Method For Solution Of Equations Research Articles

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

A Six-Point y-Function Hybrid Block Method for Direct Solution of Third Order Ordinary Differential Equations

In this article, a new continuous numerical method was developed for the numerical integration of general third-order initial value problems (IVPs) of ordinary differential equations (ODEs). The method was derived by adopting interpolation and collocation approach using a combination of power series and exponential functions as the basis function for the approximate solution to the ODEs. The approximate solution was interpolated at both nodal and off-nodal points while the differential systems from the approximate solution was collocated at all nodal points to obtain the set of methods with given step-numbers. This derived approach ensured that the hybrid points are obtained at the y-values of the methods. This class of hybrid linear multi-step method satisfied the conditions for usability and accuracy of any given method. These conditions were confirmed by testing the consistency, stability and convergence of the developed method. The method was applied to solve directly linear and non-linear third order ODEs without reduction to system of first-order ordinary differential equations. Results from the computation compared favorably with some existing numerical methods in literature with comparable properties to confirm the superiority of the newly developed method.

A Variant of Accelerated Ramadan Group Adomian Decomposition Method for Numerical Solution of Fractional Riccati Differential Equations

A Variant of Accelerated Ramadan Group Adomian Decomposition Method for Numerical Solution of Fractional Riccati Differential Equations

A Comparison of Interpolation and Least Squares Methods for the Approximate Solution of Differential Equation

A Comparison of Interpolation and Least Squares Methods for the Approximate Solution of Differential Equation

Hosoya polynomial method for the numerical solution of Volterra integral equations

AbstractThe numerical solution of Volterra integral equation using one of the graph theoretic polynomial is Hosoya polynomial. To reduce the VIEs to a system of algebraic equations by substituting collocation points. To simplify these system by the help of Matlab using Newton’s iteration technique, we get the Hosoya coefficients and substitute these coefficients in function approximation to get the required solutions as shown in tables and graphically represent the figures. The error analysis demonstrates the accuracy, stability, and consistency. The proposed method shows efficiency and validity as compared to the existing methods.

Higher-order Haar wavelet method for solution of fourth-order integro-differential equations

This paper presents a numerical approach to solve third and fourth order intego-differential equations (IDEs). In order to ascertain the numerical solution for third and fourth order IDEs of second kind, the newly introduced Higher order Haar wavelet method (HOHWM) has been employed to improve the numerical result and rate of convergence compared to classical Haar wavelet approach. Some examples available in the literature have been solved to verify the HOHWM’s effectiveness. To ensure that the approach presented is legitimate, applicable and achieves its objective, the maximum absolute error of each test problem is calculated at a test point.

Approximate solution of multi-term fractional differential equations via a block-by-block method

In this paper we introduce a block-by-block method for the numerical solution of multi-term fractional differential equations (MFDEs). The main idea is to convert a MFDE to a Volterra integral equation of weakly singular type, to which a well known block-by-block method is applied. We also provide the error analysis and convergence of the method. Finally, numerical examples involving Bagley–Torvik and relaxation-oscillation equations are given to confirm applications and the theoretical results.

A Solution Method for Nonlinear Monotone Equations via Hybrid Spectral Conjugate Gradient and Signal Recovery Problems

A Solution Method for Nonlinear Monotone Equations via Hybrid Spectral Conjugate Gradient and Signal Recovery Problems

Review of Adams-Bashforth method for numerical solution of first order ordinary differential equations

This project work presents a comprehensive review of the Adams-Bashforth method for the numerical solution of explicit first-order ordinary differential equations (ODEs). The study begins with a historical overview of the development of ordinary differential equations (ODEs), tracing back to the seminal works of Newton, Leibniz, and their contemporaries. The evolution of differential equations as a distinct mathematical discipline and their wide-ranging applications across various fields are discussed. Furthermore, the paper provides an in-depth analysis of the Adams-Bashforth method, a prominent numerical technique for solving ODEs. The method is derived and analyzed using Mathematica Software, demonstrating its efficacy and accuracy in approximating solutions to ODEs. A comparison with existing methods such as the Euler method and the Runge-Kutta method highlights the advantages of the Adams-Bashforth method, particularly in terms of computational efficiency and accuracy. Moreover, fundamental concepts related to ODEs, including the initial value problem, existence, and uniqueness of solutions, are explored in the context of numerical solution methods. The properties of linear multistep methods and their relevance to solving ODEs efficiently are also discussed. In conclusion, this project work provides valuable insights into the Adams-Bashforth method as a powerful tool for the numerical solution of first-order ODEs. By leveraging modern computational tools such as Mathematica Software, the method offers a practical and efficient approach to solving complex differential equations encountered in various scientific and engineering applications.

An Improvement Meshless Method for the Numerical Solution of Two-Dimensional Stochastic Fredholm Integral Equations

An Improvement Meshless Method for the Numerical Solution of Two-Dimensional Stochastic Fredholm Integral Equations

A multistep collocation method for approximate solution of Volterra integro-differential equations of the third kind

A multistep collocation method for approximate solution of Volterra integro-differential equations of the third kind

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step procedure, defined over a graded mesh, which is based on a truncated expansion of the vector field along the orthonormal Jacobi polynomial basis. Under mild hypotheses, the proposed procedure is capable of getting spectral accuracy. A few numerical examples are reported to confirm the theoretical findings.

FİNİTE ELEMENT ANALYSİS OF THE OUTER TUBE PART OF THE ENDOSCOPE

Even after cleaning and disinfecting the endoscope, the possibility of contamination is high and can adversely affect the health of patients. The US Centers for Disease Control report that there are approximately two million infected victims each year, and approximately 20,000 people succumb to the infection. In recent news, several deaths at UCLA Ronald Reagan Medical Center have been linked to bacterial contamination of endoscopes. Hence, there is a need to design an endoscope that will reduce or eliminate the chances of these deadly bacteria entering the body. In this article, we will look at the finite element analysis of the outer tube part of a newly designed disposable endoscope. The finite element method (SEM) is a popular method for the numerical solution of differential equations arising in engineering and mathematical modeling. Traditional areas of structural analysis include typical problem areas related to heat transfer, fluid flow, mass transport, and electromagnetic potential. Keywords: endoscope, outer tube, finite element method, mathematical modeling, analysis.

Two effective methods for solution of the Gardner–Kawahara equation arising in wave propagation

In this paper, we deal with the analytical and numerical solutions of the Gardner–Kawahara (G-K) equation, known as the extended Korteweg–de Vries equation, which describes solitary-wave propagation in media and occurs in the notion in plasmas and in notion of shallow water waves with surface tension and notion of magneto-acoustic waves. This study is split into two primary parts: in the first part, (G′/G) and exp (−ϕ(ξ)) expansion methods are employed to obtain some novel exact solutions of the G-K equation. Furthermore, some hypotheses are used to get different exact traveling wave solutions (including Hyperbolic,Trigonometric and Rational solutions). (G′/G)-expansion method is a comprehensible and noncomplex technique used to procure traveling wave solutions of partial differential equations (PDEs). By using the (G′/G)-expansion technique, solution of PDEs can be indicated as a series and the coefficients of the series can be procured by solving a set of algebraic equations. This method is highly effective, requiring only the solution of a few algebraic equations to create the traveling wave solution (Ali and Tag-eldin, 2023 [3]). The exp (−ϕ(ξ))- expansion method is used as the first time to examine the wave solution of a nonlinear dynamical system in a new double-Chain model of DNA and a diffusive predator–prey system. The method also has been used for many other nonlinear evolution equations (Abdelrahman et al., 2015 [1]). In the second part, finite element method is dedicated to deriving numerical solutions of the equation. In the same part, based on septic B-spline approximation, a collocation method has been offered and applied for numerical solutions of G-K equation conceiving different parameter values of test problem. To discretize time and space derivatives, forward finite difference and Crank–Nicolson approximations have been used, respectively. Unconditional stability is proved by using Von-Neumann method. B-spline functions have some reliable properties like smoothness, local support and ability of handling local phenomena, which make them proper to solve linear and nonlinear partial differential equations easily and precisely. Collocation technique has two great superiorities: constituting method does not involve integrations and the resulting matrix system is banded with small band width. For this reason, B-splines when associate with the collocation technique ensure a simple solution process of linear and nonlinear PDEs. An example is successfully solved by calculating the error norms L2 and L∞ for illustrating the proficiency and reliability of the method and highlighted the significance of this work. Also to reflect the efficiency of this method for solving the nonlinear equation, the results are depicted both graphically and in tabular form. Calculated results are found to be in good agreement with the exact solutions and other techniques given in the literature.

Genocchi collocation method for accurate solution of nonlinear fractional differential equations with error analysis

In this study, we introduce an innovative fractional Genocchi collocation method for solving nonlinear fractional differential equations, which have significant applications in science and engineering. The fractional derivative is defined in the Caputo sense and by leveraging fractional-order Genocchi polynomials, we transform the nonlinear problem into a system of nonlinear algebraic equations. A novel technique is employed to solve this system, enabling the determination of unknown coefficients and ultimately the solution. We derive the error bound for our proposed method and validate its efficacy through several test problems. Our results demonstrate superior accuracy compared to existing techniques in the literature, suggesting the potential for extending this approach to tackle more complex problems of critical physical significance.

Simple pendulum: period dependent on amplitude of oscillation

In this study, with a pedagogical aim suited for the undergraduate, the differential equations of motion that characterize and determine the motion of a simple pendulum were obtained considering small and large amplitudes of oscillation. These differential equations were solved through differential equation solution methods, numerical, expansion of functions and integrations. The solutions obtained using the different methods were compared. It was possible to verify, both experimentally and theoretically, that for the oscillatory movement of the simple pendulum, its oscillation period increases and its angular frequency decreases with the increase of the oscillation amplitude. The validity range of the approximation for small ranges of motion was also determined. It was verified that the theoretical and experimental results present a good agreement for angles smaller than 55°. The experimental measurements were made with “a low-cost home-built” equipment.

Gauss-Legendre type quadrature iterative method for fuzzy Fredholm integral equations

In this paper we obtain an iterative method for the numerical solution of second kind fuzzy Fredholm integral equations by applying appropriate composite fuzzy quadrature formula. We approach a general fuzzy quadrature formula and show that the three-point and four-point Gauss-Legendre type fuzzy quadrature formulas are the best choice from computational cost and accuracy point of view. The convergence of the method is proved by providing the error estimates expressed in terms of the Lipschitz constant of the Picard iterations. Some numerical results confirm the obtained theoretical result and illustrate the performances of the three point Gauss-Legendre method in comparison with the techniques generated by the fuzzy Newton-Cotes and Gauss-Lobatto quadrature formulas, the results being tested on the Simpson and on the four point Gauss-Legendre and Gauss-Lobatto quadrature formulas in particular. The numerical experiment suggests that five point or more point fuzzy quadrature formulas are useless due to the accumulation of errors and the grown computational cost.

A Modified Residual Power Series Method for the Approximate Solution of Two-Dimensional Fractional Helmholtz Equations

In this paper, we suggest a modification for the residual power series method that is used to solve fractional-order Helmholtz equations, which is called the Shehu-transform residual power series method (ST-RPSM). This scheme uses a combination of the Shehu transform (ST) and the residual power series method (RPSM). The fractional derivatives are taken with respect to Caputo order. The novelty of this approach is that it does not restrict the fractional order and reduces the need for heavy computational work. The results were obtained using an iterative series that led to an exact solution. The 3D graphical plots for different values of fractional orders are shown to compare ST-RPSM results with exact solutions.

Analysis of lateral dynamic response characteristics of drilling riser during installation of deep-water Christmas tree

In this research, in order to investigate the safety risk of drilling riser during deep-water Christmas tree lowering process and determine the basic parameters for the safety calculation of the upper part of the Christmas tree, wave-current force calculation method is explained based on modified Morison equation. According to differential element method, the dynamic equations of vertical and horizontal bending of riser under wave and current conditions are derived and equation solution method and calculation stability conditions are analyzed. By combining the upper and lower boundaries of riser during Christmas tree connection and environmental load factors, dynamic responses of riser under different wave, top tension, platform drift, operating water depth and other parameters are investigated. The obtained results show that maximum lateral displacement, corresponding water depth and bending moment of riser as well as the amplitude and period of displacement fluctuation at riser bottom are increased with the increase of wave velocity and water depth. It is also found that increase of top tension reduced the maximum lateral displacement of riser and increased corresponding water depth, but excessive tension increased the fluctuations of displacement and bending moment at riser bottom. Increase of platform drift increased maximum lateral displacement fluctuation range, bottom string displacement and bending moment of the riser and decreased corresponding maximum displacement water depth. Drift is found to increase the fluctuations of dynamic parameters at riser bottom. Increase of operating water depth increased riser maximum lateral displacement and decreased the period and amplitude of bottom bending moment fluctuations; also, maximum water depth position approached to "1/3" position on sea surface.

Sparse adaptive basis set methods for solution of the time dependent Schrodinger equation

Scalable numerical solutions to the time dependent Schrodinger equation remain an outstanding goal in theoretical chemistry. Here we present a method which utilises recent breakthroughs in signal processing to consistently adapt a dictionary of basis functions to the dynamics of the system. We show that for two low-dimensional model problems the size of the basis set does not grow quickly with time and appears only weakly dependent on dimensionality. The generality of this finding remains to be seen. The method primarily uses energies and gradients of the potential, opening the possibility for its use in on-the-fly ab initio quantum wavepacket dynamics.