Fractional Bagley-Torvik Equation Research Articles

A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition

Abstract In this work, we present a sophisticated operating algorithm, the reproducing kernel Hilbert space method, to investigate the approximate numerical solutions for a specific class of fractional Begley-Torvik equations (FBTE) equipped with fractional integral boundary condition. Such fractional integral boundary condition allows us to understand the non-local behavior of FBTE along with the given domain. The algorithm methodology depends on creating an orthonormal basis based on reproducing kernel function that satisfies the constraint boundary conditions so that the solution is finally formulated in the form of a uniformly convergent series in ϖ 3 [ a , b ] {\varpi }_{3}\left[a,b] . From a numerical point of view, some illustrative examples are provided to determine the appropriateness of algorithm design and the effect of using non-classical boundary conditions on the behavior of solutions approach.

Utilization of the Modified Adomian Decomposition Method on the Bagley-Torvik Equation Amidst Dirichlet Boundary Conditions

The Bagley-Torvik equation is an imperative differential equation that considerably arises in various branches of mathematical physics and mechanics. However, very few methods exist for the treatment of the model analytically; in fact, researchers frequently shop for semi-analytical and numerical methods in their studies. Therefore, the main goal of this research is to find theexact analytical solution for the fractional Bagley-Torvik equation fitted with Dirichlet boundary data, as well as a system of fractional Bagley-Torvik equations. Thus, this research aims to show that the modified Adomian decomposition method (MADM) via the proposed two algorithms is a very effective method for treating a class of Bagley-Torvik equations endowed with Dirichletboundary data. Certainly, MADM is a very powerful approach for solving dissimilar functional equations without the need for either linearization, discretization, perturbation, or even unnecessary restraining postulations. Additionally, the method reveals exact analytical solutions whenever obtainable or closed-form series solutions whenever exact solutions are not feasible. Lastly, some illustrative test problems of the governing model are examined to demonstrate the superiority of the proposed algorithms.

Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

This paper uses Müntz orthogonal functions to numerically solve the fractional Bagley–Torvik equation with initial and boundary conditions. Müntz orthogonal functions are defined on the interval 0 , 1 and have simple and distinct real roots on this interval. For the function f ∈ L 2 0 , 1 , we obtain the best unique approximation using Müntz orthogonal functions. We obtain the Riemann–Liouville fractional integral operator for Müntz orthogonal functions so that we can reduce the complexity of calculations and increase the speed of solving the problem, which can be seen in the process of running the Maple program. To solve the fractional Bagley–Torvik equation with initial and boundary conditions, we use Müntz orthogonal functions and consider simple and distinct real roots of Müntz orthogonal functions as collocation points. By using the Riemann–Liouville fractional integral operator that we define for the Müntz orthogonal functions, the process of numerically solving the fractional Bagley–Torvik equation that is solved using Müntz orthogonal functions is reduced, and finally, we reach a system of algebraic equations. By solving algebraic equations and obtaining the vector of unknowns, the fractional Bagley–Torvik equation is solved using Müntz orthogonal functions, and the error value of the method can be calculated. The low error value of this numerical solution method shows the high accuracy of this method. With the help of the Müntz functions, we obtain the error bound for the approximation of the function. We have obtained the error bounds for the numerical method using which we solved the fractional Bagley–Torvik equation with initial and boundary conditions. Finally, we have given a numerical example to show the accuracy of the solution of the method presented in this paper. The results of solving this example using Müntz orthogonal functions and comparing the results with other methods that have been used the solve this example show the higher accuracy of the method proposed in this paper.

Dynamical modeling and physical analysis of pipe flow in hydraulic systems based on fractional variational theory

Based on the fractional variational theory and Hamilton principle, this paper extends a semi-analytic solution and further develops a fractional-order dynamical model of pipe flow. This model facilitates the dynamical coupling of hydraulic and mechanical systems, and ensures easily have the solution for both the physical characteristics by state space equation or the damped vibration characteristics by fractional Bagley–Torvik equation. Subsequent application in an actual hydropower system reveals the potential periodic vibration of pipe flow and the effect of fractional order memory on the dashpot time constant. The comparison of these findings with the smaller alignment index (SALI) detection results reaffirms the soundness of this study.

Solving fractional Bagley-Torvik equation by fractional order Fibonacci wavelet arising in fluid mechanics

This study introduces a new fractional order Fibonacci wavelet technique proposed for solving the fractional Bagley-Torvik equation (BTE), along with the block pulse functions. To convert the specified initial and boundary value problems into algebraic equations, the Riemann–Liouville (R-L) fractional integral operator is defined, and the operational matrices of fractional integrals (OMFI) are built. This numerical scheme’s performance is evaluated and examined on particular problems to show its proficiency and effectiveness, and other methods that are accessible in the current literature are compared. The numerical results demonstrate that the approach produces extremely precise results and is computationally more decisive than previous methods.

Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation

In this paper, we formulate an efficient algorithm based on a new iterative method for the numerical solution of the Bagley-Torvik equation. The fractional differential equation arises in many areas of applied mathematics including viscoelasticity problems and applied mechanics of the oscillation process. We construct the fractional derivatives via the Caputo-type fractional operator to formulate a three-step algorithm using the MAPLE 18 software package. We further investigate the relationships between the surface area and stiffness of the spring constants of the Bagley-Torvik equation on three case problems and numerical results are presented to demonstrate the efficiency of the proposed algorithm.

Fractional Modeling and Characteristic Analysis of Hydro-Pneumatic Suspension for Construction Vehicles

The motion differential equation of hydro-pneumatic suspension is established to describe the vibration characteristics for a certain type of construction vehicle. The output force was deduced from the suspension parameters. Based on the suspension characteristics of a multi-phase medium, fractional calculus theory was introduced, and its fractional Bagley–Torvik equation was formed. The numerical computation by a low-pass filter of the Oustaloup algorithm was performed. The numerical solution of a nonlinear fractional equation was obtained to investigate the vibration characteristics of the suspension fractional system. Through the building of an equal-ratio test platform and simulation model, the fractional- integer-order model simulation and experimental data were compared. When the fractional order is 0.9, it better describes the motion characteristics of suspension system. The experiments show that the experimental data can fit the fractional-order system model well, and thereby prove the model on a hydro-pneumatic suspension system.

Necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and solution representation

In this paper, we investigate the necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and the solution representation by using the multivariate Mittag-Leffler function. First we convert fuzzy initial value problem into the cut problem (system of fractional differential equations with inequality constraints) and obtain existence results for the solution of the cut problem under (1,1)- differentiability. Next we study the conditions for the solutions of the cut problem to constitute the solution of a fuzzy initial value problem and suggest a necessary and sufficient condition for the (1,1)-solution. Also, some examples are given to verify the effectiveness of our proposed method. The necessary and sufficient condition, solution representation for (1,2)-solution of initial value problem of fuzzy fractional Bagley-Torvik equation are shown in Appendix.

A new approach for Volterra functional integral equations with non-vanishing delays and fractional Bagley-Torvik equationss

A numerical technique for Volterra functional integral equations (VFIEs) with non-vanishing delays and fractional Bagley-Torvik equation is displayed in this work. The technique depends on Bernstein polynomial approximation. Numerical examples are utilized to evaluate the accurate results. The findings for examples figs and tables show that the technique is accurate and simple to use.

A sine-cosine wavelet method for the approximation solutions of the fractional bagley-torvik equation

A sine-cosine wavelet method for the approximation solutions of the fractional bagley-torvik equation

Homotopy Sumudu transform method for solving applications in physics

In this article, we proposed an efficient method to solve the delay Fractional Bagley-Torvik equation. The proposed method is capable of reducing the volume of the computational work as compared to the classical techniques while still maintaining the high accuracy of the numerical result; the size reduction amounts to an improvement of the performance of the approach. Our concern was to provide an analytic solution to the Bagley-Torvik equation using a relatively new analytical technique, the homotopy-Sumudu transformation method. The method does not need to calculate the fractional derivatives or integrations in each term. The results compared with implicit hybrid one-step method yield the efficiency of the presented algorithm. Also, the calculations involved in the method are very simple and straightforward.

Generalized Fibonacci Operational tau Algorithm for Fractional Bagley-Torvik Equation

Generalized Fibonacci Operational tau Algorithm for Fractional Bagley-Torvik Equation

An exponential spline approximation for fractional Bagley\u2013Torvik equation

In this paper, we approximate the solution of fractional Bagley–Torvik equation by using the exponential spline function and the shifted Grünwald difference operator. The proposed methods reduce to the system of algebraic equations. The convergence analysis of the methods has been discussed. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods.

An Investigation of Fractional Bagley-Torvik Equation.

In this article, we will solve the Bagley–Torvik equation by employing integral transform method. Caputo fractional derivative operator is used in the modeling of the equation. The obtained solution is expressed in terms of generalized G function. Further, we will compare the obtained results with other available results in the literature to validate their usefulness. Furthermore, examples are included to highlight the control of the fractional parameters on he dynamics of the model. Moreover, we use this equation in modelling of real free oscillations of a one-degree-of-freedom mechanical system composed of a cart connected with the springs to the support and moving via linear rolling bearing block along a rail.

An efficient operational matrix method for the numerical solutions of the fractional Bagley–Torvik equation using wavelets

In this paper, we apply a method which comprises of two methods Bernoulli wavelet operational matrix of derivative and integration to solve Bagley–Torvik equation of fractional order arising in mathematical chemistry. We discuss convergence and comparison of this method. The obtained results indicate that the method is very efficient and accurate.

An investigation of fractional Bagley-Torvik equation

AbstractIn this paper the authors prove the existence as well as approximations of the solutions for the Bagley-Torvik equation admitting only the existence of a lower (coupled lower and upper) solution. Our results rely on an appropriate fixed point theorem in partially ordered normed linear spaces. Illustrative examples are included to demonstrate the validity and applicability of our technique.

Numerical Solution of Fraction Bagley–Torvik Boundary Value Problem Based on Chebyshev Collocation Method

The Bagley–Torvik boundary value problems are used in several branches of applied mathematics and mechanics. Therefore, there is a necessity of a numerical scheme for the solution of these problems. In this paper, we propose an efficient Chebyshev collocation scheme to solve a special kind of two-point fractional Bagley–Torvik equation considering the fractional derivative in the Caputo sense. The given problem is successfully reduced into the system of algebraic equations with a particular choice of collocation points. The error analysis and convergence criteria of the present scheme are also discussed. We get good results in the sense that just a few terms of basis functions (SCPFK) are required. The given scheme is simple and powerful in use to solve such problems since boundary conditions are taken into account automatically. The efficiency and accuracy of the present scheme are examined through numerical examples and comparison with existing methods.

On the explicit solutions of fractional Bagley-Torvik equation arises in engineering

In this work, Bagley-Torvik equation is considered with conformable derivatives. The analytical solutions will be obtained via Sine-Gordon expansion method and Bernouli equation method for the two cases of Bagley-Torvik equation. We will illustrate and discuss about the methodology and solutions therefore the proposed equation has meaning in different areas of science and engineering.

Numerical solution of the distributed-order fractional Bagley-Torvik equation

In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation. This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function. Using the composite Boole 02BC &#xs rule the distributed-order Bagley-Torvik equation is approximated by a multi-term time-fractional equation, which is then solved by the Grunwald-Letnikov method &#x0028 GLM &#x0029 and the fractional differential transform method &#x0028 FDTM &#x0029. Finally, we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.

An Application of the Gegenbauer Wavelet Method for the Numerical Solution of the Fractional Bagley-Torvik Equation

In this paper, a potentially useful new method based on the Gegenbauer wavelet expansion, together with operational matrices of fractional integral and block-pulse functions, is proposed in order to solve the Bagley–Torvik equation. The Gegenbauer wavelets are generated here by dilation and translation of the classical orthogonal Gegenbauer polynomials. The properties of the Gegenbauer wavelets and the Gegenbauer polynomials are first presented. These functions and their associated properties are then employed to derive the Gegenbauer wavelet operational matrices of fractional integrals. The operational matrices of fractional integrals are utilized to reduce the problem to a set of algebraic equations with unknown coefficients. Illustrative examples are provided to demonstrate the validity and applicability of the method presented here.