Approach For Numerical Solution Research Articles

An Approach of Numerical Solution and Uncertainty Measure of MHD Viscous Flow Over aShrinking Sheet withSecond Order Slip Flow Model Under Fuzzy Environment

An Approach of Numerical Solution and Uncertainty Measure of MHD Viscous Flow Over aShrinking Sheet withSecond Order Slip Flow Model Under Fuzzy Environment

A Finite Difference-Based Adams-Type Approach for Numerical Solution of Nonlinear Fractional Differential Equations: A Fractional Lotka–Volterra Model as a Case Study

Abstract In this paper, we developed an efficient Adams-type predictor–corrector (PC) approach for the numerical solution of fractional differential equations (FDEs) with a power law kernel. The main idea of the proposed approach is to use a linear approximation to the nonlinear problem and then implement finite difference approximations of derivatives. Numerical comparisons with the fractional Adams method are made and simulation results are demonstrated to evaluate the approximation error of the proposed approach. The efficiency of this approach has been depicted by presenting numerical solutions of some test fractional calculus models. Numerical simulation of a fractional Lotka–Volterra model is provided, as a case study, using the proposed approach. The advantage of the proposed approach lies in its flexibility in providing approximate numerical solutions with high accuracy.

Investigation of numerical solution approaches for the cavitating flow analysis of twisted hydrofoils

In this study, geometrical parameters such as twist, section thickness, and angle of attack are numerically investigated for their influence on lift and drag coefficients, as well as cavitation development. The Delft 11 Hydrofoil, a twisted version of the NACA0009 foil with a −2° angle of attack, is used as the benchmark geometry for validation studies under both non-cavitating and cavitating flow conditions. After the validation studies, the geometries of the NACA0009 and NACA0015 hydrofoils were redesigned as half-twisted and twisted. These redesigned hydrofoils, along with the original no-twist versions, were analyzed under cavitating flow conditions within an angle of attack range of −2<AoA<5°. In the study, three-dimensional, unsteady, cavitating flow is modelled by two different solution approaches, RANS and DES, with the SST Menter k-omega turbulence model. The Schnerr-Sauer cavitation model is employed to obtain cavitation formation. The same physical conditions with RANS simulations are performed using a refined mesh with the DES method, focusing particularly on accurately and precisely obtaining cavitation development, especially cavitation cycles. Additionally, DES analyses were performed for various time step values, demonstrating the specific influence of the time step on cavitation cycles. The results show that while the RANS method effectively predicts the lift and drag forces of the hydrofoil, the DES method is crucial for capturing cavitation dynamics with greater precision, particularly for obtaining cavitation cycles. The results also indicate that twisted hydrofoil geometries produce more lift and drag force than half-twisted and no-twist geometries for both NACA0009 and NACA0015 hydrofoils. In addition, the cavitation formation around the hydrofoil generally increases with the twistiness.

Positivity preserving and unconditionally stable numerical scheme for the three-dimensional modified Fisher–Kolmogorov–Petrovsky–Piskunov equation

This paper introduces a numerical approach for the practical solution of the modified Fisher–Kolmogorov–Petrovsky–Piskunov equation that describes population dynamics. The diffusion term and nonlinear term is based on the operator splitting method and interpolation method, respectively. The analytic proof of the discrete maximum principle and positivity preserving for the numerical algorithm is demonstrated. Numerical solution calculated using the proposed method remains stable without blowing up, which implies that the proposed method is unconditionally stable. Numerical studies show that the proposed method is second-order convergence in space and first-order convergence in time. The performance and applicability of the proposed scheme are studied through various computational tests that present the effects of model parameters and evolution dynamics.

Multiaxial creep–fatigue failure mechanism and life prediction of a turbine blade based on a unified numerical solution approach

AbstractExposure of turbine blades to cyclic torsional loading at high temperature, stemming from pre‐torque installation and the aerodynamic forces during operation, has the potential to induce substantial creep–fatigue damage, thereby contributing to the likelihood of premature failure. Investigating the deformation mechanisms and proposing a reliable life prediction method aiming at torsional loading is critical to ensure the structural integrity of turbine blades. This study conducted strain‐controlled fatigue and creep–fatigue tests on Inconel 718 superalloy, employing a multiaxial servo‐hydraulic testing machine. Electron backscattering diffraction elucidated deformation and damage mechanisms, forming a basis for subsequent constitutive modeling and life prediction. The lack of creep–fatigue mechanical behavior and microscopic failure mechanism when stress triaxiality equal to 0 is filled, which provides the theoretical basis and data support for the life design and damage assessment of this material under extreme service conditions. The unified viscoplasticity constitutive model effectively characterized macroscopic deformation under torsional loading. Prediction of creep–fatigue life under torsional loading, utilizing the multiaxial ductility factor‐modified strain energy density exhaustion model, demonstrated excellent alignment with experimental findings. Finally, parametric analyses of stress distribution and damage assessment under different conditions were carried out for the example of a turbine blade with relatively rarely considered aerodynamic loading as a variable. It is expected to be popularized and applied in life design and damage assessment of high‐temperature structures under multiaxial loading in engineering.

Error analysis for discontinuous Galerkin method for time‐fractional Burgers' equation

The primary goal of this paper is to suggest a fully discrete numerical solution approach for the time‐fractional Burgers' equation. This paper will consider the fractional derivative in the Caputo sense. The time derivative of this equation will be discretized using the L2‐type discretization formula. The spatial variable is approximated by using the nonsymmetric interior penalty discontinuous Galerkin method. The proposed method is globally and unconditionally stable. The accuracy of the solution is evaluated using a convergence analysis. Computational experiments further confirm the accuracy and stability of the suggested strategy.

A machine learning investigation of the ZnO–water nanofluid flow with magnetic field through convergent and divergent channels: a numerical study

This investigation focuses on how ZnO–H2O nanofluid flow affects the thermal processes in divergent/convergent channels under the effects of magnetohydrodynamics (MHD). Advanced machine learning techniques are utilised to comprehensively quantify the interactions among the implications of nanoparticles on thermal transmission and fluid actions. Our analysis encompasses the MHD and Joule emission effects, transforming the intricate system of nonlinear system of equations to ordinary differential equations (ODEs) following the strategy of non-dimensionalisation. Leveraging the integration of artificial neural networks (ANN) integrated a hybrid approach for numerical solutions. The optimum value of ANN-based fitness function is up to 5.53 × 10 − 5 . The optimal weights and biases of ANN through ABC–WCA are ranging from − 10 to 10 . These results are meticulously compared with shooting method and homotopy perturbation method (HPM) solutions. The absolute error between the ANN, shooting and HPM are equal up to four (4) decimal places. Further, the ANN-based fitness function optimised through ABC-WCA over 250 independent runs for the efficiency of algorithms. It is found that the current results are much more authentic and powerful than the existing less converging and slow analytic and numerical methods.

A Fuzzy Fractional Power Series Approximation and Taylor Expansion for Solving Fuzzy Fractional Differential Equation

Fuzzy fractional differential has the strength to capture the senses of memory and uncertainty simultaneously involved in dynamical systems. However, a solution for fuzzy fractional differential equations is not always found regularly. This paper discusses a numerical solution approach for the fuzzy fractional differential equation using power series approximation with a fuzzy fractional counterpart of Taylor’s theorem. Caputo’s definition of the fractional derivative and generalized Hukuhara difference are used to describe the fuzzy differential equation in this paper. Utilization of the generalized Hukuhara difference for the fuzzy valued function ensures the uniqueness and boundedness of the fuzzy solution in parametric form.

Application of the Improved Cuckoo Algorithm in Differential Equations

To address the drawbacks of the slow convergence speed and lack of individual information exchange in the cuckoo search (CS) algorithm, this study proposes an improved cuckoo search algorithm based on a sharing mechanism (ICSABOSM). The enhanced algorithm reinforces information sharing among individuals through the utilization of a sharing mechanism. Additionally, new search strategies are introduced in both the global and local searches of the CS. The results from numerical experiments on four standard test functions indicate that the improved algorithm outperforms the original CS in terms of search capability and performance. Building upon the improved algorithm, this paper introduces a numerical solution approach for differential equations involving the coupling of function approximation and intelligent algorithms. By constructing an approximate function using Fourier series to satisfy the conditions of the given differential equation and boundary conditions with minimal error, the proposed method minimizes errors while satisfying the differential equation and boundary conditions. The problem of solving the differential equation is then transformed into an optimization problem with the coefficients of the approximate function as variables. Furthermore, the improved cuckoo search algorithm is employed to solve this optimization problem. The specific steps of applying the improved algorithm to solve differential equations are illustrated through examples. The research outcomes broaden the application scope of the cuckoo optimization algorithm and provide a new perspective for solving differential equations.

Numerical approaches for solution of hyperbolic difference equations on circle

Abstract The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle T 1 \mathbb{T}^{1} . The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided.

Chance Constrained Distributed Optimisation for Interconnected Power Systems

Microgrids have been attracted increasingly attention due to renewable energy integration. Multiple microgrids are often interconnected for benefiting each other. A significant challenge in operating such power systems lies in the uncertainties including intermittence in renewable energy sources and load demands estimation. Sharing state variables in interconnected systems poses additional difficulties. This paper proposes a numerical solution approach to a chance constrained distributed optimisation for interconnected power systems under uncertainty. The inner-outer approximation approach is used for the solution and ALADIN for numerical implementation. An interconnected optimal power flow problem with a three-bus and a four-bus system is solved as a case study.

An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface

&lt;p&gt;We introduced a fully explicit finite difference method (FDM) designed for numerically solving the conservative Allen–Cahn equation (CAC) on a cubic surface. In this context, the cubic surface refers to the combined areas of the six square faces that enclose the volume of a cube. The proposed numerical solution approach is structured into two sequential steps. First, the Allen–Cahn (AC) equation was solved by applying the fully explicit FDM, which is computationally efficient. Following this, the conservation term is resolved using the updated solution from the AC equation to ensure consistency with the underlying conservation principles. To evaluate the effectiveness of the proposed scheme, computational tests are performed to verify that the resulting numerical solution of the CAC equation successfully conserves the discrete mass. Additionally, the solution is examined for its ability to exhibit the property of constrained motion by mass conserving mean curvature, a critical characteristic of the CAC equation. These two properties are fundamental to the integrity and accuracy of the CAC equation.&lt;/p&gt;

Large-amplitude vibration analysis of FG rectangular plates under rapid cooling

In the present research, the large-amplitude vibration behavior of rectangular plates subjected to cooling shock is investigated based on a numerical solution approach. It is considered that the plates are made of functionally graded materials as a mixture of stainless steel and low-carbon steel. The thermoelastic properties are also considered temperature-dependent which are estimated based on available experimental data. According to the Mindlin plate theory, the nonlinear governing equations of motion together with corresponding boundary conditions are derived. The temperature profile is also obtained based on a one-dimensional Fourier-type transient heat conduction equation. Moreover, two time-dependent thermal loading scenarios for cooling shock on the plate’s top face are considered. To solve the problem numerically, the well-known generalized differential quadrature technique is used for discretization considering the Chebyshev–Gauss–Lobatto grid. In addition, the governing equations are traced in time by Newmark’s time integration scheme. After showing the validity of developed approach, a parametric study is presented to investigate the stress changes along the thickness and the effects of thermal load rapidity time, geometry, magnitude of thermal load, and material properties on the nonlinear vibrations of plates with various boundary conditions subjected to sudden decrease of temperature. It is concluded that the thermal load rapidity time has a significant role in the vibrational behavior and the stress distribution of the plate.

Convergence of the L1 Two-Term Equation Scheme

Fractional derivatives have found application in modeling various processes in different fields of science. Finite difference schemes are a main approach for numerical solution of models using fractional differential equations. In this paper we investigate the convergence and order of the numerical solution of two-term ordinary fractional differential equation which uses the L1 approximation of the fractional derivative. Inequalities for the weights of L1 approximation are derived and used to prove the convergence of the L1 scheme for the two-term equation. Conditions for the parameter of the two-term equation and error estimates of L1 scheme are obtained. Experimental results for the order and error of the L1 scheme are presented in the paper.

Basis Function Approaches for Numerical Solutions of Nonlinear Partial Differential Equations

Basis Function Approaches for Numerical Solutions of Nonlinear Partial Differential Equations

Numerical Investigation of Radiative Hybrid Nanofluid Flows over a Plumb Cone/Plate

Non-Newtonian fluids play a crucial role in applications involving heat transfer and mass transfer. The inclusion of nanoparticles in these fluids improves the efficiency of heat and mass transfer processes. This study employs a numerical solution approach to examine the flow of non-Newtonian hybrid nanofluids over a plumb cone/plate surface, considering the effects of magnetohydrodynamics (MHD) and thermal radiation. Additionally, we investigate how heat and mass transfer are affected by a fluid containing microorganisms. The governing nonlinear partial differential equations are transformed into nonlinear ordinary differential equations using a similarity transformation to simplify this complex system. We then use the Keller-box finite-difference method to solve these equations. Along with a table presenting the results for skin friction, Nusselt number, Sherwood number, and microbe density number, we present graphical representations of velocity, temperature, concentration, and microorganism diffusion behavior. Our results indicate that the addition of MHD and thermal radiation improves the diffusion of microorganisms, thereby enhancing the rates of heat and mass transfer. Through a comparative analysis with prior research, we demonstrate the reliability of our conclusions.

Vibration reduction and energy harvesting of a human-seat-boat-water dynamic interaction system excited by waves

A vibration isolation means for a human-seat-boat-water dynamic interaction system subject to sea waves is proposed, for which to be numerically analyzed, the governing equations with numerical solution approaches are developed. The key idea is adopting electromagnetic isolators to replace often-used seat-damper to reduce vibrations by harvesting wave energy. The human body is modelled as a mass supported by a spring-damper connected to a seat, that through the electromagnetic isolator connected to a boat simplifying as a uniform beam-like vessel travelling in sea way. The wave force in the model is an approximate form commonly used in dynamic analysis of marine engineering, consisting of three components proportional to the relative motion of ship to the water, which adds additional mass, stiffness, and damping onto the ship. The parameters in the simulations are chosen from the practical designs or experimental/numerical reports. The simulation results reveal the effects of integrated interactions on the natural frequencies and dynamic responses of the ship subjected to short/long water waves and show very large reductions of human vibration level and effectively collecting wave energy. This confirms the proposed numerical model for ship vibration reduction and wave energy collection is attractive to simulate this complex integrated interaction system, and the proposed means based on electromagnetic isolators is no doubt to realize the aim of the paper. Some guidelines for designing this type of isolators are presented, which would be one of further develop directions for the global plan seeking/collecting green energies.

Https://www.bakumathj.org/archive/Vol2No2/j.bmj.033.pdf

In the paper, we consider an approach for numerical solution to the optimal feedback control problem for an object with distributed parameters on the basis of observation of the object’s phase state at its specific locations by the example of the rod heating process. The control actions are the power of the heat source, the values of which are defined on the class of “zonal” controls. The values of the parameters of zonal control actions are determined by subsets of the state space, to which belong the values of the process state at the measurement points at the current moment of time. The problem of determining zonal controls is reduced to a parametric optimal control problem on determining a finite-dimensional vector of values of the parameters of zonal control actions. We derive optimality conditions for the values of the parameters of zonal control actions. These conditions contain formulas for the gradient of the objective functional with respect to the optimizable parameters of zonal controls. They make it possible to solve the stated problem numerically with the application of efficient first-order optimization methods.

Traffic flow modelling for uphill and downhill highways: Analysed by soft computing-based approach

Researchers have made significant strides in understanding car-following behaviour and traffic flow, especially with the advent of intelligent and networked technologies. Diverse mathematical models analyse traffic flow, each with pros and cons. This study focuses on a sensitivity-based mathematical model for uphill and downhill highways, examining position, velocity, and acceleration profiles to predict traffic jam occurrence. The model employs an ordinary differential equation and a machine learning-based approach (machine learning procedure neural network) for numerical solutions, exhibiting high accuracy (10−8−10−10) compared to the reference Runge–Kutta method For accuracy, reliability and stability of the results are evaluated by various performance indicators and statistical terms. For multiple independent executions, mean absolute deviation, root mean square error and error in Nash–Sutcliffe efficiency are calculated. Their values are lies in range 10−8−10−14. Moreover, graphical analysis is established for better visualization of traffic flow and congestion.

Enhanced heat transfer analysis of hybrid nanofluid over a Riga plate: Incorporating Lorentz forces and entropy generation

The Riga plate, comprising a configuration of electrodes and permanent magnets, can induce a wall-parallel Lorentz force through external electric and magnetic fields, enabling effective control of fluid flow. This paper focuses on studying the flow characteristics of Al2O3-Cu/water hybrid nanofluid past a Riga plate in the presence of strong suction. The mathematical model incorporates a two-phase approach to describe the behavior of the nanofluid, while considering the Grinberg-term to account for the wall parallel Lorentz force generated by the Riga plate. Several influential factors such as entropy generation, viscous dissipation, heat flux, thermal radiation, slippage, and convective boundary conditions of temperature are considered in our investigation. Entropy generation within thermodynamic systems contributes to energy losses resulting from various factors, including frictional forces, viscosity, and chemical reactions. Minimizing entropy generation is crucial to enhancing system performance. To address the research problem, we utilize dimensionless parameters and assume strong suction to simplify the governing partial differential equations. The dimensionless model is subsequently solved numerically using the Adams-Bashforth technique. Additionally, a comprehensive graphical analysis is conducted to examine the impact of various dimensionless parameters on the flow fields. This study highlights the importance of controlling fluid flow using the Riga plate in the presence of suction and investigates the behavior of a specific nanofluid under these conditions. The results provide valuable insights into the influence of key parameters on the flow characteristics. We conclude that effective utilization of the Riga plate can lead to significant enhancements in fluid flow control. Moreover, the study contributes to reducing energy losses through minimizing entropy generation. The novelty of this work lies in its comprehensive approach, considering multiple influential factors and employing a two-phase model for the nanofluid. Additionally, the inclusion of the Grinberg-term to account for the wall parallel Lorentz force adds to the uniqueness of the study. This research expands upon previous efforts in the literature by providing a more comprehensive analysis of the fluid flow control using the Riga plate, incorporating a broader range of factors, and presenting a numerical solution approach.