Stiff Initial Value Problems Research Articles

On Some Boundary Value Methods

Circumventing order restrictions on numerical methods designed for the integration of stiff initial value problem is the concern here via Boundary Value Method. The attainable order p = k+v and linear stability properties of the methods are discussed. The numerical test on some stiff problems shows that the new methods developed, compare favourably with existing methods, with ODE15s of MATLAB used as reference numerical solution.

One-Step Embedded Hybrid Block Method for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

One-Step Embedded Hybrid Block Method for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

One-Step Embedded Hybrid Block Method for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

One-Step Embedded Hybrid Block Method for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

Rational Approximation Method for Stiff Initial Value Problems

While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor series method is not well-suited for stiff problems since it is explicit and not A-stable. In our paper, we present a numerical-analytical method based on the rational approximation of the ODE solution, which is naturally A- and A(α)-stable. We describe the rational approximation method and consider issues of order, stability, and adaptive step control. Finally, through examples, we prove the superior performance of the rational approximation method when solving highly stiff problems, comparing it with the Taylor series and Runge–Kutta methods of the same accuracy order.

On the stability functions of second derivative implicit advanced-step point methods

In the construction of efficient numerical methods for the stiff initial value problems, some second derivative multistep methods have been introduced equipping with super future point technique. In this paper, we are going to introduce a formula for the stability functions of a class of such methods. This group of methods encompasses SDBDF methods and their extensions with advanced step-point feature. This general formula, instead of obtaining the distinct stability functions for each of methods, will facilitate stability analysis of the methods.

ORDER AND CONVERGENCE OF THE ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING INITIAL VALUE PROBLEM

This paper studied an enhanced 3-point fully implicit super class of block backward differentiation formula for solving stiff initial value problems developed by Abdullahi & Musa and go further to established the necessary and sufficient conditions for the convergence of the method. The method is zero stable, A-stable and it is of order 5. The method is found to be suitable for solving first order stiff initial value problems

ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING STIFF INITIAL VALUE PROBLEMS

This paper modified an existing 3–point block method for solving stiff initial value problems. The modification leads to the derivation of another 3 – point block method which is suitable for solving stiff initial value problems. The method approximates three solutions values per step and its order is 5. Different sets of formula can be generated from it by varying a parameter ρ ϵ (-1, 1) in the formula. It has been shown that the method is both Zero stable and A–Stable. Some linear and nonlinear stiff problems are solved and the result shows that the method outperformed an existing method and competes with others in terms of accuracy

Block procedure for solving stiff initial value problems using probabilists Hermite polynomials

Block procedure for solving stiff initial value problems using probabilists Hermite polynomials

A new numerical approach for solving initial value problems with exponential growth integral curves

Some physical processes raised in aerodynamics, hydrodynamics, and theory of catalytic reactions are modeled by stiff initial value problems. This paper considers a class of stiff problems such as singularly perturbed problems for ordinary differential equations. Singularly perturbed problems have regions of rapid change in the solution, which are called contrast structures. Most often, boundary layer exhibits in such problems, but several pieces of research devoted to inner layers too. There are many numerical and asymptotic methods for solving singularly perturbed initial value problems with contrast structures. However, it is still an open field for researches because the universal method does not exist. Problems with an exponential growth rate of integral curves have a particular interest for researchers because most of the methods for solving singularly perturbed problems cannot handle these. We are going to discuss methods for solving singularly perturbed initial value problems with contrast structures. Also, we are proposing a new technique based on the method of solution continuation for problems with the exponential growth of integral curves. This technique allows explicit numerical methods to get the solution of the considered problem when it seems impossible for them. The efficiency of the new approach will be shown on the test initial value problem.

Stability analysis of third derivative multi-step methods for stiff initial value problems

In this paper we present two class of third derivative multistep methods (TDMMs) that have good stability properties. Stability analysis of this method is examined and our numerical results are compared with the results of the existing method.

Parameterized neural network training for the solution of a class of stiff initial value systems

As computational intelligence techniques become more popular in almost all scientific fields and applications nowadays, there exists an active research effort to engage them in the study of classical mathematical problems. Among these techniques, the neural networks (NN), apart from their use in classification problems, can be used to approximate the behaviour of functions and their derivatives. Towards this direction, NN solution of differential equations (DEs), in both theoretical and technical point of view, is an active scientific field for the last two decades. NN solutions for DEs, once trained, have low computational cost and can be very useful as parts of more complex algorithms where needed, as well. Among the various classes of DEs, the stiff initial value problems (IVP) reveal difficulties in their numerical treatment by classical methodologies, whereas NN solutions of stiff DEs do not seem to do so. Moreover, their continuous nature and ability to be trained to solve classes of problems makes them an interesting tool. In this study, we investigate the NN solution of Inhomogeneous Linear IVPs. We incorporate to the NN a parameter that influences problem’s stiffness and train the network for a range of this. Therefore, the trained NN solution can solve different problems than the one training for. In order to reveal the good generalization properties of the NNs solution regarding the stiffness parameter, we compare them to the solutions of standard Matlab stiff solvers. The proposed solutions perform very well, similarly and in many cases better to their competitors.

Improved Two-Point Block Backward Differentiation Formulae for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

Two numerical methods- I2BBDF2 and I22BBDF2 that compute two points simultaneously at every step of integration by first providing a starting value via fourth order Runge-Kutta method are derived using Taylor series expansion. The two-point block schemes are derived by modifying the existing I2BBDF (5) method of Mohamad et al., (2018). Convergence and stability analysis of the new methods are established with the methods being of order two and A-stable in both cases. Despite the very low order of the new methods, the accuracy of these methods on some stiff initial value problems in the literature proves their superiority over existing methods of higher orders such as I2BBDF(5), BBDF(5), E2OSB(4) among others.

An Optimized 5-Point Block Formula for Direct Numerical Solution of First Order Stiff Initial Value Problems

In this research article, we focus on the formulation of a 5-point block formula for solving first order ordinary differential equations (ODEs). The method is formulated via interpolation and collocation approach using power series expansion as the approximate solution. It has been established that the derived method is of order six. Basic properties such zero and absolute stabilities, convergence, order and error constant have also been investigated. The accuracy of the method was verified on some selected stiff IVPs, compared with some existing methods (DIBBDF, SDIBBDF, BBDF(4), BBDF(5) and odes15s) and test performance showed that the new method is viable.

Variable order nested hybrid multistep methods for stiff ODEs

A family of variable order nested hybrid multistep method for the numerical integration of stiff initial value problems of an ordinary differential equation is presented in this paper. These methods possess high order with A-stability properties making it suitable for solving stiff problems. The method has been implemented on some non-linear problems.

A class of fifth order block hybrid Adams Moultonrsquo;s method for solving stiff initial value problems

This paper is concern with the derivation of the continuous formulations of hybrid Adams-Moulton Method for step number and incorporating three, two and one off-grid points respectively by multistep collocation method using matrix inversion technique. The discrete schemes used in block form were derived from their respective continuous formulations. The convergence analysis were carried and found that the methods are convergent. The region of absolute stability of the methods were plotted in a block form and they are all - stable. Some linear stiff problems were solved using the schemes. It was observed that the schemes for step number incorporating one off-grid point perform better than the schemes for step number incorporating two and three off-grid points respectively.

Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations

In this paper, we compare a multi-step method and a multi-stage method for stiff initial value problems. Traditionally, the multi-step method has been preferred than the multi-stage for a stiff problem, to avoid an enormous amount of computational costs required to solve a massive linear system provided by the linearization of a highly stiff system. We investigate the possibility of usage of multi-stage methods for stiff systems by discussing the difference between the two methods in several numerical experiments. Moreover, the advantages of multi-stage methods are heuristically presented even for nonlinear stiff systems through several numerical tests.

Trigonometric fitted modification of RADAU5

Implicit Runge‐Kutta (RK) methods are in common use when addressing stiff initial value problems (IVP). They usually share the property of A‐stability that is of crucial importance in solving the latter type of IVP. Radau IIA family of implicit RK methods is among the preferred ones. Especially its fifth‐order representative named RADAU5 has received a lot of attention for use with lax accuracies. Here, we try the lesser possible perturbation of its coefficients. Then, we derive a trigonometric fitted modification that is intended to be applied in periodic IVPs. Numerical tests over a variety of problems with oscillatory solutions justify our effort.

A new type OF hybrid multistep multiderivative formula for solving stiff IVPs

We present a new class of higher-order multistep multiderivative methods for the numerical solution of stiff initial value problems. These methods are obtained based on free parameters and off-point. The methods have minimum error bounds. The constructed class is A-stable for orders 3 and 4, and A(alpha)-stable for orders 5 and 6. The new class is L-stable for all orders. They are suitable for solving stiff systems of initial value problems with large eigenvalues lying close to the imaginary axis. The stability regions of the new class are plotted, and some problems are solved, which show the superiority of the class in efficiency and accuracy.

BLOCK PROCEDURE WITH IMPLICIT SIXTH ORDER LINEAR MULTISTEP METHOD USING LEGENDRE POLYNOMIALS FOR SOLVING STIFF INITIAL VALUE PROBLEMS

In this paper, a discrete implicit linear multistep method in block form of uniform step size for the solution of first-order ordinary differential equations is presented using the power series as a basis function. To improve the accuracy of the method, a perturbation term is added to the approximated solution. The method is based on collocation of the differential equation and interpolation of the approximate solution using power series at the grid points. The procedure yields four linear multistep schemes which are combined as simultaneous numerical integrators to form block method. The method is found to be consistent and zero-stable, and hence convergent. The accuracy of the method is tested with some standard stiff first order initial value problems. The results are compared with the two point implicit block backward difference formula and two point implicit block extended backward difference formula. All numerical examples show that our proposed method has a better accuracy than the existing literature.

A NEW HIGHER ORDER SECOND DERIVATIVE BLENDED BLOCK LINEAR MULTISTEP METHODS FOR THE SOLUTIONS STIFF INITIAL VALUE PROBLEMS

: This paper is concerned with the accuracy and efficiency of a higher order second derivative blended block linear multistep method for the approximate solution of stiff initial value problems. The main methods were derived by blending of two linear multistep methods using continuous collocation approach. These methods are of uniform order ten. The stability analysis of the block methods indicates that the methods are A–stable, consistent and zero stable hence convergent. Numerical results obtained using the proposed new block methods were compared with those obtained by the well known ODE solver ODE 15s to illustrate the accuracy and effectiveness. The proposed block method is found to be efficient and accurate hence recommended for the solution of stiff initial value problems.