Solution Of Initial Value Problem Research Articles

Generalizations of Riemann–Liouville fractional integrals and applications

Generalizations of Riemann–Liouville fractional integrals and applications

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results.

A-stable Two Derivative Mono-Implicit Runge-Kutta Methods for ODEs

An A-stable Two Derivative Mono Implicit Runge-Kutta (ATDMIRK) method is considered herein for the numerical solution of initial value problems (IVPs) in ordinary differential equation (ODEs). The methods are of high-order A-stable for $p=q=\lbrace 2s+1\rbrace _{s=2}^{7}\ $ The $p$, $q$ and $s$ are the order of the input, output and the stages of the methods respectively. The numerical results affirm the superior accuracy of the newly develop methods compare to the existing ones.

Comparing the Performance of Two Butcher-Based Block Hybrid Algorithms for the Solution of Initial Value Problems

In this paper, we compare the performances of two Butcher-based block hybrid methods for the numerical integration of initial value problems. We compare the condition numbers of the linear system of equations arising from both methods and the absolute errors of the solution obtained. The results of the numerical experiments illustrate that the better conditioned method outperformed its less conditioned counterpart based on the absolute errors. In addition, after applying our method on some examples, it was discovered that the absolute errors in this work were better than those of a recent study in the literature. Hence, we recommend this method for the numerical solution of stiff and non-stiff initial value problems.

Numerical Solution of Eighth Order Two Point Boundary Value Problems by Taylor Series Method

A Taylor series method which is taught at undergraduate levels and which has been hitherto used for the solution of initial value problem is successfully used in this work for the solutions of eighth-order two point boundary-value problems. The method is based on successive differentiation of the governing equation to obtain high order derivatives and then evaluated at the boundary point x=a. The solution is expressed inform a Taylor series with the unknown coefficients at a point x=a. By applying boundary conditions at x=b in the Taylor series solution, the system of unknown coefficient is obtained. After solving the system, then unknown coefficient are determined. The procedure is applied on both linear and nonlinear boundary-value problems. A comparison of the results obtained by the present method with results obtained by other methods reveals that the present method is simple, effective and also is in good agreement with the previous result and exact solution as showing in the tables and figures.

A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions

In this research, we provide a novel approach to the development of effective numerical algorithms for the solution of first-order IVPs. In particular, we detail the fundamental theory behind the development of the aforementioned approaches and show how it can be applied to the Adams–Bashforth approach in three steps. The stability of the new scheme is also analyzed. We compared the performance of our novel algorithm to that of established approaches and found it to be superior. Numerical experiments confirmed that, in comparison to standard approaches to the numerical solution of Initial Value Problems (IVPs), including oscillating solutions, our approach is significantly more effective.

A basic theory for initial value problems of first order ordinary differential equations with Lp-Carathéodory functions and applications

A basic theory on the existence and uniqueness of solutions and local solutions for initial value problems (IVPs) of first order ordinary differential equations is established, where the nonlinearities are Lp-Carathéodory functions. The basic theory contains three parts: (i) the existence and uniqueness of solutions in C([a,b],[c,d]); (ii) the existence and uniqueness of local solutions in C([a,a+h1(p)];[c,d]), where the expression of the number h1(p) will be given and is a computable constant, and (iii) the existence and uniqueness of solutions and nonnegative solutions by establishing a new Bihari inequality. The classical results dealt with the above part (ii) when f is continuous or an L1-Carathéodory function, and only obtained the existence of a number h such that the IVPs had local solutions in C[a,a+h]. Our methodology is to use the results obtained on (i) to study the results mentioned in (ii) when p∈[1,p] while the classical results used the Schauder fixed point theorems directly to obtain the existence and uniqueness of local solutions when f is continuous or p=1. Our results in parts (i) and (iii) are new. The results in part (ii) are new when p∈(1,∞] and improve the classical results when p=1. Our results will enrich the classical basic theory. As applications of the basic theory, the existence of solutions and nonnegative local solutions for logistic type population models with heterogeneous environments is studied, where the local rates of changes in the population density are Lp-Carathéodory functions.

A four-stage multiderivative explicit Runge-Kutta method for the solution of first order ordinary differential equations

The development of numerical methods for the solution of initial value problems in ordinary differential equation have turned out to be a very rapid research area in recent decades due to the difficulties encountered in finding solutions to some mathematical models composed into differential equations from real life situations. Researchers have in recent times, used higher derivatives in the derivation of numerical methods to produce totally new ways of solving these equations. In this article, a new Runge-Kutta type methods with reduced number of function evaluations in the increment function is constructed, analyzed and implemented. This proposed method border on the use of higher derivatives up to the second derivative in the ki terms of Runge-Kutta method in order to achieve a higher order of accuracy. The qualitative features: local truncation error, consistency, convergence and stability of the new method were investigated and established. Numerical examples were also performed on some initial value problems to confirm the accuracy of the new method and compared with some existing methods of which the numerical results show that the new method competes favorably.

On the construction of diagonally implicit two–step peer methods with RK stability

In this paper, diagonally implicit two–step peer methods for the numerical solution of initial value problems of order ordinary differential are divided into four types including the combination of explicit and implicit methods in a sequential or parallel environments. In this class of the methods, construction of implicit methods equipped with Runge–Kutta stability property together with A– or L–stability are investigated and examples of such methods are given up to order five. Finally, the efficiency and accuracy of the proposed methods are verified by applying them on some well–known stiff problems.

Fast Fourier transform-based solutions of initial value problems for wave propagation in microelastic media

Fast Fourier transform-based solutions of initial value problems for wave propagation in microelastic media

Maclaurin Series Solution for Seventh Order Boundary Value Problems

Maclaurin series expansion which is taught at undergraduate levels and which has been previously used for the solution of initial value problem is successfully used in this paper for the solutions of seventh-order boundary value problems. The governing equation defined between boundary points 0 and 1is differentiated to get high order derivatives and the assumed solution is expanded in Maclaurin series then, the boundary conditions at x=1 are utilized to determined unknown coefficients. The method is simple and the numerical results displayed on the tables were found to be in good agreement with the solution and even better than some existing results in the literature. This agreement is also confirmed from the figures were the graph of exact solution and approximate solution overlap.

Existence results for some classes of differential systems with "maxima"

"Local existence of solutions of initial value problems associated with a new type of systems of differential equations with “maxima” are investigated. Keywords: Systems of differential equations with ""maxima"", initial value problems, local existence of solutions."

On the weighted inertia-energy approach to forced wave equations

We show convergence of minimizers of weighted inertia-energy functionals to solutions of initial value problems for a class of nonlinear wave equations. The result is given for the nonhomogeneous case under a natural growth assumption on the forcing term.

Linear Hybrid Multistep Block Method for Direct Solution of Initial Value Problems of Third Order Ordinary Differential Equations

In this article, we focus on linear hybrid multistep method for direct solution of initial value problems of third order ordinary differential equations without reduction to system of first-order ordinary differential equations. The derivation of the method involved using collocation and interpolation techniques with power series as basis function to produce a system of linear equations. The unknown parameters in the system of equations were obtained through the Gaussian elimination technique. The values of the determined parameters were then substituted and evaluated at different grid and off-grid points to produce the required continuous block method. The discrete scheme obtained from the method is self-starting with improved accuracy and a larger interval of absolute stability. Basic properties of the method were investigated. The results showed that the method is zero stable, consistent , convergent and of order seven. The performance of the method was tested by solving linear and nonlinear problems of general third order ordinary differential equations. The result were found to compare favourably with some existing methods in literature.

Initial Value Problems: a Didactic Tool in Evolution

Various science and engineering problems, generally time-dependent, are modeled with ordinary differential equations with conditions at the same value of the independent variable, called initial value problems. The use of technological resources during the learning process allows students to execute routine procedures quickly and simply, enabling them to have more time to discuss concepts. The use of these resources helps to generate a learning environment where, through the use of certain mathematical skills, students can achieve meaningful and comprehensive learning. The aim of this work is to show an update of a self-designed tool to approximate the solution of initial value problems that allows comparisons between some one-step methods and multi-step methods and the analysis of the numerical solutions obtained when varying certain parameters. The update of the tool arises to satisfy different requirements discovered after the use of the previous tool.

Numerical algorithm for solving real-life application problems of Lane–Emden type equation

This paper presents an advanced variational iteration method for finding the approximate-exact solution for initial value problems of the Lane–Emden type equation that arise in several applications by employing the quasilinearization approach to the variational iteration method. Also, the convergence analysis of the method is studied under the Lipschitz condition. To demonstrate the applicability of the method, several application problems arise in isothermal gas spheres, stellar structures, and physics with different shape factors and are successfully solved by means of this method. The comparison is made with the available exact solution and the existing methods (Mall and Chakraverty, 2014; Umesh and Kumar, 2021; El-Essawy et al., 2023; Ahmad et al., 2016). We see that the method meets our expectations and reaches an exact solution in a few iterations. The optimality of the method is investigated by the absolute errors and absolute residual errors. The present method is time-efficient, requires less computational work, and converges to an exact solution in a few iterations. We also provided the CPU time to obtain an approximate solution to all the problems for the numerical investigation.

On the accuracy, stability and computational efficiency of explicit last-stage diagonally implicit Runge–Kutta methods (ELDIRK) for the adaptive solution of phase-field problems

Runge–Kutta algorithms with adaptive step size control provide reliable tools for the solution of initial value problems. Most widespread within the vast family of Runge–Kutta methods are the diagonally implicit Runge–Kutta (DIRK) methods. In particular, the new low-order explicit last-stage diagonally implicit Runge–Kutta (ELDIRK) methods are investigated, combining implicit schemes with an additional explicit evaluation as an explicit last stage. This results in Butcher tableaus with two solutions of different convergence orders suitable for embedded methods, where the higher-order solution is achieved by additional explicit evaluations. Thus, the iterative solution of non-linear systems is omitted for the additional stage, presenting a major reduction in computational cost for the determination of a local error estimate. The key contribution is the application of the novel Butcher tableaus to phase-field problems, solved with the finite-element method, leading to substantial numerical investigations with an efficient approach for diagonally implicit Runge–Kutta schemes. The most important aspects are the extension towards fourth-order methods, the study of the convergence orders for the ELDIRK schemes, their respective stability regions and computational efficiency. A local error estimator is presented capturing the evolution of the phase-field problems, such that adaptive step size control for the new low-order embedded schemes based on an empirical approach for error estimation is achieved. A suitable parallel algorithm is presented with conclusive two-dimensional phase-field simulations based on a Kobayashi–Warren–Carter model including benchmarks for computational efficiency. The higher-order convergence suggested by the novel schemes is confirmed, and their effective results are demonstrated, resulting in a valuable semi-explicit addition to the family of Runge–Kutta time integration schemes.

Recursive solution of initial value problems with temporal discretization

We construct a continuous domain, as a model of interval analysis, for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We prove computability of the operator within the framework of effectively given domains. The operator only requires the vector field of the differential equation to be Lipschitz continuous, in contrast to the related operators in the literature which require the vector field to be at least continuously differentiable. Within the same framework, we also analyze temporal discretization and computability of another variant of the Euler operator formulated according to Runge-Kutta theory. We prove that, compared with this variant, the second-order operator that we formulate directly, not only imposes weaker assumptions on the vector field, but also exhibits superior convergence rate. We implement the first-order, second-order, and Runge-Kutta Euler operators using arbitrary-precision interval arithmetic, and report on some experiments. The experiments confirm our theoretical results. In particular, we observe the superior convergence rate of our second-order operator compared with the Runge-Kutta Euler and the common (first-order) Euler operators.

Explicit two-step peer methods with reused stages

Two-step peer methods for the numerical solution of Initial Value Problems (IVP) combine the advantages of Runge-Kutta (RK) and multistep methods to obtain high stage order and provide in a natural way a dense output. In general, explicit s-stage peer methods require s evaluations of the vector field at each step. Nevertheless, Klinge and coworkers (BIT Numer Math, 2018) have shown that some methods use less function calls se<s, here called effective stages, by re-using sr=s−se previously computed stages (shifted stages) from the previous steps in the current one.In this paper we propose a new approach, different from the one used by Klinge and coworkers, to re-use previously computed stages, that we call peer methods with reused stages, showing that methods with reused stages and se effective stages are equivalent to three-step peer methods with se stages. Then, we analyze all the families of methods with two effective stages, obtaining methods with s=3 and orders 4 and 5 in which the free parameters of the families have been used to minimize the coefficient of the leading error term as well as to maximize the absolute stability interval. We have also studied one family of peer methods with s=4 and three effective stages, obtaining a method with order 6, superconvergent of order 7, and optimized leading error term as well as absolute stability interval. Some numerical experiments show the performance of the obtained methods by comparing them with other previously obtained peer methods as well as other standard Runge-Kutta and multistep methods.

Nonlinear Integral Inequalities Involving Tempered Ψ-Hilfer Fractional Integral and Fractional Equations with Tempered Ψ-Caputo Fractional Derivative

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered Ψ-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered Ψ-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered Ψ-Caputo fractional derivative of order 0&lt;α&lt;1, where the right side may depend on time, the solution, or its tempered Ψ-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the Ψ-exponential stability of trivial solutions are proven for these kinds of differential equations.