Linear Initial Value Problems Research Articles

Extended artificial neural networks approach for solving two-dimensional fractional-order Volterra-type integro-differential equations

Over the years, there has been a sharp increase in employing artificial neural networks in modeling and simulation of complicated real-world problems. The main aim of the present paper is to apply an extension of artificial neural networks approach for approximating solution of a two-dimensional fractional-order linear Volterra-type integro-differential initial value problem. For this aim, a suitable four-layered feed forward neural architecture is designed to approximate the solution function with the help of two distinct famous learning algorithms, namely the steepest descent and the quasi-Newton rules, to any desired degree of accuracy. Finally, the applicability and validity of preset techniques are described and tested through some numerical examples. Numerical and simulation results obtained here are also compared with the results obtained by the optimal Homotopy asymptotic method. This practice determined that the quasi-Newton rule provides far more accurate solutions than the other two mentioned above.

Accuracy Improvement of Block Backward Differentiation Formulas for Solving Stiff Ordinary Differential Equations Using Modified Versions of Euler's Method

In this study, the fully implicit 2-point block backward differentiation formulas (BBDF) method has been successfully utilized for solving stiff ordinary differential equations (ODEs) by taking into account the uses of new starting methods namely, modified Euler's method (MEM), improved modified Euler's method (IMEM), and new Euler's method (NEM). The reason of proposing the BBDF is that the method has been proven useful for stiff ODEs due to its A-stable properties. Furthermore, the method is able to approximate the solutions at two points simultaneously at each step. The proposed method is also implemented through Newton's iteration procedure, which involves the calculation of the Jacobian matrix. Accuracy of the method is evaluated based on its performance in solving linear and non-linear initial value problems (IVPs) of first order stiff ODEs with transient and steady-state solutions. Some comparisons are made with the conventional BBDF approach for indicating the reliability of the proposed method. Numerical results indicate that not only classical Euler's method provides accurate solutions for BBDF, but also the numerous modified versions of Euler's methods improve the accuracy of BBDF, in terms of absolute error at certain step size and stage of iteration.

Runge–Kutta–Nyström methods of eighth order for addressing Linear Inhomogeneous problems

Second order Linear Inhomogeneous Initial Value Problems with constant coefficients are considered here. Runge–Kutta–Nyström (RKN) methods are amongst the most effective for addressing these problems. For attaining a certain algebraic order of accuracy, we need to solve a reduced set of order conditions (in comparison with conventional RKN) with respect to the coefficients. Here, using a seven stages method we manage to solve these equations up to eighth order. Differential Evolution technique was used for achieving this. Thus, our main contribution is that we saved one stage since conventional eight order RKN methods use eight stages for advancing a step. This new method is combined at no cost with a fifth and third order companion formulas. As consequence, a triplet of orders 8(5)3 is given for estimating local errors and adjust effectively the step lengths. The new triplet of methods is applied to various relevant problems and is found to outperform standard RKN pairs from the literature.

Double-scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems

Double-scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems

Subharmonic eigenvalue orbits in the spectrum of pulsating Poiseuille flow

Spectral degeneracies where eigenvalues and eigenvectors simultaneously coalesce, also known as exceptional points, are a natural consequence of the strong non-normality of the Orr–Sommerfeld operator describing the evolution of infinitesimal disturbances in parallel shear flows. While the resonances associated with these points give rise to algebraic growth, the development of non-modal stability theory exploiting specific perturbation structures with much larger potential for transient energy growth has led to waning interest in spectral degeneracies. The appearance of subharmonic eigenvalue orbits, recently discovered in the periodic spectrum of pulsating Poiseuille flow, can be traced back to the coalescence of eigenvalues at exceptional points. We present a thorough analysis of the spectral properties of the linear operator to identify exceptional points and accurately map the prevalence of subharmonic eigenvalue orbits for a large range of pulsation amplitudes and frequencies. This information is then combined with solutions of the linear initial value problem to analyse the impact of the appearance of these orbits on the temporal evolution of linear disturbances in pulsating Poiseuille flow. The periodic amplification phases are shown to be heralded by repeated non-normal growth bursts that are intensified by the formation of subharmonic orbits involving the leading eigenvalues. These bursts are associated with the change of alignment of the perturbation from the decaying towards the amplified branch of the subharmonic eigenvalue orbits in a so-called branch transition process.

On solution of ordinary differential equations by using HWCM, ADM and RK4

The Haar wavelet collocation approach (HWCM) is an impressive numerical method for solving linear initial value problems when compared to the existing numerical methods (Adomian decomposition method (ADM) & Runge–Kutta method (RK4)). The objective of this study is to use the Haar-wavelet technique, Adomian decomposition technique (ADM) and Runge–Kutta (RK4) method to achieve the numerical solution of second-order ordinary differential equations. The proposed methods are applied to three different problems and the numerical results show that the HWCM has better agreement with analytic solutions than the other numerical methods.

One-step Hybrid Block Method for Directly Solving Fifth-order Initial Value Problems of Ordinary Differential Equations

An effective one-step hybrid block for getting the approximate solution of a fifth-order IVP with applications to problems in the sciences and engineering is constructed in this study. The mathematical formulation of the method is based on the principle of interpolation and collocation of the trial solution and its derivatives at the chosen equidistant grid and off-grid points. The basic properties of the derived method are examined, and it has an order greater than one, zero stable, consistent, and hence convergent. The derived method is applied to solve five different linear and nonlinear fifth-order initial value problems. Comparison of the absolute errors obtained using the derived method with a few existing ones in the literature supports its good performance.

Optimized Hybrid Block Adams Method for Solving First Order Ordinary Differential Equations

Multistep integration methods are being extensively used in the simulations of high dimensional systems due to their lower computational cost. The block methods were developed with the intent of obtaining numerical results on numerous points at a time and improving computational efficiency. Hybrid block methods for instance are specifically used in numerical integration of initial value problems. In this paper, an optimized hybrid block Adams block method is designed for the solutions of linear and nonlinear first-order initial value problems in ordinary differential equations (ODEs). In deriving the method, the Lagrange interpolation polynomial was employed based on some data points to replace the differential equation function and it was integrated over a specified interval. Furthermore, the convergence properties along with the region of stability of the method were examined. It was concluded that the newly derived method is convergent, consistent, and zero-stable. The method was also found to be A-stable implying that it covers the whole of the left/negative half plane. From the numerical computations of absolute errors carried out using the newly derived method, it was found that the method performed better than the ones with which we compared our results with. The method also showed its superiority over the existing methods in terms of stability and convergence.

A quadrature-based scheme for numerical solutions to Kirchhoff transformed Richards' equation

<p style='text-indent:20px;'>In this work we propose a new numerical scheme for solving Richards' equation within Gardner's framework and accomplishing mass conservation. In order to do so, we resort to Kirchhoff transformation of Richards' equation in mixed form, so to exploit specific Gardner model features, obtaining a linear second order partial differential equation. Then, leveraging the mass balance condition, we integrate both sides of the equation over a generic grid cell and discretize integrals using trapezoidal rule. This approach provides a linear non-homogeneous initial value problem with respect to the Kirchhoff transform variable, whose solution yields the sought numerical scheme. Such a scheme is proven to be <inline-formula><tex-math id="M1">\begin{document}$ l^{2} $\end{document}</tex-math></inline-formula>-stable and convergent to the exact solution under suitably conditions on step-sizes, retaining the order of convergence from the underlying quadrature formula.</p>

On Solution of First Order Initial Value Problems using Laplace Transform in Fuzzy Environment

This study aimed at solving a nonhomogeneous linear first order initial value problem by means of Laplace transform method in fuzzy environment. The conditions for a fuzzy function to be H−differentiable and gH−differentiability are well established. Finally, example is constructed to test the applicability or otherwise of the established results.

Optimal homotopy asymptotic method for solving several models of first order fuzzy fractional IVPs

In this work, the Optimal Homotopy Asymptotic Method (OHAM) is prolifically implemented to find the optimal solutions of fractional order of fuzzy differential equations. We inspect the competence of the method by examining fractional order time-dependent first order fuzzy initial value problems in the Caputo derivative sense. The concepts of fuzzy sets theory and fractional calculus were used to present a new general fuzzification formulation of the OHAM method followed by fuzzy analysis for the proposed fractional problems are presented in detail. OHAM is recognized as a feasible and convergent method for solving linear and nonlinear models, and it provides a convenient technique for controlling the convergence of approximate solutions optimally. The method's capability is demonstrated by obtaining approximate solutions to linear and nonlinear fuzzy initial value problems in various orders of fractional derivative. The numerical results acknowledge that OHAM is a feasible and transformation mechanism for solving linear and nonlinear fuzzy fractional initial value problems. Also, the results ensure that the method used was succinct, efficient, and simple to use, to deal with a more comprehensive fuzzy fractional differential equation.

Maximum Principles and Applications for Fractional Differential Equations with Operators Involving Mittag-Leffler Function

Abstract In this paper, we formulate and prove two maximum principles to nonlinear fractional differential equations. We consider a fractional derivative operator with Mittag-Leffler function of two parameters in the kernel. These maximum principles are used to establish a pre-norm estimate of solutions, and to derive certain uniqueness and positivity results to related linear and nonlinear fractional initial value problems.

The B-spline collocation method for solving conformable initial value problems of non-singular and singular types

In the present study, the uniform cubic B-spline collocation method is constructed, employed, and experimented to given smooth approximations for the numerical solution of a class of linear and nonlinear fractional initial value problems of singular and non-singular types based on the conformable fractional derivative. As a special case, the singular Lane-Emden type model is presented such that the source fractional differential equation is transformed at the point of singularity. The converted issue is then solved by using third B-splines on a uniform mesh approximation procedure with help of Mathematica 11. Several illustrative examples are incorporated to demonstrate the efficiency, validity, and applicability of the proposed technique for such conformable problems.

Application of Laplace residual power series method for approximate solutions of fractional IVP’s

In this study, different systems of linear and non-linear fractional initial value problems are solved analytically utilizing an attractive novel technique so-called the Laplace residual power series approach, and which is based on the coupling of the residual power series approach with the Laplace transform operator to generate analytical and approximate solutions in fast convergent series forms by using the concept of the limit with less time and effort compared with the residual power series technique. To confirm the simplicity, performance, and viability of the proposed technique, three problems are tested and simulated. Analysis of the obtained results reveals that the aforesaid technique is straightforward, accurate, and suitable to investigate the solutions of the non-linear physical and engineering problems.

Linear stability analysis of two fluid columns of different densities and viscosities in a gravity field

The linear stability of a vertical interface separating two miscible fluid columns of different densities and viscosities under the influence of gravity is investigated. This flow possesses a time-dependent reference state (each column accelerates at different rates owing to their different densities) and the interface thickness grows as the square root of time (by diffusion). Numerical integration of the linear initial-value problem is carried out and discussed in detail as a function of vertical and spanwise wavenumbers and the flow parameters. Adjoint-based optimization is performed in order to determine initial conditions that lead to maximum growth of disturbances in finite time. Results indicate that the rate of growth of the perturbation energy at small wavenumbers (less affected by viscosity initially) is dominated by two-dimensional modes (no spanwise variation). Substantial transient growth is observed at higher wave modes initially, followed by asymptotic decay of the perturbations at large time. Sensitivity of perturbation growth with respect to initial time, density and viscosity ratios is investigated. This work is complementary to previous inviscid analysis of this configuration, which showed that the interface was unconditionally unstable at all wave modes, even in the presence of surface tension, and that instability grew as the exponential of time squared.

An order verification method for truncated asymptotic expansion solutions to initial value problems

The focus of this paper is to obtain explicit solutions to initial value problems, where numerical methods cannot provide one, and to verify the accuracy orders of the explicit solutions. One of available methods to obtain an explicit solution is the asymptotic (formal) expansion method. However, we must be sure with the accuracy order of the explicit solution. In this paper, an order verification method is proposed for truncated asymptotic formal expansion solutions to initial value problems. A least-squares fit of error data is used in the existing order verification method. The method that we propose does not involve any application of least-squares fit of error data, so is simpler, yet produces accurate expected accuracy orders of solutions of explicit truncated asymptotic formal expansions. With our proposed method, we are successful in verifying the accuracy orders of solutions of truncated asymptotic formal expansions to the linear and nonlinear initial value problems accurately.

Adam’s block with first and second derivative future points for initial value problems in ordinary differential equations

In this paper, we develop Adam’s Block with first and second derivative future points for solving linear and non-linear first order initial value problems in ordinary differential equations. The derivation of the method is based on Taylor series approach. The region of absolute stability of the method is investigated using the boundary locus method and this family of methods have been found to be A-stable for r=2,3,4 and 5. Numerical experiments are demonstrated with the method and computational comparisons are presented with some existing numerical methods. The computational comparison depicts the efficiency of the methods on initial value problems in ordinary differential equations (ODEs).

A study on single-iteration sobolev descent for linear initial value problems

Mahavier and Montgomery construct a Sobolev space for approximate solution of linear initial value problems in a finite difference setting in single-iteration sobolev descent for linear initial value problems, Mahavier, Montgomery, MJMS, 2013. Their Sobolev space is constructed so that gradient-descent converges to a solution in a single iteration, demonstrating the existence of a best Sobolev gradient for finite difference approximation of solutions of linear initial value problems. They then ask if there is a broader class of problems for which convergence in a single iteration in an appropriate Sobolev space occurs. We use their results to show the existence of single-step iteration to solution in a lower dimensional Sobolev space for their examples and then a class of problems for single-step convergence.

Direct Solution of [formula omitted] Using Three Point Block Method of Order Eight with Applications

This study aims to construct an implicit block method with three-point to tackle general second-order ordinary differential equations (ODEs) directly. Hermite Interpolating Polynomial is used as the fundamental function to obtain the proposed method which involved the first and second derivatives of f(t,u,u′). From the investigation done, it was found that the proposed method is consistent and zero-stable, hence it is convergent. The proposed method’s efficiency was obtained and a comparison was made in terms of accuracy to some existing methods with similar order or higher than it. This new method is able to solve linear and nonlinear initial value problems of the general second-order ODEs and outperformed existing methods with impressive results. Applications of the new method such as in the Fermi-Pasta-Ulam problem and van der Pol oscillator are discussed.

Solving System of Integro-Differential Equations using a New Hybrid Semi-Analytical Method

This paper presents a new hybrid method that combines Kharrat-Toma transform technique with the homotopy perturbation method for the solutions of linear and nonlinear initial value problems represented by integro-differential equations systems with initial conditions. Where, the advantages of this hybrid approach are rapid convergence to an approximate or exact solution, reduces the computational steps and integrals which making it a highly efficient and applicable method compared with the classical methods. Therefore, this proposed new hybrid approach (Kharrat-Toma transform and homotopy perturbation method) can be apply to find the analytical solutions of integro-differential equations systems arising in many applications. To show the accuracy and effectiveness of the presented technique several examples are introduced.