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In this contribution, the solution of Mass-Spring-Damper Systems in the fractional context by using Caputo derivative and Orthogonal Collocation Method is investigated. For this purpose, different case studies considering constant and periodic sources are evaluated. The dimensional consistency of the model is guaranteed by introducing an auxiliary parameter. The obtained results are compared with those found by using both the analytical solution and the predictor-corrector method of Adams–Bashforth–Moulton type. The influence of the fractional order on the mechanical system is evaluated.

In the present contribution, an extension of the Orthogonal Collocation Method to solve fractional differential equations is proposed.

In general, the proposed methodology was able to solve a classical mechanical engineering problem with different characteristics.

The development of a new numerical method to solve fractional differential equations is the major contribution.

The Hermite collocation method of discretization can be used to determine highly accurate solutions to the steady‐state one‐dimensional convection‐diffusion equation (which can be used to model the transport of contaminants dissolved in groundwater). This accuracy is dependent upon sufficient refinement of the finite‐element mesh as well as applying upstream or downstream weighting to the convective term through the determination of collocation locations which meet specified constraints. Owing to an increase in computational intensity of the application of the method of collocation associated with increases in the mesh refinement, minimal mesh refinement is sought. Very often this optimization problem is the one where the feasible region is not connected and as such requires a specialized optimization search technique. This paper aims to focus on this method.

An original hybrid method that utilizes a specialized adaptive genetic algorithm followed by a hill‐climbing approach is used to search for the optimal mesh refinement for a number of models differentiated by their velocity fields. The adaptive genetic algorithm is used to determine a mesh refinement that is close to a locally optimal mesh refinement. Following the adaptive genetic algorithm, a hill‐climbing approach is used to determine a local optimal feasible mesh refinement.

In all cases the optimal mesh refinements determined with this hybrid method are equally optimal to, or a significant improvement over, mesh refinements determined through direct search methods.

Further extensions of this work could include the application of the mesh refinement technique presented in this paper to non‐steady‐state problems with time‐dependent coefficients with multi‐dimensional velocity fields.

The present work applies an original hybrid optimization technique to obtain highly accurate solutions using the method of Hermite collocation with minimal mesh refinement.

The purpose of this paper is to present the analytical solution to the Hermite collocation discretization of a quadratically forced steady‐state convection‐diffusion equation in one spatial dimension with constant coefficients, defined on a uniform mesh, with Dirichlet boundary conditions. To improve the accuracy of the method “upstream weighting” of the convective term is used in an optimal way. The authors also provide a method to determine where the forcing function should be optimally sampled. Computational examples are given, which support and illustrate the theory of the optimal sampling of the convective and forcing term.

The authors: extend previously published results (which dealt only with the case of linear forcing) to the case of quadratic forcing; prove the theorem that governs the quadratic case; and then illustrate the results of the theorem using computational examples.

The algorithm developed for the quadratic case dramatically decreases the error (i.e. the difference between the continuous and numerical solutions).

Because the methodology successfully extends the linear case to the quadratic case, it is hoped that the method can, indeed, be extended further to more general cases. It is true, however, that the level of complexity rose significantly from the linear case to the quadratic case.

Hermite collocation can be used in an optimal way to solve differential equations, especially convection‐diffusion equations.

Since convection‐dominated convection‐diffusion equations are difficult to solve numerically, the results in this paper make a valuable contribution to research in this field.

This paper aims to present a new method for the approximate solution of two-dimensional nonlinear Volterra–Fredholm partial integro-differential equations with boundary conditions using two-dimensional Chebyshev wavelets.

For this purpose, an operational matrix of product and integration of the cross-product and differentiation are introduced that essentially of Chebyshev wavelets. The use of these operational matrices simplifies considerably the structure of the computation used for a set of the algebraic system has been obtained by using the collocation points and solved.

Theorem for convergence analysis and some illustrative examples of using the presented method to show the validity, efficiency, high accuracy and applicability of the proposed technique. Some figures are plotted to demonstrate the error analysis of the proposed scheme.

This paper uses operational matrices of two-dimensional Chebyshev wavelets and helps to obtain high accuracy of the method.

This paper aims to solve the Falkner–Skan problem over an isothermal moving wedge using the combination of the quasilinearization method and the fractional order of rational Chebyshev function (FRC) collocation method on a semi-infinite domain.

The quasilinearization method converts the equation into a sequence of linear equations, and then by using the FRC collocation method, these linear equations are solved. The governing nonlinear partial differential equations are reduced to the nonlinear ordinary differential equation by similarity transformations.

The entropy generation and the effects of the various parameters of the problem are investigated, and various graphs for them are plotted.

Very good approximation solutions to the system of equations in the problem are obtained, and the convergence of numerical results is shown by using plots and tables.

The study of heat transfer mechanisms is an area of great interest because of various applications that can be developed. Mathematically, these phenomena are usually represented by partial differential equations associated with initial and boundary conditions. In general, the resolution of these problems requires using numerical techniques through discretization of boundary and internal points of the domain considered, implying a high computational cost. As an alternative to reducing computational costs, various approaches based on meshless (or meshfree) methods have been evaluated in the literature. In this contribution, the purpose of this paper is to formulate and solve direct and inverse problems applied to Laplace’s equation (steady state and bi-dimensional) considering different geometries and regularization techniques. For this purpose, the method of fundamental solutions is associated to Tikhonov regularization or the singular value decomposition method for solving the direct problem and the differential Evolution algorithm is considered as an optimization tool for solving the inverse problem. From the obtained results, it was observed that using a regularization technique is very important for obtaining a reliable solution. Concerning the inverse problem, it was concluded that the results obtained by the proposed methodology were considered satisfactory, as even with different levels of noise, good estimates for design variables in proposed inverse problems were obtained.

In this contribution, the method of fundamental solution is used to solve inverse problems considering the Laplace equation.

In general, the proposed methodology was able to solve inverse problems considering different geometries.

The association between the differential evolution algorithm and the method of fundamental solutions is the major contribution.

The purpose of this paper is to demonstrate the eligibility of the Weighted Residual Methods (WRMs) applied to magneto hydro dynamic (MHD) nanofluid flow in divergent and convergent channels. Selecting the most appropriate method among the WRMs and discussing about Jeffery-Hamel flow's treatment in divergent and convergent channels are the other important purposes of the present research.

Three analytical methods (Collocation, Galerkin and Least Square Method) and numerical method have been applied to solve the governing equations. The reliability of the methods is also approved by a comparison made between the fourth-order Runge-Kutta numerical method.

The obtained solutions revealed that WRMs can be simple, powerful and efficient techniques for finding analytical solutions in science and engineering non-linear differential equations.

It could be considered as a first endeavor to use the solution of the MHD nanofluid flow in divergent and convergent channels using these kinds of analytical methods along with the numerical approach.

The purpose of this paper is to demonstrate the eligibility of the weighted residual methods (WRMs) applied to Jeffery-Hamel Flow. Selecting the most appropriate method among the WRMs and discussing about Jeffery-Hamel flow's treatment in divergent and convergent channels are the other important purposes of the present research.

Three analytical methods (collocation, Galerkin and least square method) have been applied to solve the governing equations. The reliability of the methods is also approved by a comparison made between the forth order Runge-Kutta numerical method.

The obtained solutions revealed that WRMs can be simple, powerful and efficient techniques for finding analytical solutions in science and engineering non-linear differential equations.

It could be considered as a first endeavor to use the solution of the Jeffery-Hamel flow using these kind of analytical methods along with the numerical approach.

The purpose of the paper is to study the variation of optimal burnout angle at the end of the ascent phase and the optimal control deflection during the glide phase, that would maximize the downrange performance of a hypersonic boost-glide waverider, with variation in heat rate and integrated heat load limit.

The approach used is to model the boost phase so as to optimize the burnout conditions. The nonlinear, multiphase, constraint optimal control problem is solved using an hp-adaptive pseudospectral method.

The constraint heat load results for the waverider configuration reveal that the integrated heat load can be reduced by more than half with only 10 per cent penalty in the overall downrange of the hypersonic boost-glide vehicle, within a burnout speed range of 3.7 to 4.3 km/s. The angle-of-attack trim control requirements increase with stringent heat rate and integrated heat load bounds. The normal acceleration remains within limits.

The trajectory results imply lower thermal protection system weight because of reduced heat load trajectory profile and therefore lower thermal protection system cost.

The research provides further study on the trajectory design to the hypersonic boost-glide vehicles for medium range application.

The purpose of this paper is to develop rationalized Haar functions to approximate the solutions of the integro‐differential equations.

Properties of rationalized Haar functions are first presented, and the operational matrix of the product of two rationalized Haar functions vector is utilized to reduce the computation of integro‐differential equations to some algebraic equations.

Numerical results support the theoretical results.

Presents a method for solving integro‐differential equations.