Fourth-order Ordinary Differential Equation Research Articles

A Chebyshev Generated Block Method for Directly Solving Nonlinear and Ill-posed Fourth-Order ODEs

This study presents a block method induced by Chebyshev polynomials of first kind for directly solving initial value problems of fourth-order ordinary differential equations without reducing the problems to a system of first-order differential equations. The method was developed by applying interpolation and collocation procedures to a Chebyshev approximate polynomial. The unknown parameters were obtained using the Gaussian elimination method, then substituted into the approximate solution to get the continuous scheme and evaluated at the selected point to give the discrete scheme. The method was zero stable, consistent, and convergent, p-stable as shown by the region of absolute stability and has an order of seven. The accuracy and usability of the developed method were tested by applying it to solve six numerical examples. The method was found to be efficient as it gives minimal error. The numerical results of the method were compared with other works cited in the literature and found to be better as it gives minor errors.

On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Symmetric Boundary Conditions for One Nonlinear Fourth-Order Ordinary Differential Equation

On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Symmetric Boundary Conditions for One Nonlinear Fourth-Order Ordinary Differential Equation

Properties of eigenfunctions of a boundary value problem for ordinary differential equations of fourth-order with boundary conditions depending on the spectral parameter

In this paper, we consider an eigenvalue problem for fourth-order ordinary differential equations with a spectral parameter contained in all boundary conditions. We characterize the location of eigenvalues on the real axis, find their multiplicities, and study the oscillatory properties of eigenfunctions. Moreover, we obtain refined asymptotic formulas for the eigenvalues, as well as for the values at the endpoints of the interval of eigenfunctions and their derivatives. Then, using these results and the operator-theoretic formulation, we establish sufficient conditions for the system of eigenfunctions of this problem to form a basis in the space Lp,1<p<∞, after removing four functions.

Analytical solution to a coupled system including tuned liquid damper and single degree of freedom under free vibration with modal decomposition method

This research focuses on employing a linear analytical approach to transform free surface waves and velocities into mode coordinates, with the aim of investigating the free vibration behavior of a coupled system consisting of a Single Degree of Freedom and a sloshing tank. Through a series of manipulations and simplifications of the coupled equations, a fourth-order ordinary differential equation is derived to showcase the overall response of the system, highlighting the contribution of each odd mode. Key concepts explored include system stability, mode-specific natural periods, establishment of initial boundary conditions, and formulation of the complete system response. The analytical method applied to study Tuned Liquid Dampers, a type of elevated sloshing tank, reveals that in higher modes, the lower frequency aligns with the structural natural frequency, while the higher frequency is approximately n times the structural natural frequency (where n is the odd mode number). This approach also elucidates why the system's response does not exhibit a higher-frequency component in higher modes. The study further investigates concepts such as employing an initial perturbation to excite higher frequencies and the potential for approximating the system through the first mode. Additionally, a numerical model was developed using variable separation and modal decomposition methods to complement and validate the analytical approach. Finally, further verification of the model was performed using the Preismann scheme applied to the relevant equations and the central upwind applied to nonlinear equations.

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.

A Chebyshev polynomial based block integrator for the direct numerical solution of fourth order ordinary differential equations

This paper introduces an innovative method for numerically integrating fourth-order initial value problems by utilizing Chebyshev polynomials as the fundamental basis function. The block integrator base on Chebyshev polynomial demonstrates significant improvements in accuracy and stability, rendering it a valuable tool across various scientific and engineering fields. By leveraging the characteristics of Chebyshev polynomials, this approach accurately estimates solutions for fourth-order differential equations without reducing it to a system of first order Ordinary Differential Equations while at the same time effectively managing error accumulation within a block integration framework and thereby enhancing its accuracy over extended intervals. Through rigorous numerical experiments, the effectiveness and reliability of the new integrator are demonstrated and compared with existing methods. The new method is consistent, zero stable and convergent. The method also shows an appreciable error constants. The new method performed better in terms of accuracy than the existing methods in the literature in both linear and nonlinear problems.

Application of pseudo-parameter iteration method to nonlinear deflection analysis of an infinite beam with variable beam cross-sections on a nonlinear elastic foundation

In this study, the nonlinear deflection of an infinite beam with variable beam cross-sections on a nonlinear elastic foundation was analyzed using the pseudo-parameter iteration method (PIM), which is a novel iterative semi-analytic method for solving ordinary/partial differential equations. To do this, we set six types of infinite beams with concave and convex shapes under static loading conditions. To calculate the nonlinear deflection of the infinite beam with variable cross-sections, the Bernoulli-Euler beam equation (fourth-order ordinary differential equation) considering changing beam flexural rigidity was introduced, and the PIM was adopted to this equation. Through the numerical experiment, it was confirmed that the nonlinear deflections calculated via the PIM are quite close to the exact solution within a few iterations. In addition, the graph of error quickly reaches the steady state error for all cases as the number of iterations increases.

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

Initial Value Solvers for Direct Solution of Fourth Order Ordinary Differential Equations in a Block from Using Chebyshev Polynomial as Basis Function

The numerical computation of fourth order ordinary differential equations cannot be gloss over easily due to its significant and importance. There have been glowing needs to find an appropriate numerical method that will handle effectively fourth order ordinary differential equations without resolving such an equation to a system of first order ordinary differential equations. To this end, this presentation focuses on direct numerical computation to fourth order ordinary differential equations without resolving such equations to a system of first order ordinary differential equations. The method is not predictor – corrector one due to its limitation in the level of accuracy. The method is order wise christened “Block Method” which is a self-starting method. In order to achieve this objective, Chebyshev polynomial is hereby used as basis function.

On a boundary problem for the fourth order equation with the third derivative with respect to time

In this paper, we consider a boundary value problem in a rectangular domain for a fourth-order homogeneous partial differential equation containing the third derivative with respect to time. The uniqueness of the solution of the stated problem is proved by the method of energy integrals. Using the method of separation of variables, the solution of the considered problem is sought as a multiplication of two functions X (x) and Y (y). To determine X (x),we obtain a fourth-order ordinary differential equation with four boundary conditions at the segment boundary [0,p], and for a Y (y) – third-order ordinary differential equation with three boundary conditions at the boundary of the segment [0,q]. Imposing conditions on the given functions, we prove the existence theorem for a regular solution of the problem. The solution of the problem is constructed in the form of an infinite series, and the possibility of term-by-term differentiation of the series with respect to all variables is substantiated. When substantiating the uniform convergence, it is shown that the “small denominator” is different from zero.

TRAVELING WAVES IN HEMODYNAMIC EQUATIONS WITH THE KOITER SHELL

We consider a one-dimensional model of hemodynamics for a tube defined by a Koiter shell, and investigate traveling wave solutions. The system of partial differential equations is reduced to a fourth-order ordinary differential equation for such solutions. We find a single equilibrium point and determine the condition for the changing type of the equilibria for the corresponding system of first-order differential equations. The conditions for changing the type of the equilibrium point are formulated. Then we introduce the classification of blood flow regimes relative to the type of the equilibrium point. Numerical experiments are carried out to confirm and analyze the obtained results, various regimes of blood flow are considered.

KANTOROVICH VARIATIONAL METHOD FOR THE ELASTIC BUCKLING ANALYSIS OF KIRCHHOFF PLATES WITH TWO OPPOSITE SIMPLY SUPPORTED EDGES

We present the Kantorovich variational method for the elastic buckling solutions of Kirchhoff plates with two opposite simply supported edges. Uniform compressive loading is applied on the two simply supported edges, x = 0 and x = a. Three cases of edge support considered along the y = 0 and y = b edges are (i) both are clamped (ii) both are simply supported, and (iii) edge y = 0 is simply supported while edge y = b is free. The Kantorovich method which is based on seeking to minimize the total potential energy functional P with respect to the unknown deflection w(x, y) assumes the deflection in variable separable form as an infinite series of the product of an unknown function fm(y) and a known sinusoidal function that satisfies the Dirichlet boundary conditions along the simply supported edges. Euler-Lagrange differential equation is used to obtain the fourth order ordinary differential equation which fm(y) must satisfy to minimize P. The general solution for fm(y) is obtained as a combination of hyperbolic and trigonometric functions with four unknown integration constants. The conditions for nontrivial solutions are applied to find the characteristic stability equations in each considered case. Exact solutions are obtained for the elastic buckling equations which are identical with previously obtained solutions in the literature. The elastic buckling equations are solved by iteration methods and by closed form solution methods for the Kirchhoff plate to obtain the elastic buckling loads expressed in terms of the elastic buckling load coefficients. It was found that the method yields exact elastic buckling load and exact elastic buckling load coefficient for each boundary condition considered in this study.

An Effective Approach Based on Generalized Bernstein Basis Functions for the System of Fourth-Order Initial Value Problems for an Arbitrary Interval

The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.

Coherent hybrid block method for approximating fourth-order ordinary differential equations

Conventionally, the most used method of solving fourth-order initial value problems of ordinary differential is to first reduce to a system of first-order differential equations. This approach affects the effectiveness and convergence of the numerical method due to the transformation. This paper comprises the derivation, analysis, and implementation of a new hybrid block method for direct solution of fourth-order equations. The method is derived by collocation and interpolation of an assumed basis function. The basic properties of the block method, including zero stability, error constants, consistency, order, and convergence, were analyzed. From the analysis, the block method derived was found to be zero-stable, consistent, and convergent. Errors were computed for the proposed method, and they were proven to produce approximations that agree with exact solutions and as such this shows improvement with those of existing works.

Generalized and multi-oscillation solitons in the nonlinear Schrödinger equation with quartic dispersion.

We study different types of solitons of a generalized nonlinear Schrödinger equation (GNLSE) that models optical pulses traveling down an optical waveguide with quadratic as well as quartic dispersion. A traveling-wave ansatz transforms this partial differential equation into a fourth-order nonlinear ordinary differential equation (ODE) that is Hamiltonian and has two reversible symmetries. Homoclinic orbits of the ODE that connect the origin to itself represent solitons of the GNLSE, and this allows one to study the existence and organization of solitons with advanced numerical tools for the detection and continuation of connecting orbits. In this paper, we establish the existence of new types of connecting orbits, namely, PtoP connections from one periodic orbit to another. As we show, these global objects provide a general mechanism that generates additional families of two types of solitons in the GNLSE. First, we find generalized solitons with oscillating tails whose amplitude does not decay but reaches a nonzero limit. Second, PtoP connections in the zero energy level can be combined with EtoP connections from the origin to a selected periodic orbit to create multi-oscillation solitons; their characterizing property is to feature several episodes of different oscillations in between decaying tails. As is the case for solitons that were known previously, generalized solitons and multi-oscillation solitons are shown to be an integral part of the phenomenon of truncated homoclinic snaking.

Analysis of wave energy device dynamics and power optimization

As an important renewable resource, wave energy has considerable development value and application prospects. How to design a wave energy conversion device to maximize the output power is an urgent problem in its practical application. In this paper, we mainly study the principle of wave energy device capacity, the motion and power model of the constructed wave energy device, and finally design a specific scheme to achieve the maximum output power of wave energy through kinetic analysis, the fourth-order ordinary differential equation solved by the Longacurta method, and the Simpson integral method. At the end of the paper, the developed model is objectively evaluated and the potential value of the developed model in practical wave energy device applications is illustrated.

Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip

The investigation focuses on the effects of wall velocity slip on the solution of a viscous, laminar, incompressible channel flow subjected to small-scaled contraction and expansion of the weakly permeable walls. The study of such flow systems is often contextual for fluid transport in biological organisms. In the considered flow configuration, the vertically moving porous walls enable the fluid to enter or exit with a constant rate. The tangential slip velocity of the flow at the porous walls is modeled with the Navier slip boundary condition. The flow dynamics inside the channel is governed by the full Navier–Stokes equations. The Lie symmetry analysis and the invariant method are adopted to reduce the number of independent variables in the system of governing equations. Consequently, a single fourth-order ordinary differential equation is obtained, which is solved analytically by the double perturbation method and the variation of iteration method. The solutions are compared for different arrangements. Furthermore, the approximated analytical solutions are likened to the numerical solutions obtained from a fourth-order Runge–Kutta solver embedding the Shooting method to check the accuracy. It is observed that the boundary layers are formed, and the flow rapidly turns near the walls, when suction and wall contraction coexist. Alternatively, if injection and wall expansion are paired, the flow adjacent to the walls is delayed. The existence of wall velocity slip advances the near-wall velocity and cuts down the speed of centerline velocity. It results in a change in the volumetric flow rate and shear rate. The overall pressure is also varied by higher wall velocity slip. The results are explored for different values of the permeation Reynold number and the dimensionless wall dilation rate to capture all possible impacts of the flow parameters. The current analysis rectifies the existing errors in the work of Boutros et al. [“Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,”Appl. Math. Model. 31(6), 1092–1108 (2007)] with the no-slip boundary condition and discusses the overall influences of slip boundary condition on the Lie symmetry solution of the flow system.

Similarity reduction and multiple novel travelling and solitary wave solutions for the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients

In this study, the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients (vcBK) which describes the interaction of Riemann waves has been solved two times by using the balance technique. First, the balance procedure was used to reduce the vcBK equation into a fourth-order nonlinear ordinary differential equation and by its implementation again composed with the auxiliary method, many novel periodic, kink and solitary wave solutions were obtained for the vcBK equation. Second, the balance method was used to construct the Bäcklund transformation for the vcBK equation, by which n-soliton solutions were obtained. Finally, some figures were given to show the soliton interaction for different values of the variable coefficients.

Lie symmetry analysis, particular solutions and conservation laws of a (2+1)-dimensional KdV4 equation.

In this paper, a (2+1)-dimensional KdV4 equation is considered. We obtain Lie symmetries of this equation by utilizing Lie point symmetry analysis method, then use them to perform symmetry reductions. By using translation symmetries, two fourth-order ordinary differential equations are obtained. Solutions of one fourth order ordinary differential equation are presented by using direct integration method and $ (G'/G) $-expansion method respectively. Furthermore, the corresponding solutions are depicted with appropriate graphical representations. The other fourth-order ordinary differential equation is solved by using power series technique. Finally, two kinds of conserved vectors of this equation are presented by invoking the multiplier method and Noether's theorem respectively.

On indirect representability of fourth order ordinary differential equation in form of Hamilton - Ostrogradsky equations

On indirect representability of fourth order ordinary differential equation in form of Hamilton - Ostrogradsky equations