Linear Initial Value Problems Research Articles

An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Systems of First-Order Linear Fuzzy Initial Value Problems and Their Applications

Systems of First-Order Linear Fuzzy Initial Value Problems and Their Applications

Three-step Collocation Method for Solution of Third Derivative Initial Value Problem

In this research paper, we introduce a novel three-step block method designed to directly tackle third-order initial value problems. This method is crafted through an interpolation and collocation approach, leveraging power series analysis. We conduct a comprehensive examination of the proposed method, ensuring it meets all requisite conditions for rigorous analysis. To assess its efficacy and validity, we employ the method on both highly stiff linear and nonlinear initial value problems, juxtaposing our findings with established approaches in the literature. Our results highlight that the new method exhibits faster convergence compared to existing methods, underscoring its superior performance. Initial value problems, Interpolation and collocation procedure, Three step, Third order

A Block Hybrid Method with Equally Spaced Grid Points for Third-Order Initial Value Problems

In this paper, we extend the block hybrid method with equally spaced intra-step points to solve linear and nonlinear third-order initial value problems. The proposed block hybrid method uses a simple iteration scheme to linearize the equations. Numerical experimentation demonstrates that equally spaced grid points for the block hybrid method enhance its speed of convergence and accuracy compared to other conventional block hybrid methods in the literature. This improvement is attributed to the linearization process, which avoids the use of derivatives. Further, the block hybrid method is consistent, stable, and gives rapid convergence to the solutions. We show that the simple iteration method, when combined with the block hybrid method, exhibits impressive convergence characteristics while preserving computational efficiency. In this study, we also implement the proposed method to solve the nonlinear Jerk equation, producing comparable results with other methods used in the literature.

Investigation of a one-step hybrid algorithm towards the solution of first order linear and nonlinear initial-value problems of ordinary differential equations (FOLNIVP)

The aim of developing any numerical method is to complement the challenges inherent in obtaining the analytical solution of a differential equation, if at all a closed form solution does exist. In this study we present a one-step implicit code of order eight block algorithms for the purpose of utilizing data at points other than a whole step number. The major advantage of hybrid method is that they possess remarkably small error constants which translate to better approximation. These methods constitute a class of methods whose computational potentialities have probably not yet been fully exploited. Therefore, the performance of the derived block hybrid algorithm is investigated using some numerical examples for the purpose of demonstrating its validity and applicability. The results obtained revealed that the algorithm is suitable for solving first order linear and nonlinear initial value problems (IVPs) of ordinary differential equations.

An Hermite–Obreshkov method for 2nd-order linear initial-value problems for ODE

An Hermite–Obreshkov method for 2nd-order linear initial-value problems for ODE

Solutions of initial and boundary value problems using invariant curves

<p>The purpose of this study is to investigate the solutions of initial and boundary value problems of ordinary differential equations by employing Lie symmetry generators. In this investigation, it shown that invariant curves, which obtained by symmetry generators, also be utilized to find solutions to initial and boundary value problems. A method, involving invariant curves, presented to find solutions to initial and boundary value problems. Solutions to many linear and nonlinear initial and boundary value problems discussed by applying the proposed method.</p>

Numerical treatments of nonlinear Burgers–Fisher equation via a combined approximation technique

A combined spectral matrix collocation strategy is presented to solve the time-dependent nonlinear Burgers–Fisher equation pertaining to various important physical mechanisms such as advection, diffusion, and logistic reaction. The nonlinearity of the model is first tackled by a time-marching scheme based on the Taylor formula. Hence in each time step, we solve a linear initial boundary value problem (IBVP) by using the spectral collocation technique based on novel Boole polynomials. Various numerical computations are carried out to indicate the pertinent features and testify the applicability of the presented combined technique. Comparisons are made between our results and the exact analytical solutions and some available numerical outcomes in the literature to show the validity of the method.

Well-Conditioned and Ill Conditioned Linear First Order Initial Value Problems in Ordinary Differential Equation

We construct a linear first order ordinary differential equation with the parameter λ. We show that our constructed equation is well conditioned for λ< 0 and ill conditioned forλ > 0. We also state and prove some related theorems.

Explicit Runge–Kutta–Nyström methods for the numerical solution of second order linear inhomogeneous IVPs

Runge–Kutta–Nyström (RKN) methods for the numerical solution of inhomogeneous linear initial value problems with constant coefficients are considered.A general procedure to construct explicit s-stage RKN methods with maximal order p=s+1, similar to the developed by the authors (Montijano et al., 2023) for the class of second order IVP under consideration, depending on the nodes ci,i=1,…,s is presented. This procedure requires only the solution of successive linear equations in the elements aij, 1≤j<i≤s of the matrix of coefficients A of the RKN method and avoids the solution of non linear equations.The remarkable fact is that using as free parameters the nodes ci,i=1,…,s with a quadrature relation, the s(s−1)/2 elements of matrix A can be computed by solving successively linear systems with coefficients depending on the nodes, so that if they are non-singular we get a unique s-stage method with maximal order s+1.We obtain an optimized six-stage seventh-order RKN method in the sense that the nodes are chosen so that minimize the leading term of the local error. Finally, some numerical experiments are presented to test the behaviour of the optimized RKN method with others with Radau and Lobatto nodes.

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

PurposeThe purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.Design/methodology/approachThe authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.FindingsThe authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.Originality/valueSince psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

A new approach for solving boundary and initial value problems by coupling the he method and Sawi transform

This paper discusses and implements a newly developed technique using the He method with Sawi Transform. The main aim is to solve some initial and boundary problems. This combination exhibits an accurate strategy to obtain a precise solution for linear and nonlinear problems. To validate the proposed Hybrid method, a 4- examples are discussed, these including: Burger’s equation, telegraph equation, Kelin-Gordan equation, Duffing oscillator with cubic nonlinear term. The obtained results improve the exactness and the accuracy of the proposed combinations, and the proposed method is capable to solve a large number of linear and nonlinear initial and boundary value problems.

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Accurate analytical calculation of the rate coefficient for the diffusion-controlled reactions due to hyperbolic diffusion.

Using an approach based on the diffusion analog of the Cattaneo-Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski's boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski's theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice's formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper bound of the exact value. Here, the obtained formula seems to be of great significance for bridging a known gap between an analytically estimated rate coefficient on the one hand and molecular dynamics simulations together with experimentally observed results for the short times regime on the other hand. A particular emphasis has been placed on the rigorous mathematical treatment and important properties of the relevant initial boundary value problems in parabolic and hyperbolic diffusion theories.

Explicit Runge–Kutta methods for the numerical solution of linear inhomogeneous IVPs

Runge–Kutta methods for the numerical solution of inhomogeneous linear initial value problems with constant coefficients are considered.A general procedure to construct explicit s-stage RK methods with order s, for the class of IVP under consideration, depending on the nodes ci,i=1,…,s is presented. This procedure only requires the solution of successive linear equations in the elements of the matrix A and avoids the solution of non linear equations.We obtain several RK schemes with number of stages s=5,…,8 and maximal order p=s for the class of problems under consideration. Finally, some numerical experiments to test the behaviour of the new RK schemes are presented.

Properties of Degenerate Systems of Linear Integro-Differential Equations and Initial Value Problems for These Equations

We consider systems of higher order integro-differential equations in which the matrix multiplying the highest derivative of the unknown vector function is identically singular in the domain where the system is considered. We give criteria for the solvability of such systems and initial value problems for these systems as well as examples illustrating the theoretical results.

Properties of Degenerate Systems of Linear Integro-Differential Equations and Initial Value Problems for These Equations

Properties of Degenerate Systems of Linear Integro-Differential Equations and Initial Value Problems for These Equations

Exponential Euler and Backward Euler Methods for Nonlinear Heat Conduction Problems

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved. We compare this method to the backward Euler method combined with nonlinear iterations. For both methods we show monotonicity and boundedness of the solutions and give sufficient conditions for convergence of the nonlinear iterations. Numerical tests are presented to examine performance of the two schemes. The presented exponential Euler scheme is implemented based on restarted Krylov subspace methods and, hence, is essentially explicit (involves only matrix-vector products).

Экспоненциальная и неявная схемы Эйлера для решения нелинейных задач теплопроводности

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved. We compare this method to the backward Euler method combined with nonlinear iterations. For both methods we show monotonicity and boundedness of the solutions and give sufficient conditions for convergence of the nonlinear iterations. Numerical tests are presented to examine performance of the two schemes. The presented exponential Euler scheme is implemented based on restarted Krylov subspace methods and, hence, is essentially explicit (involves only matrix-vector products).

Forecasting the behavior of fractional order Bloch equations appearing in NMR flow via a hybrid computational technique

In this paper, we present an efficient computational approach named as Sumudu residual power series method (SRPSM) to solve fractional Bloch equations appearing in an NMR flow. This method is a copulation of the residual power series method (RPSM) and the Sumudu transform to construct approximate solution in shapes of sharp convergent series by adopting the notion of limit with a view of smooth calculations and time saving strategy as compared to the classical RPSM in which computations of fractional derivatives are required. This work also verifies and compares the solution obtained by the proposed technique with the classical RPSM. To assure the applicability, performance, and reliability of the introduced method, a system of fractional Bloch equations are examined along with numerical simulations. The aspect of application supported with computer simulations demonstrates that the suggested approach is precise, and appropriate to explore the solutions of linear & nonlinear initial value problems with fractional-order derivatives.