. 74, no. 4, 2021, pp. 441–465) (obtained by significantly more complex hodograph-plane methods). In all the tested cases the agreement is good, for long periods of time. This model provides reliable insights and fast quantitative predictions recommended for use in research and engineering problems where a fair approximation is sufficient for the application.

Approximate Analytical Solution of Unstable Ordinary Differential Equation Using Differential Evolution Algorithm

The application of evolutionary optimization algorithms in problem solving is currently gaining wide popularity. Use of Differential Evolution (DE) algorithm in obtaining analytically approximate solution of unstable second–order initial value Ordinary Differential Equation (ODE) is presented in this work. The methodology involves solving an associated problem of optimization with constrains to get an analytically approximate solution for the ODE under consideration. Three test cases were used to demonstrate the efficiency of our method. In comparison with other methods discussed in the literature, our method gave significant improvement on the accuracy of the obtained results.

Remark on Algorithm 982: Explicit Solutions of Triangular Systems of First-order Linear Initial-value Ordinary Differential Equations with Constant Coefficients

Algorithm 982: Explicit solutions of triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients provides an explicit solution for an homogeneous system, and a brief description of how to compute a solution for the inhomogeneous case. The method described is not directly useful if the coefficient matrix is singular. This remark explains more completely how to compute the solution for the inhomogeneous case and for the singular coefficient matrix case.

Model reduction for initial value ODEs

Many physical phenomena in biology and physiology are described by mathematical models that comprise a system of initial value ordinary differential equations. Each differential equation may often be written as the sum of several terms, where each term represents a different physical entity. A wide range of techniques, ranging from heuristic observation to mathematically rigorous asymptotic analysis, may be used to simplify these equations allowing the identification of the key phenomena responsible for a given observed behaviour. In this study we extend an algorithm for automatically simplifying systems of initial value ordinary differential equations (Whiteley (2010)) that is based on a posteriori analysis of the full system of equations. Our extensions to the algorithm make the following contributions: (i) each equation in a system of differential equations may be written as a finite sum of contributions (including the derivative term), and any one of these terms may be neglected (if it is appropriate to do so) in the simplified model; and (ii) a simplified model is generated that allows accurate prediction of one or more components of the solution at all times. These extensions are illustrated using examples drawn from enzyme kinetics and cardiac electrophysiology.

A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems

In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.

A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems

In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.

On the Numerical Solution of Ordinary, Interval and Fuzzy Differential Equations by Use of F-Transform

An interesting property of the inverse F-transform f ^ of a continuous function f on a given interval [ a , b ] says that the integrals of f ^ and f on [ a , b ] coincide. Furthermore, the same property can be established for the restrictions of the functions to all subintervals [ a , p k ] of the fuzzy partition of [ a , b ] used to define the F-transform. Based on this fact, we propose a new method for the numerical solution of ordinary differential equations (initial-value ordinary differential equation (ODE)) obtained by approximating the derivative x · ( t ) via F-transform, then computing (an approximation of) the solution x ( t ) by exact integration. For an ODE, a global second-order approximation is obtained. A similar construction is then applied to interval-valued and (level-wise) fuzzy differential equations in the setting of generalized differentiability (gH-derivative). Properties of the new method are analyzed and a computational section illustrates the performance of the obtained procedures, in comparison with well-known efficient algorithms.

Numerical solution of initial value problems by rational interpolation method using Chebyshev polynomials

In this research, a modified rational interpolation method for the numerical solution of initial value problem is presented. The proposed method is obtained by fitting the classical rational interpolation formula in Chebyshev polynomials leading to a new stability function and new scheme. Three numerical test problems are presented in other to test the efficiency of the proposed method. The numerical result for each test problem is compared with the exact solution. The approximate solutions are show competitiveness with the exact solutions of the ODEs throughout the solution interval.Keywords and Phrases: Chebyshev polynomial, Rational Interpolation, Minimaxpolynomial, Initial Value Problems and Ordinary Differential Equations (ODEs)

An Adaptive, Highly Accurate and Efficient, Parker-Sochacki Algorithm for Numerical Solutions to Initial Value Ordinary Differential Equation Systems

An Adaptive, Highly Accurate and Efficient, Parker-Sochacki Algorithm for Numerical Solutions to Initial Value Ordinary Differential Equation Systems

Nodal Unsteady Inverse Elastohydrodynamic Lubrication: Axisymmetric Normal Approach

An “inverse” formulation is described for general problems of unsteady elastohydrodynamic lubrication (EHL). Spatial discretization gives an explicit initial-value ordinary differential equation (ODE) problem with (interior) nodal film thicknesses as state variables. Numerical results are compared with published experimental results for normal approach of a spherical surface to an elastic foundation.

Finite Difference Method for Solving the Initial Value Ordinary Differential Equations (ODE)

Finite Difference Method for Solving the Initial Value Ordinary Differential Equations (ODE)

Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

Algorithm 982

A method to compute explicit solutions of homogeneous triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients is described. It is suitable for the limited case of well separated eigenvalues, or for multiple zero eigenvalues provided the entire column corresponding to a zero eigenvalue is zero. The solution for the case of constant inhomogeneity is described. The method requires only the computation of a constant matrix using a simple recurrence. Computing the solutions of the system from that matrix, for values of the independent variable, requires one to exponentiate only the diagonal of a matrix. It is not necessary to compute the exponential of a general triangular matrix. Although this work was motivated by a study of nuclear decay without fission or neutron absorption, which is used throughout as an example, it has wider applicability.

Artificial neural network approach for a class of fractional ordinary differential equation

The essential characteristic of artificial neural networks which against the logistic traditional systems is a data-based approach and has led a number of higher education scholars to investigate its efficacy, during the past few decades. The aim of this paper was concerned with the application of neural networks to approximate series solutions of a class of initial value ordinary differential equations of fractional orders, over a bounded domain. The proposed technique uses a suitable truncated power series of the solution function and transforms the original differential equation in a minimization problem. Then, the minimization problem is solved using an accurate neural network model to compute the parameters with high accuracy. Numerical results are given to validate the iterative method.

Numerical Solutions of Initial Value Ordinary Differential Equations Using Finite Difference Method

Initial value ordinary differential equations arise in formulation of problems in various fields such as physics and Engineering. The present paper shows the method how to solve the initial value ordinary differential equation on some interval by using finite difference method in a very accurate manner with the formulation of error estimation.

Automatic simplification of systems of reaction–diffusion equations by a posteriori analysis

Many mathematical models in biology and physiology are represented by systems of nonlinear differential equations. In recent years these models have become increasingly complex in order to explain the enormous volume of data now available. A key role of modellers is to determine which components of the model have the greatest effect on a given observed behaviour. An approach for automatically fulfilling this role, based on a posteriori analysis, has recently been developed for nonlinear initial value ordinary differential equations [J.P. Whiteley, Model reduction using a posteriori analysis, Math. Biosci. 225 (2010) 44–52]. In this paper we extend this model reduction technique for application to both steady-state and time-dependent nonlinear reaction–diffusion systems. Exemplar problems drawn from biology are used to demonstrate the applicability of the technique.