First-order Linear Ordinary Differential Equations Research Articles

Assessment of Numerical Performance of Some Runge-Kutta Methods and New Iteration Method on First Order Differential Problems

This research focuses on the assessment of the numerical performance of some Runge-Kutta methods and New Iteration Method “NIM”  for solving first-order differential problems. The assessment is conducted through extensive numerical experiments and comparative  analyses. Accuracy, efficiency, and stability are among the key factors considered in evaluating the performance of the methods. A range  of first-order differential problems with diverse characteristics and complexity levels is employed to thoroughly examine the methods'  capabilities and limitations. The numerical investigation that is defined in the study as well as the results that are stated in the Tables,  demonstrates that all the approaches produce extremely accurate results. However, the “NIM” was shown to be the most effective of the  three methods used in this study. Conclusively, the “NIM” should be employed to solve first-order nonlinear and linear ordinary  differential equations in place of Runge-Kutta Fourth order method (RK4M) and Butcher Runge-Kutta Fifth order method (BRK5M). In  addition, BRK5M is more applicable and efficient than RK4M when solving first order ordinary differential problems.  

STOCHASTIC GALERKIN METHOD AND PORT-HAMILTONIAN FORM FOR LINEAR FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

We consider linear first-order systems of ordinary differential equations (ODEs) in port-Hamiltonian (pH) form. Physical parameters are remodeled as random variables to conduct an uncertainty quantification. A stochastic Galerkin projection yields a larger deterministic system of ODEs, which does not exhibit a pH form in general. We apply transformations of the original systems such that the stochastic Galerkin projection becomes structure-preserving. Furthermore, we investigate meaning and properties of the Hamiltonian function belonging to the stochastic Galerkin system. A large number of random variables implies a high-dimensional stochastic Galerkin system, which suggests itself to apply model order reduction (MOR) generating a low-dimensional system of ODEs. We discuss structure preservation in projection-based MOR, where the smaller systems of ODEs feature pH form again. Results of numerical computations are presented using two test examples.

Comparison of numerical and exact analyses in a neutron activation and decay chain

A method for solving a set of coupled linear first-order ordinary differential equations as it applies to neutron activation and subsequent radioisotope decay chains was developed that is easy to implement and accurate enough for most preliminary engineering design calculations. The method is applied to a specific radioisotope production scheme to evaluate the yield of the desired isotope as well as production of secondary and tertiary contaminant radioisotopes. An analytical closed-form solution was also derived for the same activation and decay chain to provide a basis for comparison; the analytical solution was benchmarked against several limiting cases to validate the equations. Results of the simulation involving the numerical approach were compared to the exact analytical solution. The numerical solution, while obviously not exact, is found to be close enough for engineering purposes. A discussion of the numerical precision of the result for the particular case are discussed.

"TA’LIM JARAYONIDA BIRINCHI TARTIBLI CHIZIQLI ODDIY DIFFERENSIAL TENGLAMALARNING YECHIMINI MAPLE DASTURIDA TOPISH"

"TA’LIM JARAYONIDA BIRINCHI TARTIBLI CHIZIQLI ODDIY DIFFERENSIAL TENGLAMALARNING YECHIMINI MAPLE DASTURIDA TOPISH"

On the stability of systems of two linear first-order ordinary differential equations

On the stability of systems of two linear first-order ordinary differential equations

Analytic solution of the exact Daum–Huang flow equation for particle filters

State estimation for nonlinear systems, especially in high dimensions, is a generally intractable problem, despite the ever-increasing computing power. Efficient algorithms usually apply a finite-dimensional model for approximating the probability density of the state vector or treat the estimation problem numerically. In 2007 Daum and Huang introduced a novel particle filter approach that uses a homotopy-induced particle flow for the Bayesian update step. Multiple types of particle flows were derived since with different properties. The exact flow considered in this work is a first-order linear ordinary time-varying inhomogeneous differential equation for the particle motion. An analytic solution in the interval [0,1] is derived for the scalar measurement case, which enables significantly faster computation of the Bayesian update step for particle filters.

Students’ thinking about problem solving tasks: the case of initial value problems

Contributing to research on undergraduate students’ thinking about problem solving tasks, the present study reports on students’ reasoning about two initial-value problems i.e. first-order linear ordinary differential equations with initial conditions. A qualitative analysis of task-based interviews and work written by 34 students revealed that although the students knew the algebraic steps they had to use to solve the two problems, 16 students reported that steps involving integral calculus were particularly challenging. In fact, a majority of these students exhibited lack of facility with the -substution integration technique, something that severed as a stumbling block for them when calculating the integrating factor in their attempt to solve the two problems. Only seven students were successful in mathematizing one of the problems. Five students created and used diagrams to support their thinking when solving one of the problems. These students reported that the creation of the diagrams helped them to solve one of the two problems. Overall, findings of this study suggest that helping students develop greater facility with integration techniques, and more broadly integral calculus, could increase students’ success when solving initial-value problems. Implications for instruction and directions for future research are discussed.

Stability criteria for systems of two first-order linear ordinary differential equations

Abstract The Riccati equation method is used to establish stability criteria for systems of two first-order linear ordinary differential equations. Examples are presented in which the obtained result is compared with the results obtained by the Lyapunov and Bogdanov methods, by a method involving estimates of solutions in the Lozinskii logarithmic norms and by the freezing method.

New reducibility criteria for systems of two linear first-order ordinary differential equations

New reducibility criteria for systems of two linear first-order ordinary differential equations

Properties of solutions of quaternionic Riccati equations

In this paper we study properties of regular solutions of quaternionic Riccati equations. The obtained results we use for study of the asymptotic behavior of solutions of two first-order linear quaternionic ordinary differential equations.

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.

Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘ε’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, EN,Δt and rate of convergence, Pε N,Δt. The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘  ’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, N t , E  and rate of convergence, N t , P  . The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

An Observation about First-Order Linear ODEs

Although it is not usually stated this way, every student of calculus studies a simple case of a first-order linear ordinary differential equation, namely, y′=q(x). Since students are generally tau...

On the reducibility systems of two linear first-order ordinary differential equations

Some global solvability criteria for the scalar Riccati equations are used to establish new reducibility criteria for systems of two linear first-order ordinary differential equations. Some examples are presented.

On the reducibility of systems of two linear first-order ordinary differential equations

On the reducibility of systems of two linear first-order ordinary differential equations

On the reducibility of systems of two linear first-order ordinary differential equations

On the reducibility of systems of two linear first-order ordinary differential equations

Unification of the common methods for solving the first-order linear ordinary differential equations

<p style="text-indent:20px;">A good understanding of the mathematical processes of solving the first-order linear ordinary differential equations (ODEs) is the foundation for undergraduate students in science and engineering programs to progress smoothly to advanced ODEs and/or partial differential equations (PDEs) later. However, different methods for solving the first-order linear ODEs are presented in various textbooks and resources, which often confuses students in their choice of the method for solving the ODEs. This special tutorial note presents the practices the author used to address such confusions in solving the first-order linear ODEs for students engaged in the bachelorette engineering studies at a regional university in Australia in recent years. The derivation processes of the four commonly adopted methods for solving the first-order linear ODEs, including three explicit methods and one implicit method presented in many textbooks, are presented first, followed by the logical interconnections that unify these four methods to clarify student's confusions on different presentations of the procedures and the solutions in different sources. Comparisons among these methods are also made.

Formulation of a New Implicit Method for Group Implicit BBDF in Solving Related Stiff Ordinary Differential Equations

This paper proposed a new alternative approach of the implicit diagonal block backward differentiation formula (BBDF) to solve linear and nonlinear first-order stiff ordinary differential equations (ODEs). We generate the solver by manipulating the numbers of back values to achieve a higher-order possible using the interpolation procedure. The algorithm is developed and implemented in C ++ medium. The numerical integrator approximates few solution points concurrently with off-step points in a block scheme over a non-overlapping solution interval at a single iteration. The lower triangular matrix form of the implicit diagonal causes fewer differentiation coefficients and ultimately reduces the execution time during running codes. We choose two intermediate points as off-step points appropriately, which are proven to guarantee the method's zero stability. The off-step points help to increase the accuracy by optimizing the local truncation error. The proposed solver satisfied theoretical consistency and zero-stable requirements, leading to a convergent multistep method with third algebraic order. We used the well-known and standard linear and nonlinear stiff IVP problems used in literature for validation to measure the algorithm's accuracy and processor time efficiency. The performance metrics are validated by comparing them with a proven solver, and the output shows that the alternative method is better than the existing one.

Stability Criterion for Systems of Two First-Order Linear Ordinary Differential Equations

The method of Riccati’s equation is applied to find a stability criterion for systems of two first-order linear ordinary differential equations. The obtained result is compared for a particular example with results obtained by the Lyapunov and Bogdanov methods, by using estimates of solutions of systems in terms of the Losinskii logarithmic norms, and by the freezing method.