First-order Initial Value Problems Research Articles

Reduced Element for Adaptive Finite Element Analysis of First-Order IVP with Built-in Error Estimator in Maximum Norm

This paper proposes a novel yet simple approach to the adaptive finite element (FE) analysis of the first-order Initial Value Problems (IVPs) in the maximum norm by introducing the reduced element technique. In the present approach, the FE solution uh of the conventional Galerkin element of degree m + 1 is decomposed into two parts: a reduced solution uRh from the reduced element of degree m obtained by ignoring the highest degree term of uh, and a built-in point-wise error estimator εRh directly given by the ignored term. Since the end node solutions of the reduced element are inherited from the full order element, it gains O(h2m+2) accuracy and achieves a nodal/element accuracy ratio as high as two, which greatly enhances its adaptive capability regarding solving IVPs on long time domains. The related error analysis is addressed and a complete adaptivity algorithm is given. Typical numerical examples of both linear and nonlinear IVPs of both single and systems of equations are presented to show the validity and effectiveness of the proposed approach.

A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions: III the Role of the Derivative of the Phase Lag and the Derivative of the Amplification Factor

A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions: III the Role of the Derivative of the Phase Lag and the Derivative of the Amplification Factor

A novel A-stable method of order of accuracy three, based on optimization for solving initial value problems

This paper presents a novel A-stable nonlinear method of order of accuracy three, based on optimization for solving the first-order initial value problems. The A-stability is proven and the consistency of order three is investigated as well. The convergence of the method is established under certain assumptions. Additionally, a second-order method is derived based on this approach, and a variable step size formulation is implemented using the second and third-order methods to enhance efficiency. Numerical examples provided in the paper demonstrate the validity of the theory and underscore the advantages of the method compared to three other A-stable methods of order three, including an implicit RK method, a Rosenbrock method, and an explicit (nonstandard) method.

New computational methods using seventh-derivative type for the solution of first-order initial value problems

This study uses finite power series as the basis function and interpolation and collocation techniques to study a class of implicit block methods of a seventh-derivative type. Discrete schemes are implicit two-point block methods that are obtained by selecting collocation points carefully and unevenly in order to improve the stability of the methods through testing. Nevertheless, in contrast to other current numerical equations, these methods require seventh-derivative functions. The novel techniques are identified, examined, and shown to be A-stable and convergent. Newton Raphson’s approach is used to accomplish method implementation. Trials demonstrated the effectiveness and precision of the derived equations in terms of computational time and absolute errors on a variety of first-order initial value issues, such as first-order, second-order linear differential systems and SIR model. When compared to similar methods that are currently in the literature, the suggested methods produce better numerical results.

Efficient Multistep Algorithms for First-Order IVPs with Oscillating Solutions: II Implicit and Predictor–Corrector Algorithms

This research introduces a fresh methodology for creating efficient numerical algorithms to solve first-order Initial Value Problems (IVPs). The study delves into the theoretical foundations of these methods and demonstrates their application to the Adams–Moulton technique in a five-step process. We focus on developing amplification-fitted algorithms with minimal phase-lagor phase-lag equal to zero (phase-fitted). The request of amplification-fitted (zero dissipation) is to ensure behavior like symmetric multistep methods (symmetric multistep methods are methods with zero dissipation). Additionally, the stability of the innovative algorithms is examined. Comparisons between our new algorithm and traditional methods reveal its superior performance. Numerical tests corroborate that our approach is considerably more effective than standard methods for solving IVPs, especially those with oscillatory solutions.

New Computational Methods Using Seventh Derivative Type for the Solution of First Order Initial Value Problems

In this research, a class of implicit block methods of a seventh derivative type are examined through interpolation and collocation techniques using finite power series as the basis function. The discrete schemes, which are implicit two-point block methods, are obtained by carefully and unevenly choose collocation points that ensure better methods’ stability via test. However, these schemes require seventh derivative functions unlike other existing numerical formulae. The new methods are found, investigated and proven to be convergent and A-stable. The implementation of methods is achieved by using Newton Raphson’s method. Experiments show the efficiency and accuracy of the developed formulae on different class of first-order initial value problems, including SIR, growth models and Prothero-Robinson oscillatory problem and with comparison to such existing methods. In addition, it is observed that uneven and positioning of collocation points greatly influence the efficiency and accuracy of numerical methods.

A One-Step Hybrid Obrechkoff-Type Block Method for First-Order Initial Value Problems in Ordinary Differential Equations

In this paper, we propose a one-step hybrid block method of the Obrechkoff-type block method for solving initial value problems in ordinary differential equations using Taylor series expansion. The new method was implemented with some selected initial value problems to determine the efficiency and accuracy of the method. The convergence and stability properties were also examined. It was discovered that the new method compared favorably with other existing methods in the selected literature.

MHD nanofluid flow through Darcy medium with thermal radiation and heat source

In this analysis, we have considered heat transmission in two-dimensional steady laminar nanofluid flow past a wedge. Magnetohydrodynamic (MHD), Brownian motion, viscous dissipation and thermophoresis effects are considered over the porous surface. Similarity transformations have been used to change the governing partial differential equations (PDEs) into nonlinear higher-order ordinary differential equations (ODEs). Governing ODEs with boundary conditions are then converted to the system of first-order initial value problem. After that the modeled system is solved numerically by RK4 technique. Impact of the magnetic number, Eckert number, Prandtl number, Lewis number, Brownian motion, thermophoresis and permeability parameters on the flow domain is analyzed graphically as well as in tabular form. It is noted that magnitude of Nusselt number for the flow regime increases with the increase of nondimensional parameter [Formula: see text] while opposite behavior is observed in case of R.

A New Hybrid Block Method for Solving First-Order Differential System Models in Applied Sciences and Engineering

This paper presents a new hybrid block method formulated in variable stepsize mode to solve some first-order initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method is implemented considering an adaptive stepsize strategy to maintain the estimated error in each step within a specified tolerance. In order to evaluate the performance and usefulness of the proposed technique in real-world applications, several differential problems from applied sciences and engineering, such as the SIR model, Jacobi elliptic function problem, and chemical reactions problems, are solved numerically. The results of numerical simulations in this work demonstrate that the proposed method is more efficient than other existing numerical methods used for comparisons.

Derivation of Five-Stage Implicit Runge-Kutta Method of Order 10 via Gauss-Legendre Quadrature for Ordinary Differential Equations

This paper presents developing and implementing a five-stage implicit Runge-Kutta method of order ten via Gauss-Legendre quadrature for stiff or oscillatory first-order initial value problems (IVPs) of ordinary differential equations (ODEs). Using continuous collocation and interpolation techniques, the implicit Runge-Kutta method was developed by combining Legendre polynomials and exponential functions as the basis function. The properties of the method were investigated, and it was shown that it is consistent and A-stable. The new method was evaluated on two sampled problems involving stiffness and oscillation. The numerical results demonstrate that the new implicit Runge-Kutta is computationally efficient and outperforms previous methods of similar derivations.

A New Hybrid Block Method via Combined Hermite Polynomials and Exponential Functions as Basis Function

This paper presents the derivation and implementation of a hybrid block method for solving stiff and oscillatory first-order initial value problems of ordinary differential equations (ODEs). The hybrid block method was derived by continuous collocation and interpolation using combined Hermite polynomials and exponential functions as the basis function to produce a continuous implicit Linear Multistep Method (LMM) of order nine and implement in block form. The basic properties of the derived method were studied, and the hybrid block integrator was demonstrated to be zero-stable, convergent, consistent, and to have an A-stable region of absolute stability, which made it suitable for stiff and oscillatory ordinary differential equations. The use of a combined basis in the generation of LMMs is worthy of universal acceptance. The technique indicates that, utilizing an interpolation and collocation approach, continuous LMMs can be derived from combinations of any polynomials and exponential functions. On two sampled stiff and oscillatory problems, the new integrator was tested. The numerical results indicate that our new hybrid block integrator is computationally efficient and outperforms existing methods in terms of accuracy

Stability Analysis of Dual Solutions of Convective Flow of Casson Nanofluid past a Shrinking/Stretching Slippery Sheet with Thermophoresis and Brownian Motion in Porous Media

This article considered the steady two-dimensional boundary layer flow of incompressible viscous Casson nanofluids over a permeable, convectively heated, shrinking/stretching slippery sheet surface. The achievements of this work are extremely relevant, both theoretically with respect to the mathematical modeling of non-Newtonian nanofluid flow with heat transfer in engineering systems and with respect to engineering cooling applications. The combined impacts of suction/injection, viscous dissipation, convective heating, and chemical reactions were considered. The governing modeled partial differential equations with boundary conditions are transformed into nonlinear ordinary differential equations using similarity transformations and finally converted to the first-order initial value problem. Then, the technique of the fourth-fifth order Runge–Kutta–Fehlberg with the shooting method is used to obtain numerical solutions. Moreover, the effects of different involving parameters on the dimensionless temperature, velocity, and concentration, as well as, from an engineering viewpoint, local Nusselt number, the skin friction, and local Sherwood number are illustrated and presented in graphs and tabular forms. For critical shrinking parameter λ c , the existence of a dual solution within the interval λ c < λ < 0 is revealed, and this range escalates with the suction and slipperiness parameters; hence, both control the flow stability. The increment in the values of the porous media, Casson, Forchheimer, slipperiness, and convective heating parameters reduces the local skin friction and intensifies the rates of mass and heat transfer. For the Newtonian flow (that is, as the Casson parameter β gets to infinity ∞ ), the thermal boundary layer thickness, temperature profile, and skin friction diminish, whereas the concentration profile, mass, and heat transfer rates increase compared to the non-Newtonian Casson nanofluid. These results excellently agree with the existing ones.

Odd Order Integrator with Two Complex Functions Control Parameters for Solving Systems of Initial Value Problems

In this study, a numerical integrator that is based on a nonlinear interpolant, for the local representation of the theoretical solution is presented. The resulting integrator aims to solve second and higher-order initial value problems as systems of first-order initial value problems. The method is designed to have two complex functions as control parameters. The control parameters may become real, depending on the nature of the second-order initial value problems to be solved. The generalization and properties of the scheme are also presented.

Overlapping Grid-Based Optimized Single-Step Hybrid Block Method for Solving First-Order Initial Value Problems

This study presents a new variant of the hybrid block methods (HBMs) for solving initial value problems (IVPs). The overlapping hybrid block technique is developed by changing each integrating block of the HBM to incorporate the penultimate intra-step point of the previous block. In this paper, we present preliminary results obtained by applying the overlapping HBM to IVPs of the first order, utilizing equally spaced grid points and optimal points that maximize the local truncation errors of the main formulas at the intersection of each integration block. It is proven that the novel method reduces the local truncation error by at least one order of the integration step size, O(h). In order to demonstrate the superiority of the suggested method, numerical experimentation results were compared to the corresponding HBM based on the standard non-overlapping grid. It is established that the proposed method is more accurate than HBM versions of the same order that have been published in the literature.

Higher-Order Asymptotic Numerical Solutions for Singularly Perturbed Problems with Variable Coefficients

For the purpose of solving a second-order singularly perturbed problem (SPP) with variable coefficients, a mth-order asymptotic-numerical method was developed, which decomposes the solutions into two independent sub-problems: a reduced first-order linear problem with a left-end boundary condition; and a linear second-order problem with the boundary conditions given at two ends. These are coupled through a left-end boundary condition. Traditionally, the asymptotic solution within the boundary layer is carried out in the stretched coordinates by either analytic or numerical method. The present paper executes the mth-order asymptotic series solution in terms of the original coordinates. After introducing 2(m+1) new variables, the outer and inner problems are transformed together to a set of 3(m+1) first-order initial value problems with the given zero initial conditions; then, the Runge–Kutta method is applied to integrate the differential equations to determine the 2(m+1) unknown terminal values of the new variables until they are convergent. The asymptotic-numerical solution exactly satisfies the boundary conditions, which are different from the conventional asymptotic solution. Several examples demonstrated that the newly proposed method can achieve a better asymptotic solution. For all values of the perturbing parameter, the method not only preserves the inherent asymptotic property within the boundary layer but also improves the accuracy of the solution in the entire domain. We derive the sufficient conditions, which terminate the series of asymptotic solutions for inner and outer problems of the SPP without having the spring term. For a specific case, we can derive a closed-form asymptotic solution, which is also the exact solution of the considered SPP.

Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions

Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions

Block Trigonometrically Fitted Backward Differentiation Formula for the Initial Value Problem with Oscillating Solutions

This paper presents a fitted backward differentiation formula (BTBDF) whose coefficients are functions of a fixed fitting frequency and step length especially designed for the numerical integration of first-order Initial Value Problems (IVPs) with oscillatory results. The BTBDF is a product of three discrete formulas which are obtained from a continuous second derivative trigonometrically fitted method (CSDTFM). The BTBDF is applied in a block-wise form that makes it enjoys the advantages of the self-starting formula. The convergence of the BTBDF is discussed and its superiority is demonstrated in some numerical experiments to illustrate accuracy advantage.

Fast Computation of Highly Oscillatory ODE Problems: Applications in High-Frequency Communication Circuits

The paper demonstrates symmetric integral operator and interpolation based numerical approximations for linear and nonlinear ordinary differential equations (ODEs) with oscillatory factor x′=ψ(x)+χω(t), where the function χω(t) is an oscillatory forcing term. These equations appear in a variety of computational problems occurring in Fourier analysis, computational harmonic analysis, fluid dynamics, electromagnetics, and quantum mechanics. Classical methods such as Runge–Kutta methods etc. fail to approximate the oscillatory ODEs due the existence of oscillatory term χω(t). Two types of methods are presented to approximate highly oscillatory ODEs. The first method uses radial basis function interpolation, and then quadrature rules are used to evaluate the integral part of the solution equation. The second approach is more generic and can approximate highly oscillatory and nonoscillatory initial value problems. Accordingly, the first-order initial value problem with oscillatory forcing term is transformed into highly oscillatory integral equation. The transformed symmetric oscillatory integral equation is then evaluated numerically by the Levin collocation method. Finally, the nonlinear form of the initial value problems with an oscillatory forcing term is converted into a linear form using Bernoulli’s transformation. The resulting linear oscillatory problem is then computed by the Levin method. Results of the proposed methods are more reliable and accurate than some state-of-the-art methods such as asymptotic method, etc. The improved results are shown in the numerical section.

Asymptotic Numerical Solutions for Second-Order Quasilinear Singularly Perturbed Problems

For a second-order quasilinear singularly perturbed problem under the Dirichlet boundary conditions, we propose a new asymptotic numerical method, which involves two problems: a reduced problem with a one-side boundary condition and a novel boundary layer correction problem with a two-sided boundary condition. Through the introduction of two new variables, both problems are transformed to a set of three first-order initial value problems with zero initial conditions. The Runge–Kutta method is then applied to integrate the differential equations and to determine two unknown terminal values of the new variables until they converge. The modified asymptotic numerical solution satisfies the Dirichlet boundary conditions. Some examples confirm that the newly proposed method can achieve a better asymptotic solution to the quasilinear singularly perturbed problem. For most values of the perturbing parameter, the present method not only preserves the inherent asymptotic property within the boundary layer but also improves the accuracy within the entire domain.

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.