Problems For Ordinary Differential Equations Research Articles

Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations

This article introduces two numerical methods to address boundary value problems associated with secondorder and fractional differential equations. These methods employ two parameters related to shifted Gegenbauer polynomials as their basis functions. The process involves establishing a differentiation operational matrix for the shifted Gegenbauer polynomials. Subsequently, the initial/boundary value problems for ordinary and fractional differential equations are transformed into a system of equations through the Galerkin, collocation, and tau methods. The convergence analysis is ensured by leveraging theorems pertaining to the shifted Gegenbauer polynomials. To validate the accuracy of the approach, numerous numerical examples are presented.

Development of a fully implicit ODE-solver for containment analysis code

The thermal–hydraulic dynamics in containment are governed by a system of stiff ordinary differential equations (ODEs). A fully implicit discretization scheme is adopted to discretize these ODEs in order to mitigate the effects of stiffness. In comparison with explicit or semi-implicit discretization schemes that are subject to Courant limits on time steps, the fully implicit discretization scheme is more suitable for a containment analysis code that focuses on predicting both short-term and long-term thermal–hydraulic parameters after an accident. This study introduces a general-purpose ODE solver for the containment analysis code. The outline of the solver is as follows: The fully implicit discrete equations lead to a large set of nonlinear equations that need to be solved using Newton’s iterative method. The partial derivative components in the Jacobi matrix are calculated by the perturbation method using finite difference approximation, which avoids the complicated derivation of partial derivatives. The scaling modification technique is incorporated into this ODE solver to deal with significant differences in unknown variable magnitudes, and the line search method is introduced to address the difficulty of obtaining an accurate root estimate with Newton’s method when the initial guess is far from the actual root. This proposed ODE solver was applied to two typical stiff ODE problems to test its stiffness-suppressed ability and to demonstrate that this proposed solver can perform calculations with a very large time step. Then, the CASSIA code, a containment analysis code developed by China Nuclear Power Technology Research Institute Co., Ltd (CNPRI), equipped with this ODE solver, was applied to the CSNI (Committee on the Safety of Nuclear Installations) benchmark problem and the Carolinas Virginia Tube Reactor (CVTR) test 3 problem to preliminarily demonstrate that the proposed ODE solver can perform containment thermal–hydraulic analysis correctly. This study could provide references for the development of a home-made containment analysis code.

Fractal quintic spline solutions for singularly perturbed reaction-diffusion boundary-value problems

In this article, with the help of fractal quintic spline, numerical solutions of singularly perturbed boundary value problem of ordinary differential equations are obtained. To check the accuracy of the proposed method, convergence analysis is derived. The developed method has fourth order convergence. To check the correctness of the proposed method, numerical examples are provided.

One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations

In this paper, a one-step block of optimized hybrid schemes for the numerical integration of second-order initial value problems (IVP) of ordinary differential equations (ODE) is constructed via collocation techniques. The developed scheme is obtained by considering two intra-step nodal points as hybrid points, which are chosen in order to achieve optimized errors of the main formulae approximating the solution such that 0 < v1 < v 2 < 1 where v1 and v2 are defined as hybrid points. The characteristics of the developed scheme are analyzed. Application of the new scheme on some second-order IVPs shows the accuracy and effectiveness of the scheme compared with some existing methods.

Numerical Analysis for Sturm–Liouville Problems with Nonlocal Generalized Boundary Conditions

For the generalized Sturm–Liouville problem (GSLP), a new formulation is undertaken to reduce the number of unknowns from two to one in the target equation for the determination of eigenvalue. The eigenparameter-dependent shape functions are derived for using in a variable transformation, such that the GSLP becomes an initial value problem for a new variable. For the uniqueness of eigenfunction an extra condition is imposed, which renders the right-end value of the new variable available; a derived implicit nonlinear equation is solved by an iterative method without using the differential; we can achieve highly precise eigenvalues. For the nonlocal Sturm–Liouville problem (NSLP), we consider two types of integral boundary conditions on the right end. For the first type of NSLP we can prove sufficient conditions for the positiveness of the eigenvalue. Negative eigenvalues and multiple solutions may exist for the second type of NSLP. We propose a boundary shape function method, a two-dimensional fixed-quasi-Newton method and a combination of them to solve the NSLP with fast convergence and high accuracy. From the aspect of numerical analysis the initial value problem of ordinary differential equations and scalar nonlinear equations are more easily treated than the original GSLP and NSLP, which is the main novelty of the paper to provide the mathematically equivalent and simpler mediums to determine the eigenvalues and eigenfunctions.

Derivative block methods for the solution for fourth-order boundary value problems of ordinary differential equations

The application of block methods in the study of dynamical system is one area which needs more research. Therefore, this work presents block methods and its application to solve some problems in solid and fluid mechanics. The method was developed directly using the approaches of collocation and interpolation, and using power series polynomial as a trial solution. First, the system of linear equations is solved to obtain the unknown coefficients. The coefficients gotten are then substituted into the approximate solution to obtain continuous scheme. The continuous scheme, its first, second and third derivatives are evaluated at all the grid points to generate the block methods. The derived methods were applied to solve fourth-order boundary value problems of ordinary differential equations arising from beams and chemical problems. The results demonstrate the reliability and efficiency of the proposed method.

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

Effect of radiation and slip parameter on MHD mixed convective Casson fluid flow along a vertical stretching sheet

The flow behavior of Casson fluids along a vertical stretching sheet is important for a wide range of industrial processes where such fluids are encountered. For example, knowledge of this study might be used in the design of coating processes or in biomedical applications involving blood flow. Aims of this study are the optimization of engineering processes, ensuring efficiency and quality in various applications and analysis of boundary layer along stretching sheet. Mathematical formulation of the problem consists parial differential equations and later on converted to a set of ordinary differential equations (ODEs) with boundary conditions and solved using “Bvp4c solver” in MATLAB. The Bvp4c solver is a numerical technique used to solve boundary value problems (BVPs) for ODEs. For various values of physical parameters the velocity, temperature, and concentration distribution are presented through figures. Skin friction coefficient, Nusselt number, and Sherwood number are tabulated and analyzed through bar graphs. It is noted that for assisting flow [Formula: see text], velocity profile decreases near the sheet while it increases away from the sheet for growing inputs of Casson fluid parameter ([Formula: see text]), while opposite seniro has been observed for Hartmann number ([Formula: see text]). Coefficient of surface drag is lowered with growing inputs of Casson fluid parameter and heat transfer rate is enhanced with increased inputs of Hartmann number in both assisting and opposing flow. The temperature profile enhances for raising inputs of radiation parameter ([Formula: see text]) in both assisting and opposing flow. The concentration profiles also fall, when chemical reaction parameter ([Formula: see text]) and Schmidt number ([Formula: see text]) are enhanced.

ON THE UNIQUE SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR DIFFERENTIAL EQUATIONS WITH PARAMETER

A linear boundary value problem for a differential equation with a parameter is investigated on a finite interval by the parameterization method. The studied boundary value problem with parameter is reduced to an equivalent multipoint boundary value problem with parameters by splitting the interval, introducing additional parameters at the points of splitting and new functions. The obtained equivalent boundary value problem contains Cauchy problems for ordinary differential equations with respect to new functions. By substituting the solution representation of the Cauchy problem into the boundary conditions and continuity conditions of the solution, a system of linear algebraic equations with respect to the introduced parameters is compiled. An algorithm for finding a solution to the boundary value problem with parameters is constructed. The formulation of the theorem on sufficient conditions of unique solvability of the boundary value problem with parameters is given. Sufficient conditions of its unique solvability are obtained in terms of the data of the original boundary value problem. An example showing the fulfillment of the conditions of the theorem is given.

Surface wrinkling of a film coated to a graded substrate

We study the surface wrinkling of a stiff thin elastic film bonded to a compliant graded elastic substrate subject to compressive stress generated either by compression or growth of the bilayer. Our aim is to clarify the influence of the modulus gradient on the onset and surface pattern in this bilayer. Within the framework of finite elasticity, an exact bifurcation condition is obtained using the Stroh formulation and the surface impedance matrix method. Further analytical progress is made by focusing on the case of short wavelength limit for which the Wentzel–Kramers–Brillouin method can be used to resolve the eigenvalue problem of ordinary differential equations with variable coefficients. An explicit bifurcation condition is obtained from which the critical buckling load and the critical wavelength are derived asymptotically. In particular, we consider two distinct situations depending on the ratio β of the shear modulus at the substrate surface to that at infinity. If β is of O(1) or small, the parameters related to modulus gradient all appear in the higher-order terms and play an insignificant role in the bifurcation. In that case, it is the modulus ratio between the film and substrate surface that governs the onset of surface wrinkling. If, however, β≫1, the modulus gradient affects the critical condition through leading-order terms. Through our analysis we unravel the influence of different material and geometric parameters, including the modulus gradient, on the bifurcation threshold and the associated wavelength which can be of importance in many biological and technological settings.

An Efficient Third-Order Scheme Based on Runge–Kutta and Taylor Series Expansion for Solving Initial Value Problems

In this paper, we propose a new numerical scheme based on a variation of the standard formulation of the Runge–Kutta method using Taylor series expansion for solving initial value problems (IVPs) in ordinary differential equations. Analytically, the accuracy, consistency, and absolute stability of the new method are discussed. It is established that the new method is consistent and stable and has third-order convergence. Numerically, we present two models involving applications from physics and engineering to illustrate the efficiency and accuracy of our new method and compare it with further pertinent techniques carried out in the same order.

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.

Numerical Solution of Fourth-order Initial Value Problems Using Novel Fourth-order Block Algorithm

In this paper, a one-step fourth-order block scheme for solving fourth order Initial Value Problems (IVPs) of Ordinary Differential Equations (ODE) is developed using interpolation and collocation techniques. The derived schemes contain two hybrid points which are chosen such that 0 < w1 < w2 < 1 where w1 and w2 are defined as hybrid points. The characteristics of the developed schemes are analyzed. The obtained schemes are applied in block form to solve some fourth-order IVPs and the numerical results show the accuracy and effectiveness of the block scheme compared with some existing methods.

Numerical methods for solving second-order initial value problems of ordinary differential equations with Euler and Runge-Kutta fourth-order methods

This paper presents two standard numerical methods for solving second order initial value problems for ordinary differential equations (ODEs). The Euler and the Runge-Kutta fourth-order methods are applied without any discretization or restrictive assumptions for solving ODEs. The numerical solutions obtained by the two methods are in good agreement with the exact solutions. The convergence and error analysis which are discussed demonstrate the effectiveness of the methods. The results obtained from the two numerical methods show that the RK4 method is appropriate, consistent, convergent, quite stable, and more accurate than the Euler's method.

Transient Thermal Analysis of Space Rectangular Thin-Walled Structures by Carrera Unified Formulation

The accurate prediction of transient thermal fields of space structures is of paramount significance for the analysis of geometrical stability and thermal management systems. This paper proposes a novel method to solve the transient thermal field of the thin-walled cross section of beams in outer space. High-order beam elements are derived by means of the Carrera unified formulation (CUF), where the longitudinal direction of the beam is discretized by one-dimensional two-node and four-node elements, whereas arbitrary high-order expansion functions are used for the description of the temperature expansion within the cross section. The governing differential equations of the transient temperature field are derived according to the transient thermal conduction theory and the weighted residual method. The solution of the initial value problem for ordinary differential equations is obtained through the application of the Adams–Bashforth method, leading to the determination of the transient thermal field. In order to demonstrate the accuracy of the temperature results, a convergence analysis is conducted. Finally, temperature distributions within the cross section and along the longitudinal direction of the model are discussed to show the effects of several factors, including material properties, angle of solar radiation, and shadows induced by the shelter of other components.

A New Fixed Coefficient Diagonally Implicit Block Backward Differentiation Formula for Solving Stiff Initial Value Problems

Stiff initial value problems in ordinary differential equations occur when solution components evolve at varying rates, posing challenges for traditional computational methods. Specialized techniques are crucial for maintaining accuracy and stability during rapid transitions, emphasizing their significance in developing reliable numerical algorithms across scientific and engineering applications. This study aims to develop a new fixed coefficient 3-point diagonally implicit block backward differentiation formula for the numerical solution of first order stiff initial value problems. The method is constructed by integrating a triangular matrix into the coefficient matrix of an existing extended 3-point super class of block BDF for solving stiff initial value problems. The selection of a fixed coefficient within the interval accompanies this integration  to ensure optimal stability. The method is found to order five. Stability analysis indicates that the method is consistent, zero-stable, and almost A-stable, validating its applicability to stiff initial value problems. Implementation of the method involves Newton’s iteration, and a code in the C programming language is devised to demonstrate its effectiveness. Comparative examination of numerical outcomes with the existing 3BBDF and 3ESBBDF methods highlights the proposed method's enhanced accuracy and reduced computation time.

Implicit Block and Runge- Kutta type Methods for Solution of Second-Order Ordinary Differential Equations.

In this research, implicit discrete schemes which form our block integrators were developed for solving Initial Valued Problems of Ordinary Differential Equations. The equivalent second order Runge-Kutta type Methods (RKTM) were also constructed for the same purpose. Both methods were demonstrated on linear and nonlinear problems of Ordinary Differential Equations. Numerical results obtained from RKTM show that the method is competitive with the existing one.

One-step three-parameter optimized hybrid block method for solving first order initial value problems of ordinary differential equations

A one-step three-parameter optimized hybrid block method and second derivative hybrid block method with optimized points were proposed to solve first-order ordinary differential equations. The techniques of interpolation and collocation were adopted for the derivation of the methods using a three-parameter approximation. The hybrid points were obtained by optimizing the local truncation error of the method. The schemes obtained were reformulated to reduce the number of occurrences of the source term. The hybrid points were used in the derivation of the second derivative hybrid block method. The discrete schemeswere produced as a by-product of the continuous scheme and used to simultaneously solve initial value problems (IVPs) in block mode. The resulting schemes are self-starting, do not require the creation of individual predictors, and are consistent, zero-stable, and convergent. The accuracy and efficiency of the methods were ascertained using several numerical test problems. The numerical results were favourably compared to some techniques from the cited literature.

Multidimensional Diffusion-Wave-Type Solutions to the Second-Order Evolutionary Equation

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties.

Numerical Identification of Boundary Conditions for Richards’ Equation

A time stepping quasilinearization approach to the mixed (or coupled) form of one and two dimensional Richards’ equations is developed. For numerical solution of the linear ordinary differential equation (ODE) for 1D case and elliptic for 2D case, obtained after this semidiscretization, a finite volume method is used for direct problems arising on each time level. Next, we propose a version of the decomposition method for the numerical solution of the inverse ODE and 2D elliptic boundary problems. Computational results for some soil types and its related parameters reported in the literature are presented.