•Two new algorithms based on DQM have been developed to the solution of BVPs.•Some error bounds have been obtained for the first and second order differentiation.•By defining special end conditions ...for cubic spline, the optimal convergence obtained.•Well-known problems like 2D and 3D coupled Burgers’ equations have been solved.•The methods have been compared with many other existing methods.
A new differential quadrature method based on cubic B-spline is developed for the numerical solution of differential equations. In order to develop the new approach, the B-spline basis functions are used on the grid and midpoints of a uniform partition. Some error bounds are obtained by help of cubic spline collocation, which show that the method in its classic form is second order convergent. In order to derive higher accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. A new fourth order method is developed for the numerical solution of systems of second order ordinary differential equations. By solving some test problems, the performance of the proposed methods is examined. Also the implementation of the method for multi-dimensional time dependent partial differential equations is presented. The stability of the proposed methods is examined via matrix analysis. To demonstrate the applicability of the algorithms, we solve the 2D and 3D coupled Burgers’ equations and 2D sine-Gordon equation as test problems. Also the coefficient matrix of the methods for multi-dimensional problems is described to analyze the stability.
Scanning electron micrographs of (a) ash and (b) nFe-A adsorbents.
The main goal of this study was to evaluate the performance of two new adsorbents, ash and Fe nanoparticles loaded ash (nFe-A) for ...the removal of Pb(II) from aqueous solution. Batch experiments were performed as a function of various experimental parameters. The adsorption studies included both equilibrium adsorption isotherms and kinetics. Equilibrium data fitted very well with the Langmuir isotherm model. Kinetics studies showed better applicability for pseudo-second-order model for both adsorbents. The regeneration studies showed promising regeneration potential of these adsorbents. The breakthrough capacities of ash and nFe-A for Pb(II) removal was found 25 and 30mgg−1, respectively.
•We examine GPDEA models in relation to the MCDEA framework.•GPDEA models perform poorly in weight dispersion and discrimination power.•The proposed BiO-MCDEA model outperforms GPDEA models in all ...datasets tested.
Lack of discrimination power and poor weight dispersion remain major issues in Data Envelopment Analysis (DEA). Since the initial multiple criteria DEA (MCDEA) model developed in the late 1990s, only goal programming approaches; that is, the GPDEA-CCR and GPDEA-BCC were introduced for solving the said problems in a multi-objective framework. We found GPDEA models to be invalid and demonstrate that our proposed bi-objective multiple criteria DEA (BiO-MCDEA) outperforms the GPDEA models in the aspects of discrimination power and weight dispersion, as well as requiring less computational codes. An application of energy dependency among 25 European Union member countries is further used to describe the efficacy of our approach.
The idea of differential quadrature is used to construct a new algorithm for the solution of differential equations. To determine the weighting coefficients of DQM, B-spline basis functions of degree ...r are used as test functions. The method is constructed on a set of points mixed from grid points and mid points of a uniform partition. Using the definition of B-splines interpolation as alternative, some error bounds are obtained for DQM. The method is successfully implemented on nonlinear boundary value problems of order m. Also the application of the proposed method to approximate the solution of multi-dimensional elliptic PDEs is included in the paper. As test problem, some examples of biharmonic and Poisson equations are solved in 2D and 3D. The results are compared with some existing methods to show the efficiency and performance of the proposed algorithm. Also some examples of time dependent PDEs are solved to compare the results with other existing spline based DQ methods.
Using the idea of differential quadrature, two new methods are constructed to approximate the solution of elliptic partial differential equations in higher dimensions. To obtain the weighting ...coefficients of the first method, a mixture of grid points and mid points of the uniform partition is used. The order of convergence obtained by the first algorithm is non-optimal, so we mixed the idea with spline collocation to obtain higher order approximations. Using the weighting coefficients of the non-optimal algorithm, some new weighting coefficients are obtained which are used to obtain higher accuracy. For the first time, a sixth order approximation is obtained to the solution of well-known high order multi-dimensional elliptic PDEs such as biharmonic and triharmonic problems. We found a way to increase the order of convergence and the accuracy without increasing the CPU runtime. The block form of the coefficients matrix for the linear case is presented to have a better vision on the system arised from the method. Some examples of biharmonic and triharmonic problems are solved to show the good performance and applicability of the proposed algorithms.
•We examine DEA models with asymmetric inputs-outputs in the energy context.•The model accounts for both crisp and fuzzy efficiency measures across α-levels.•The model handles undesirable outputs ...without producing overly optimistic results.•The computation of the proposed model requires fewer procedures.
Data Envelopment Analysis (DEA) is a powerful analytical technique for measuring the relative efficiency of alternatives based on their inputs and outputs. The alternatives can be in the form of countries who attempt to enhance their productivity and environmental efficiencies concurrently. However, when desirable outputs such as productivity increases, undesirable outputs increase as well (e.g. carbon emissions), thus making the performance evaluation questionable. In addition, traditional environmental efficiency has been typically measured by crisp input and output (desirable and undesirable). However, the input and output data, such as CO2 emissions, in real-world evaluation problems are often imprecise or ambiguous. This paper proposes a DEA-based framework where the input and output data are characterized by symmetrical and asymmetrical fuzzy numbers. The proposed method allows the environmental evaluation to be assessed at different levels of certainty. The validity of the proposed model has been tested and its usefulness is illustrated using two numerical examples. An application of energy efficiency among 23 European Union (EU) member countries is further presented to show the applicability and efficacy of the proposed approach under asymmetric fuzzy numbers.
Fig sawdust was used as a precursor for the production of activated carbon by chemical activation with H3PO4. The developed Fig sawdust activated carbon (FSAC) was used as a biosorbent for the ...removal of Pb(II) from aqueous solution. Highest adsorption of Pb(II) (95.8%) was found at pH 4. Equilibrium data fitted very well with the Langmuir isotherm model. Maximum adsorption capacity was determined 80.645mgg−1 at pH 4. Kinetic studies demonstrated that the adsorption followed a pseudo second order kinetics model. The negative value of ΔG° confirmed the feasibility and spontaneity of FSAC for Pb(II) adsorption.
Two numerical procedures are developed to approximate the solution of one-dimensional parabolic equations using extrapolated collocation method. By defining two different end conditions and forcing ...cubic spline to satisfy the interpolation conditions along with one of the end conditions, we obtain fourth- (CBS4) and sixth-order (CBS6) approximations to the solution in spatial direction. Also in time direction, a weighted finite difference discretization is used to approximate the solution at each time level. The convergence analysis is discussed in detail and some error bounds are obtained theoretically. Finally, some different examples of Burgers’ equation with applications in fluid mechanics as well as convection–diffusion problems with applications in transport are solved to show the applicability and good performance of the procedures.
Measurements of spinal segment ranges of motion (RoMs), movement coordination, and three-dimensional kinematics during occupational activities have implications in occupational/clinical biomechanics. ...Due to the large amount of adipose tissues, obese individuals may have different RoMs, lumbopelvic coordination, and kinematics than normal-weight ones. We aimed to measure/compare trunk, lumbar, and pelvis primary RoMs in all anatomical planes/directions, lumbopelvic ratios (lumbar to pelvis rotations at different trunk angles) in all anatomical planes/directions and three-dimensional spine kinematics during twelve symmetric/asymmetric statics load-handling activities in healthy normal-weight and obese individuals. Kinematics/motion data were collected from nine healthy young male normal-weight and nine age/height/sex matched obese individuals via a ten-camera Vicon motion capture system. Obese individuals had significantly smaller (p < 0.05) lumbar flexion (~9° in average) and larger pelvis right lateral bending (~5°) RoMs as well as smaller lumbopelvic ratios (~37%) in lateral bending and axial rotation movements as compared to normal-weight individuals. Moreover, the two groups had generally non-significant different segmental orientations (<20° and in most cases < 10°) in load-handling tasks that depended on the magnitude of load asymmetry angle (p < 0.05). Differences were larger for tasks performed near the floor, away from body, and at larger load asymmetry angles. Biomechanical models simulating pure lateral bending, axial rotation, or tasks involving large load asymmetry may therefore need subject-specific, rather than population-based, motion analysis due to the effects from body weight. In clinical applications, it should be noted that healthy obese individuals may have different RoMs and lumbopelvic rhythms than healthy normal-weight individuals in some anatomical planes/directions.
Two numerical procedures are developed to approximate the solution of one-dimensional parabolic equations using extrapolated collocation method. By defining two different end conditions and forcing ...cubic spline to satisfy the interpolation conditions along with one of the end conditions, we obtain fourth- (CBS4) and sixth-order (CBS6) approximations to the solution in spatial direction. Also in time direction, a weighted finite-difference discretization is used to approximate the solution at each time level. The convergence analysis of the methods is discussed in detail and some error bounds are obtained theoretically. Finally, some different examples of Burgers’ equations with applications in fluid mechanics as well as convection–diffusion problems with applications in transport problems are solved to show the applicability and good performance of the procedures.
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