Chebyshev Wavelet Method Research Articles

Numerical study of hybrid nanofluid and thermal transport in sun-powered energy ship within the application of parabolic trough solar collectors

PurposeRecent advancements in technology have led to the exploration of solar-based thermal radiation and nanotechnology in the field of fluid dynamics. Solar energy is captured through sunlight absorption, acting as the primary source of heat. Various solar technologies, such as solar water heating and photovoltaic cells, rely on solar energy for heat generation. This study focuses on investigating heat transfer mechanisms by utilizing a hybrid nanofluid within a parabolic trough solar collector (PTSC) to advance research in solar ship technology. The model incorporates multiple effects that are detailed in the formulation.Design/methodology/approachThe mathematical model is transformed using suitable similarity transformations into a system of higher-order nonlinear differential equations. The model was solved by implementing a numerical procedure based on the Wavelets and Chebyshev wavelet method for simulating the outcome.FindingsThe velocity profile is reduced by Deborah's number and velocity slip parameter. The Ag-EG nanoparticles mixture demonstrates less smooth fluid flow compared to the significantly smoother fluid flow of the Ag-Fe3O4/EG hybrid nanofluids (HNFs). Additionally, the Ag-Ethylene Glycol nanofluids (NFs) exhibit higher radiative performance compared to the Ag-Fe3O4/Ethylene Glycol hybrid nanofluids (HNFs).Practical implicationsAdditionally, the Oldroyd-B hybrid nanofluid demonstrates improved thermal conductivity compared to traditional fluids, making it suitable for use in cooling systems and energy applications in the maritime industry.Originality/valueThe originality of the study lies in the exploration of the thermal transport enhancement in sun-powered energy ships through the incorporation of silver-magnetite hybrid nanoparticles within the heat transfer fluid circulating in parabolic trough solar collectors. This particular aspect has not been thoroughly researched previously. The findings have been validated and provide a highly positive comparison with the research papers.

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A numerical solution for a quasi solution of the time-fractional stochastic backward parabolic equation

In the present study, we consider a time-fractional stochastic backward parabolic equation driven by standard Brownian motion. In this problem, the fractional derivative is considered in the Caputo sense. Using the minimization of a least-squares functional, stochastic variational formulation, Fréchet differentiability and utility theorems adopted directly from deterministic fractional backward equations, the existence and uniqueness theorems for a quasi solution of the proposed problem are proved. To approximate the quasi solution, a numerical technique based on 2D Chebyshev wavelets is applied. We employ the Levenberg–Marquardt regularization technique since the derived equivalent system of linear equations is ill-posed. Also, the convergence analysis for this numerical algorithm is investigated. Our results provide a new insight to find quasi solutions and apply adapted deterministic methods for some fractional stochastic backward equations. Moreover, a numerical example is provided to indicate the accuracy and efficiency of the Chebyshev wavelet method in solving the mentioned problem.

Comparative work for the source identification in parabolic inverse problem based on Taylor and Chebyshev wavelet methods

In this article, we study wavelet collocation methods based on Taylor and Chebyshev wavelets for source identification in the parabolic inverse problem. In the proposed method, the highest order derivative is written in terms of the Taylor and Chebyshev wavelet series, and required unknown terms are obtained using successive integration. Taylor series approximation has been utilized to obtain the source control parameter. Convergence analysis is carried out in order to guarantee the accuracy of the method. Numerical results have been obtained based on the proposed methods, and it is shown that the Taylor wavelet method provides us with a better result than the Chebyshev wavelet method. CPU time has also been shown to ensure the efficiency of the method.

Improvement of mechanical energy using thermal efficiency of hybrid nanofluid on solar aircraft wings: an application of renewable, sustainable energy

ABSTRACT The modern world uses sun-based thermal radiation and nanotechnology to promote new technologies. In addition to solar thermal aircraft, photovoltaic cells, and sun-based hybrid nanofluids, solar energy is the primary source of heat derived from the absorption of sunlight. Researchers are currently investigating the application of nanotechnology to sun-based thermal radiation with the intent of enhancing the efficiency of aircraft flight. This study is focused on the research of heat transport via employing hybrid nanofluid on the interior of solar wings using a parabolic trough solar collector (PTSC) to enrich the investigations of the solar aircraft wing. In addition, the thermodynamics of entropy production for steady tangent hyperbolic fluid is examined for its energy balance and usefulness for significant physical influenced parameters. This study examines the viscoelastic properties of the thermal transfer process of two distinct kinds of nano solid particles, zirconium dioxide and copper (Cu), with non-Newtonian EG (ethylene glycol) as the base fluid. To achieve the model's aims, a novel solution, namely wavelets and the Chebyshev wavelets method (CWM), was employed to solve the velocity, energy equation, and entropy generation. .

Numerical Solutions of Inverse Nodal Problems for a Boundary Value Problem

In this paper, we study inverse nodal problems for a boundary value problem. A uniqueness result for the potential function and a reconstruction method are obtained. By using the nodal points as input data, we compute the approximation solution of the potential function for the boundary value problem by the first kind Chebyshev wavelet method. Two numerical examples show that the first kind Chebyshev wavelet method for solving the inverse nodal problems for the boundary value problem is valid.

Solving an inverse nodal problem with Herglotz–Nevanlinna functions in boundary conditions using the second‐kind Chebyshev wavelets method

This study considers Sturm–Liouville (S–L) equation with a special function which includes spectral parameter at boundary conditions and expresses asymptotic forms of inverse problem solution, nodal points, and lengths. Furthermore, it proves some uniqueness theorems. Doing this, approximate solution of inverse S–L problem is obtained by second‐kind Chebyshev polynomials whereby the second‐kind Chebyshev wavelets (SCW) method is used to solve these types of problems. Finally, effectiveness of the method is demonstrated in a few examples.

Entropy generation and mixed convection of CuO–water near an oblique stagnation point: modified Chebyshev wavelets approach

In this article, an analysis of CuO–water nanofluid over a moving wall with oblique stagnation point has been studied under the influence of a heat source. CuO nanoparticles having shapes of platelet, brick and spheres were used in this study. The aim of this study is to explore the impact of the shapes of nanoparticles on the flow properties near an oblique stagnation point flow. The flow system was governed by a set of nonlinear partial differential equations (PDEs). These PDEs were converted to nonlinear ordinary differential equations, which were further solved numerically using a modified Chebyshev wavelet method. The effects of parameters appeared in analysis on skin friction, local Nusselt number, velocity, temperature and entropy are plotted graphically. Streamlines are also plotted to analyze patterns of flow for different levels of stretching and different angles of strike. Tabular results are generated to specify numerical values of free parameter A, components of skin friction, local Nusselt number and stagnation point. It is found that platelets shaped nanoparticles show more efficient results for inertial and thermal boundary layer. Entropy of system is influenced by stretching, angle of intact and mixed convection parameter.

Analysis of the Fractional-Order Delay Differential Equations by the Numerical Method

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. The fractional derivative is defined in Caputo manner. The proposed method is simple, effective, and straightforward as compared to other numerical techniques. To check the validity and accuracy of the proposed method, some illustrative examples are solved by using the present scenario. The obtained results have confirmed the greater accuracy than the modified Laguerre wavelet method, the Chebyshev wavelet method, and the modified wavelet-based algorithm. Moreover, based on the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations.

An application of Chebyshev wavelet method for the nonlinear time fractional Schrödinger equation

In the present manuscript, we will deal with time fractional Schrödinger equation having appropriate initial and boundary conditions with Chebyshev wavelet method numerically. The Chebyshev wavelet method will be utilized successfully for two test problems. In order to find out efficiency and accuracy of this method, the widely used error norms L2 and L∞ of the newly found results have been compared with some of the other approximate results in the literature. The results have been given in tables and figures to show the compatibility between the new results and those in other articles.

Higher prices for specialty carbon blacks plus surcharges for European products

In this paper, we have applied an efficient shifted second kind Chebyshev wavelet method (S2KCWM) to vibrating dynamical models arising in mechanical systems such as vibration of circular membrane, damped spring mass system and ship oscillatory motions. To the best of our knowledge, until now there is no rigorous wavelet based solution has been reported for the vibrating dynamical models. The power of the manageable method is confirmed. The wavelet solutions are compared with numerical simulations by MATLAB. Good agreement between the solutions is presented in this paper. Some numerical examples are given to demonstrate the validity and applicability of the proposed method. Moreover the use of Chebyshev wavelets is found to be simple, efficient, flexible, convenient, less computation costs and computationally attractive.

Numerical solutions of inverse time fractional coupled Burgers’ equations by the Chebyshev wavelet method

Numerical solutions of inverse time fractional coupled Burgers’ equations by the Chebyshev wavelet method

Investigation of shape effects of Cu-nanoparticle on heat transfer of MHD rotating flow over nonlinear stretching sheet

In the current study, we focused on the shape effects of copper (Cu) nano-particles on heat transmission of three-dimensional magnetohydrodynamic (MHD) nano-fluid. A particular type of flow is considered, i.e., rotating flow over an exponentially stretching sheet. Significant actions of thermal radiation and Grashof number are also formulated to study. The modified Chebyshev wavelets method is introduced to examine the numerical solutions of the accomplished model. Error analysis and further comparison with existing results reflect the appropriateness of the proposed modification. Graphical behavior of dimensionless velocities, temperature, Nusselt number, and skin friction under the inspiration of several parameters is also presented. The attained solutions propose that the modification is beneficial and can be protracted to other highly nonlinear problems. The modified Chebyshev wavelets technique lowers computing time, according to the study. The research takes into account the form impacts of nano-particles. There are three kinds of nano-particles to examine (spherical, laminar, and cylinder). The behavior of various particles varies depending on the flow and temperature profile. For microparticles with a laminar structure, velocity and temperature have higher values.

Fractional‐order Chebyshev wavelet method for variable‐order fractional optimal control problems

A new alternative numerical method for solving variable‐order fractional optimal control problems (VO‐FOCPs) is introduced. We use fractional‐order Chebyshev wavelets for solving these VO‐FOCPs. By applying the regularized beta functions, an exact value for the fractional integration of the given wavelets is provided. By applying this formula and the given wavelets, we reduce the given VO‐FOCP to a system of algebraic equations which can be solved by using known methods. For the function approximation of our method, we also give the convergence analysis. By using several numerical examples, we show that this method is more accurate than the other existing methods in the literature.

Fourth kind Chebyshev Wavelet Method for the solution of multi-term variable order fractional differential equations

PurposeMulti-term variable-order fractional differential equations (VO-FDEs) are powerful tools in accurate modeling of transient-regime real-life problems such as diffusion phenomena and nonlinear viscoelasticity. In this paper the Chebyshev polynomials of the fourth kind is employed to obtain a numerical solution for those multi-term VO-FDEs.Design/methodology/approachTo this end, operational matrices for the approximation of the VO-FDEs are obtained using the Fourth kind Chebyshev Wavelets (FKCW). Thus, the VO-FDE is condensed into an algebraic equation system. The solution of the system of those equations yields a coefficient vector, the coefficient vector in turn yields the approximate solution.FindingsSeveral examples that we present at the end of the paper emphasize the efficacy and preciseness of the proposed method.Originality/valueThe value of the paper stems from the exploitation of FKCWs for the numerical solution of multi-term VO-FDEs. The method produces accurate results even for relatively small collocation points. What is more, FKCW method provides a compact mapping between multi-term VO-FDEs and a system of algebraic equations given in vector-matrix form.

Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

Solving fractional optimal control problems with inequality constraints by a new kind of Chebyshev wavelets method

In this paper, a new orthonormal wavelets based on the sixth-kind Chebyshev polynomials is constructed to obtain the solution of fractional optimal control problems (FOCPs) with inequality constraints. The convergence analysis and error estimate of the sixth-kind Chebyshev wavelets function expansion are investigated. By using the relationship between the second-kind and sixth-kind Chebyshev polynomials, the exact formula of Riemann-Liouville fractional integral operator of Chebyshev wavelet is derived. For solving FOCPs, positive slack functions are added to inequality conditions, and then the problem is simplified to the problem of solving algebraic equations by fractional integral operational matrix and collocation method. The applicability and validity of the proposed method are verified by several examples and the results are compared with those reported in literature.

Second Chebyshev wavelets (SCWs) method for solving finite-time fractional linear quadratic optimal control problems

In this paper, we present an indirect computational procedure based on the truncated second kind Chebyshev wavelets for finding the solutions of Caputo fractional time-invariant linear optimal control systems which the functional cost consists of the finite-time quadratic cost function. Utilizing the operational matrices of second Chebyshev wavelets (SCWs) of the Riemann–Liouville fractional integral, the corresponding linear two-point boundary value problem (TPBVP), obtained from the fractional Euler–Lagrange equations, is reduced to a coupled Sylvester-type matrix equation. An equivalent linear matrix form using the Kronecker product is constructed. The upper bound of the error of the SCWs approximation and the convergence of the proposed method are investigated. Low computational complexity and flexible accuracy are two important superiorities of this approach. Numerical experiments provide satisfactory results compared to the exiting techniques.

New reproducing kernel Chebyshev wavelets method for solving a fractional telegraph equation

In this paper, a new reproducing kernel Chebyshev wavelets method of solving a fractional telegraph equation is proposed. For solving the equation, reproducing kernel Chebyshev wavelets bases is constructed based on Chebyshev polynomials with a parameter. We choose an improved differential quadrature method with fourth-order truncation error to approximate second-order derivative term of the equation. Subsequently, the fractional telegraph equation is transformed into integral equation and the best approximate solution is obtained by searching the minimum of $$\varepsilon $$ -approximate solutions. It is satisfied that the accuracy of errors provided by examples is very high.

Chebyshev wavelet approximation of functions having first derivative of H$$\ddot{\text {o}}$$lder’s class

In this paper, the Chebyshev wavelet approximations of a uniformly continuous function f and a functions f whose first derivative $$f'$$ of class $$H^\alpha [0,1), 0 < \alpha \le 1$$ , have been determined. These wavelet estimators are sharper, better and best possible in Wavelet analysis. A new method for solving differential equations by the Chebyshev wavelet method has been proposed. Lane-Emden and third-order pantograph non-linear differential equations have been solved by this method. The solutions of these equations have been compared by their exact solution. It is found that the exact solutions and solutions by the Chebyshev Wavelet method are nearly same. This is a significant achievement in wavelet analysis.