Journal articles: 'Numerical simulation of solutions'

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Debrie, Ayalew. "Simulation of Harmonic motion." J. of Physical and Chemical Sciences 6, no. 3 (2018): 02. https://doi.org/10.5281/zenodo.1254645.

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            In this work, we study   the characteristics of simple harmonic motion and to solve physical problems related to simple harmonic motion (SHM) using MATLAB computer program. To achieve the objectives, the necessary techniques had been done using both analytical and numerical mathematical computations. Among the numerical methods, we had concentrate Runge Kutta forth order method. The solutions would be given by figures. For the undamped and damped SHM we solve numerically using the above numerical methods with applying the differential algorithm. With this respect, by adjusting the set of equidistant points (the step size) the numerical solution is comparable with the analytical one. For both cases, the undamped and damped SHM the numerical solution obtained using Runge-kutta is almost the same as to the solution obtained using analytical method.

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Skinner, J. W., and J. Y.-K. Cho. "Numerical convergence of hot-Jupiter atmospheric flow solutions." Monthly Notices of the Royal Astronomical Society 504, no. 4 (2021): 5172–87. http://dx.doi.org/10.1093/mnras/stab971.

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ABSTRACT We perform an extensive study of numerical convergence for hot-Jupiter atmospheric flow solutions in simulations employing a setup commonly used in extrasolar planet studies – a resting state thermally forced to a prescribed temperature distribution on a short time-scale at high altitudes. Convergence is assessed rigorously with (i) a highly accurate pseudospectral model that has been explicitly verified to perform well under hot-Jupiter flow conditions and (ii) comparisons of the kinetic energy spectra, instantaneous (unaveraged) vorticity fields and temporal evolutions of the vorticity field from simulations that are numerically equatable. In the simulations, the (horizontal as well as vertical) resolution, dissipation operator order, and viscosity coefficient are varied with identical physical and initial setups. All of the simulations are compared against a fiducial reference simulation at high horizontal resolution and dissipation order (T682 and ∇ 16, respectively) – as well as against each other. Broadly, the reference solution features a dynamic, zonally (east–west) asymmetric jet with a copious amount of small-scale vortices and gravity waves. Here, we show that simulations converge to the reference simulation only at T341 resolution and with ∇ 16 dissipation order. Below this resolution and order, simulations either do not converge or converge to unphysical solutions. The general convergence behaviour is independent of the vertical range of the atmosphere modelled, from $\sim 2 \times 10^{-3}$MPa to $\sim 2 \times 10^1$ MPa. Ramifications for current extrasolar planet atmosphere modelling and observations are discussed.

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Zhang, Cheng Li, and Yun Zeng. "A Simple Numerical Simulation Method for Lorenz System Families." Applied Mechanics and Materials 444-445 (October 2013): 786–90. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.786.

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Lorenz system families contain Lorenz system, Chen system and Lu system, their accurate analytical solutions are not yet obtained now. The segmenting recursion method was put forward in this paper, the equations of Lorenz system families were reasonably linearized within small segment, the recursion formulas were obtained by solving the approximate analytical solutions within small segment, and all numerical solutions were got by the recursion formulas. The chaotic motion of Lorenz system families were numerically simulated by means of the segmenting recursion method, the simulation results were compared with Runge-Kutta method. The comparative results show that the segmenting recursion method is very effective to numerically simulate Lorenz system families, not only method is simple, programming is easy, but result is accurate. this method is a universal new method to numerically simulate similar system.

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Mamman, Ali Bulama, Haruna Usman Idriss, and Bashir Mai Umar. "Numerical Solution of linear Second Order Partial Differential Equation." International Journal of Research and Innovation in Applied Science X, no. IV (2025): 951–65. https://doi.org/10.51584/ijrias.2025.10040081.

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This study explores the analytical and numerical solutions of partial differential equations (PDEs), focusing on parabolic (heat). The first part presents their analytical solutions using initial and boundary conditions and delves into the finite difference method (FDM), discussing forward, backward, and central difference schemes. These methods are applied to numerically solve one- and two-dimensional heat. The Crank-Nicolson method, recognized for its unconditional stability, is employed to improve the accuracy of heat equation solutions, overcoming limitations of explicit and implicit schemes. We then analyze the performance, strengths, and weaknesses of FDM through numerical simulations of one-dimensional heat. Due to computational constraints, Crank-Nicolson for 1D simulation, was not executed. Results indicate that the implicit backward difference method demonstrates superior stability by allowing unrestricted step sizes compared to the explicit forward difference method. These findings contribute to a deeper understanding of numerical PDE solutions and stability considerations in computational mathematics.

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Yokuş, Asıf, and Doğan Kaya. "Comparison exact and numerical simulation of the traveling wave solution in nonlinear dynamics." International Journal of Modern Physics B 34, no. 29 (2020): 2050282. http://dx.doi.org/10.1142/s0217979220502823.

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The traveling wave solutions of the combined Korteweg de Vries-modified Korteweg de Vries (cKdV-mKdV) equation and a complexly coupled KdV (CcKdV) equation are obtained by using the auto-Bäcklund Transformation Method (aBTM). To numerically approximate the exact solutions, the Finite Difference Method (FDM) is used. In addition, these exact traveling wave solutions and numerical solutions are compared by illustrating the tables and figures. Via the Fourier–von Neumann stability analysis, the stability of the FDM with the cKdV–mKdV equation is analyzed. The [Formula: see text] and [Formula: see text] norm errors are given for the numerical solutions. The 2D and 3D figures of the obtained solutions to these equations are plotted.

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Tai, Irfan, Marie Ann Giddins, and Ann Muggeridge. "Improved Calculation of Wellblock Pressures for Numerical Simulation of Non-Newtonian Polymer Injection." SPE Journal 26, no. 04 (2021): 2352–63. http://dx.doi.org/10.2118/205339-pa.

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Summary The viability of any enhanced-oil-recovery project depends on the ability to inject the displacing fluid at an economic rate. This is typically evaluated using finite-volume numerical simulation. These simulators calculate injectivity using the Peaceman method (Peaceman 1978), which assumes that flow is Newtonian. Most polymer solutions exhibit some degree of non-Newtonian behavior resulting in a changing polymer viscosity with distance from the injection well. For shear-thinning polymer solutions, conventional simulations can overpredict injection-well bottomhole pressure (BHP) by several hundred psi, unless a computationally costly local grid refinement is used in the near-wellbore region. We show theoretically and numerically that the Peaceman pressure-equivalent radius, based on Darcy flow, is not correct when fluids are shear thinning, and derive an analytical expression for calculating the correct radius. The expression does not depend on any particular functional relationship between polymer-solution viscosity and velocity. We test it using the relationship described by the Meter equation (Meter and Bird 1964) and the Cannella et al. (1988) correlation. Numerical tests indicate that the solution provides a significant improvement in the accuracy of BHP calculations for conventional numerical simulation, reducing or removing the need for expensive local grid refinement around the well when simulating the injection of fluids with shear-thinning non-Newtonian rheology.

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Pérez-Benítez, J. A., P. Martínez-Ortiz, and J. Aguila-Muñoz. "A Review of Formulations, Boundary Value Problems and Solutions for Numerical Computation of Transcranial Magnetic Stimulation Fields." Brain Sciences 13, no. 8 (2023): 1142. http://dx.doi.org/10.3390/brainsci13081142.

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Since the inception of the transcranial magnetic stimulation (TMS) technique, it has become imperative to numerically compute the distribution of the electric field induced in the brain. Various models of the coil-brain system have been proposed for this purpose. These models yield a set of formulations and boundary conditions that can be employed to calculate the induced electric field. However, the literature on TMS simulation presents several of these formulations, leading to potential confusion regarding the interpretation and contribution of each source of electric field. The present study undertakes an extensive compilation of widely utilized formulations, boundary value problems and numerical solutions employed in TMS fields simulations, analyzing the advantages and disadvantages associated with each used formulation and numerical method. Additionally, it explores the implementation strategies employed for their numerical computation. Furthermore, this work provides numerical expressions that can be utilized for the numerical computation of TMS fields using the finite difference and finite element methods. Notably, some of these expressions are deduced within the present study. Finally, an overview of some of the most significant results obtained from numerical computation of TMS fields is presented. The aim of this work is to serve as a guide for future research endeavors concerning the numerical simulation of TMS.

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Noshad, Mahlagha, Amir Pishkoo, and Maslina Darus. "Solving Conformable Fractional Differential Equations with ``EJS Software and Visualization of Sub-diffusion Process." European Journal of Pure and Applied Mathematics 15, no. 4 (2022): 1738–49. http://dx.doi.org/10.29020/nybg.ejpam.v15i4.4547.

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In this work, we numerically solve ordinary and conformable fractional differential equations using Easy Java Simulations software. Their solutions, including homogeneous and non-homogeneous parts, are compared in various time intervals. Using software’s visualization andsimulation features, we may better examine, compare, and evaluate solutions of analytical and numerical fractional differential equations. A kind of oscillatory behavior is seen in long enough times. In simulation of diffusion and sub-diffusion processes, two intriguing events have been observed.

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Risher, D. W., L. M. Schutte, and C. F. Runge. "The Use of Inverse Dynamics Solutions in Direct Dynamics Simulations." Journal of Biomechanical Engineering 119, no. 4 (1997): 417–22. http://dx.doi.org/10.1115/1.2798288.

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Previous attempts to use inverse dynamics solutions in direct dynamics simulations have failed to replicate the input data of the inverse dynamics problem. Measurement and derivative estimation error, different inverse dynamics and direct dynamics models, and numerical integration error have all been suggested as possible causes of inverse dynamics simulation failure. However, using a biomechanical model of the type typically used in gait analysis applications for inverse dynamics calculations of joint moments, we produce a direct dynamics simulation that exactly matches the measured movement pattern used as input to the inverse dynamic problem. This example of successful inverse dynamics simulation demonstrates that although different inverse dynamics and direct dynamics models may lead to inverse dynamics simulation failure, measurement and derivative estimation error do not. In addition, inverse dynamics simulation failure due to numerical integration errors can be avoided. Further, we demonstrate that insufficient control signal dimensionality (i.e., freedom of the control signals to take on different “shapes”), a previously unrecognized cause of inverse dynamics simulation failure, will cause inverse dynamics simulation failure even with a perfect model and perfect data, regardless of sampling frequency.

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Bretti, Gabriella. "Differential Models, Numerical Simulations and Applications." Axioms 10, no. 4 (2021): 260. http://dx.doi.org/10.3390/axioms10040260.

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Differential models, numerical methods and computer simulations play a fundamental role in applied sciences. Since most of the differential models inspired by real world applications have no analytical solutions, the development of numerical methods and efficient simulation algorithms play a key role in the computation of the solutions to many relevant problems. Moreover, since the model parameters in mathematical models have interesting scientific interpretations and their values are often unknown, estimation techniques need to be developed for parameter identification against the measured data of observed phenomena. In this respect, this Special Issue collects some important developments in different areas of application.

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Khuri, Suheil A., and Ali M. Sayfy. "NUMERICAL SOLUTION OF A CLASS OF NONLINEAR SYSTEM OF SECOND-ORDER BOUNDARY-VALUE PROBLEMS: A FOURTH-ORDER CUBIC SPLINE APPROACH." Mathematical Modelling and Analysis 20, no. 5 (2015): 681–700. http://dx.doi.org/10.3846/13926292.2015.1091793.

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A cubic B-spline collocation approach is described and presented for the numerical solution of an extended system of linear and nonlinear second-order boundary-value problems. The system, whether regular or singularly perturbed, is tackled using a spline collocation approach constructed over uniform or non-uniform meshes. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order. The efficiency and applicability of the technique are demonstrated by applying the scheme to a number of linear and nonlinear examples. The numerical solutions are contrasted with both analytical and other existing numerical solutions that exist in the literature. The numerical results demonstrate that this method is superior as it yields more accurate solutions.

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Wu, G., Eric Wai Ming Lee, and Gao Li. "Numerical solutions of the reaction-diffusion equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 2 (2015): 265–71. http://dx.doi.org/10.1108/hff-04-2014-0113.

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Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.

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Cao, Lulu, Zexin Lin, Kay Chen Tan, and Min Jiang. "Interpretable Solutions for Multi-Physics PDEs Using T-NNGP." Proceedings of the AAAI Conference on Artificial Intelligence 39, no. 13 (2025): 14212–20. https://doi.org/10.1609/aaai.v39i13.33556.

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Multiphysics simulation aims to predict and understand interactions between multiple physical phenomena, aiding in comprehending natural processes and guiding engineering design. The system of Partial Differential Equations (PDEs) is crucial for representing these physical fields, and solving these PDEs is fundamental to such simulations. However, current methods primarily yield numerical outputs, limiting interpretability and generalizability. We introduce T-NNGP, a hybrid genetic programming algorithm that integrates traditional numerical methods with deep learning to derive approximate symbolic expressions for multiple unknown functions within a system of PDEs. T-NNGP initially obtains numerical solutions using traditional methods, then generates candidate symbolic expressions via deep reinforcement learning, and finally optimizes these expressions using genetic programming. Furthermore, a universal decoupling strategy guides the search direction and addresses coupling problems, thereby accelerating the search process. Experimental results on three types of PDEs demonstrate that our method can reliably obtain human-understandable symbolic expressions that fit both the PDEs and the numerical solutions from traditional methods. This work advances multiphysics simulation by enhancing our ability to derive approximate symbolic solutions for PDEs, thereby improving our understanding of complex physical phenomena.

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Erwina, Novry, Didit Adytia, Sri Redjeki Pudjaprasetya, and Toni Nuryaman. "Staggered Conservative Scheme for 2-Dimensional Shallow Water Flows." Fluids 5, no. 3 (2020): 149. http://dx.doi.org/10.3390/fluids5030149.

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Simulating discontinuous phenomena such as shock waves and wave breaking during wave propagation and run-up has been a challenging task for wave modeller. This requires a robust, accurate, and efficient numerical implementation. In this paper, we propose a two-dimensional numerical model for simulating wave propagation and run-up in shallow areas. We implemented numerically the 2-dimensional Shallow Water Equations (SWE) on a staggered grid by applying the momentum conserving approximation in the advection terms. The numerical model is named MCS-2d. For simulations of wet–dry phenomena and wave run-up, a method called thin layer is used, which is essentially a calculation of the momentum deactivated in dry areas, i.e., locations where the water thickness is less than the specified threshold value. Efficiency and robustness of the scheme are demonstrated by simulations of various benchmark shallow flow tests, including those with complex bathymetry and wave run-up. The accuracy of the scheme in the calculation of the moving shoreline was validated using the analytical solutions of Thacker 1981, N-wave by Carrier et al., 2003, and solitary wave in a sloping bay by Zelt 1986. Laboratory benchmarking was performed by simulation of a solitary wave run-up on a conical island, as well as a simulation of the Monai Valley case. Here, the embedded-influxing method is used to generate an appropriate wave influx for these simulations. Simulation results were compared favorably to the analytical and experimental data. Good agreement was reached with regard to wave signals and the calculation of moving shoreline. These observations suggest that the MCS method is appropriate for simulations of varying shallow water flow.

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Haberland, Maria, and Lars Hohmuth. "Coping with Randomness in Highly Complex Sys-tems Using the Example of Quantum-Inspired Traffic Flow Optimization." SUMO Conference Proceedings 4 (June 29, 2023): 65–74. http://dx.doi.org/10.52825/scp.v4i.216.

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Developing new solutions to complicated large-scale problems typically requires large-scale numerical simulation. Therefore, traffic simulations often run against randomized simulations instead of real-world traffic situations. This paper demonstrates a method to calculate the statistical significance of numerical simulations and optimizations in the presence of numerous random variables in complex systems using one-sided paired t-tests. While the paper covers a specific Fujitsu traffic-optimization project which uses SUMO for simulating the traffic situation, the method can be applied to many similar projects where a complete investigation of the solution space is not feasible due to the size of the solution space.

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Chen, Yi Bin, Jian Zhong Zhou, Shu Huang, and Yue Qing Sun. "FE Simulation of Complex Contour Forming of Sheet Metals Based on Laser Bending." Materials Science Forum 575-578 (April 2008): 696–701. http://dx.doi.org/10.4028/www.scientific.net/msf.575-578.696.

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Laser bending is a flexible forming process which forms sheet metal by means of stresses induced by external heat instead of external forces. In this paper, a three-dimensional coupled thermal-mechanical model for numerical simulation is established with finite element code ABAQUS. Some key problems about the simulation of laser bending are investigated in detail, and the reasonable solutions are presented. Taking AISI-1008 steel as an example, numerical simulations are carried out for the complex contour forming of sheet by using Sequentially Coupled Thermal-Stress Analysis technique. Then the corresponding experiments are performed to validate the simulation results. Good correlation between the numerical simulation and the experimental results was demonstrated.

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Srivastava, H. M., and Khaled M. Saad. "Numerical Simulation of the Fractal-Fractional Ebola Virus." Fractal and Fractional 4, no. 4 (2020): 49. http://dx.doi.org/10.3390/fractalfract4040049.

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In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter ρ is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of ρ=k=1, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of ρ and k. All calculations in this work are accomplished by using the Mathematica package.

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Han, Weimin, Wenxiang Cong, and Ge Wang. "Mathematical Study and Numerical Simulation of Multispectral Bioluminescence Tomography." International Journal of Biomedical Imaging 2006 (2006): 1–10. http://dx.doi.org/10.1155/ijbi/2006/54390.

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Multispectral bioluminescence tomography (BLT) attracts increasingly more attention in the area of optical molecular imaging. In this paper, we analyze the properties of the solutions to the regularized and discretized multispectral BLT problems. First, we show the solution existence, uniqueness, and its continuous dependence on the data. Then, we introduce stable numerical schemes and derive error estimates for numerical solutions. We report some numerical results to illustrate the performance of the numerical methods on the quality of multispectral BLT reconstruction.

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Čemeljić, M. "“Atlas” of numerical solutions for star-disk magnetospheric interaction." Astronomy & Astrophysics 624 (April 2019): A31. http://dx.doi.org/10.1051/0004-6361/201834580.

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Aims. I report results in numerical simulations of star-disk magnetospheric interaction. A thin accretion disk with a corona above a rotating stellar surface is simulated, and a parameter study is performed to find trends in the angular momentum flux. The results are presented for young stellar objects, but they can be rescaled to other objects with similar geometry. Methods. I performed resistive and viscous magnetohydrodynamic simulations that reached a quasi-stationary state for cases with different parameters. I computed angular momentum fluxes in the different components of the flow to compare the results. Results. I present the simulation results with the matter density distribution and a sample of the magnetic field lines, gathered in an “Atlas” of solutions. The torque exerted on the star is computed, together with the angular momentum flux that is expelled from the system in the cases with a conical outflow. I find trends in the components of the flow in the part of parameter space with a slowly rotating star.

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Su Jin, Ouyang Jie, and Wang Xiao-Dong. "Micro-macro numerical simulation of rod-like polymeric solutions." Acta Physica Sinica 59, no. 5 (2010): 3362. http://dx.doi.org/10.7498/aps.59.3362.

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Yazdani, Ali, and Brij B. Maini. "Pitfalls and Solutions in Numerical Simulation of VAPEX Experiments." Energy & Fuels 23, no. 8 (2009): 3981–88. http://dx.doi.org/10.1021/ef900200f.

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Kierzkowska-Pawlak, Hanna, and Andrzej Chacuk. "Numerical simulation of CO2 absorption into aqueous methyldiethanolamine solutions." Korean Journal of Chemical Engineering 29, no. 6 (2012): 707–15. http://dx.doi.org/10.1007/s11814-011-0244-9.

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Peshkov, I. M. "Numerical simulation of discontinuous solutions in nonlinear elasticity theory." Journal of Applied Mechanics and Technical Physics 50, no. 5 (2009): 858–65. http://dx.doi.org/10.1007/s10808-009-0116-9.

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Azis, Mohammad Ivan, Adi Maulana, and Muh Nur. "Convective Heat Conduction in Anisotropic FGMs: A Numerical Simulation." Journal of Advanced Research in Numerical Heat Transfer 31, no. 1 (2025): 104–25. https://doi.org/10.37934/arnht.31.1.104125.

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A combination of the Laplace transform (LT) and domain-boundary element method (DBEM) was employed to find numerical solutions for an unsteady diffusion-convection (DC) equation with spatial and temporal coefficients, considering arbitrary initial conditions and source terms. This method addresses problems in a unique class of anisotropic functionally graded materials (FGMs). The procedure involves converting the variable coefficient equation to a constant coefficient equation, Laplace transform and then deriving a domain-boundary integral equation. Numerical solutions were obtained using the standard domain-boundary element method and the Stehfest algorithm. This study examines cases involving compressible or incompressible flows, showcasing the precision of the numerical solutions.

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Li, Long-yuan, and Brian Tighe. "Numerical simulation of corneal transport processes." Journal of The Royal Society Interface 3, no. 7 (2005): 303–10. http://dx.doi.org/10.1098/rsif.2005.0085.

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This paper presents a numerical study on the transport of ions and ionic solution in human corneas and the corresponding influences on corneal hydration. The transport equations for each ionic species and ionic solution within the corneal stroma are derived based on the transport processes developed for electrolytic solutions, whereas the transport across epithelial and endothelial membranes is modelled by using phenomenological equations derived from the thermodynamics of irreversible processes. Numerical examples are provided for both human and rabbit corneas, from which some important features are highlighted.

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Iqbal, Naveed, Muhammad Tajammal Chughtai, and Nehad Ali Shah. "Numerical simulation of fractional-order two-dimensional Helmholtz equations." AIMS Mathematics 8, no. 6 (2023): 13205–18. http://dx.doi.org/10.3934/math.2023667.

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<abstract><p>In this paper, we investigate the exact solutions of several fractional-order Helmholtz equations using the homotopy perturbation transform method. We specify sufficient requirements for its convergence and provide error estimations. The homotopy perturbation transform method yields a quickly converging succession of solutions. Solutions for various fractional space derivatives are compared to present approaches and explained using figures. Appropriate parameter selection produces approximations identical to the exact answer. Test examples are provided to demonstrate the proposed approach's precision and competence. The results demonstrate that our system is appealing, user-friendly, dependable, and highly effective.</p></abstract>

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Pincevičius, A., R. J. Rakauskas, and G. Misevičius. "The Numerical Simulation in Ballistics." Nonlinear Analysis: Modelling and Control 6, no. 1 (2001): 89–104. https://doi.org/10.15388/na.2001.6.1.15219.

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In this paper we examine the movement of the solid body thrown with some angle to the horizon (for example the shot mine). Such movement is described by non-linear system of equations. This system is being approximated by linear system, in segments. The experiment results have been approximated and the dependence of air resistance coefficient from mean value of velocity along the trajectory was found. From the point of view of mathematics the incorrect problem must be solved because the initial conditions of system corresponding to fixed values of solutions (the coordinates of target points) must been estimated. In the case of linear system it is possible to examine the influence of non-large increment of initial conditions to the final result. In this work the probability of destruction of some fixed target and the mean square deviation of shooting regression. On the other hand has been estimated the possibility of destruction of group target using the method of Monte-Carlo.

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Samuel, Adamu, Anjikwi Favour, Bukar Hassan, Aduroja Olamiposi Ojo, and Tahir Alhaji. "A Study on Some Numerical Methods for Simulating Mathematical Models of Ordinary Differential Equations." UMYU Scientifica 4, no. 2 (2025): 7–15. https://doi.org/10.56919/usci.2542.002.

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Study’s Excerpt:• The study evaluates numerical methods for ODEs to guide suitable model simulations.• Adomian Decomposition suits decay/growth models; block method excels in all problems, including SIR.• The study highlights numerical methods' ease and precision in simulating mathematical models.Full Abstract:This study presents a comparative analysis on some numerical methods for simulating mathematical models of ordinary differential equations, including Euler, Classical Runge-Kutta, Adomian Decomposition, Block, and Simulink. We examine each method's accuracy, stability, and consistency through a series of test cases. These methods are applied to simulate some selected mathematical models, and the results are shown in tables. The graph of each table is depicted in figures for discussion and comparative analysis. The results show that, while straightforward, the Euler method demonstrates significant limitations in accuracy compared to the Classical Runge-Kutta method, which provides reliable and precise results. The Adomian Decomposition Method solves the problems and yields results very close to the analytical solution, but the block method performs better due to its multistep approach. Simulink offers a more robust approach for modelling and simulation with visible and interpretable solutions for good understanding. This study revealed that numerical methods can easily be used to better simulate mathematical models that may not have analytical solutions and thus provide approximate solutions.

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Soliman, A. A. "Numerical Simulation of the FitzHugh-Nagumo Equations." Abstract and Applied Analysis 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/762516.

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The variational iteration method and Adomian decomposition method are applied to solve the FitzHugh-Nagumo (FN) equations. The two algorithms are illustrated by studying an initial value problem. The obtained results show that only few terms are required to deduce approximated solutions which are found to be accurate and efficient.

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Pourabdian, Majid, Pourya Omidvar, and Mohammad Reza Morad. "Multiphase simulation of liquid jet breakup using smoothed particle hydrodynamics." International Journal of Modern Physics C 28, no. 04 (2017): 1750054. http://dx.doi.org/10.1142/s0129183117500541.

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This paper deals with numerical modeling of two-phase liquid jet breakup using the smoothed particle hydrodynamics (SPH) method. Simulation of multiphase flows involving fluids with a high-density ratio causes large pressure gradients at the interface and subsequently divergence of numerical solutions. A modified procedure extended by Monaghan and Rafiee is employed to stabilize the sharp interface between the fluids. Various test cases such as Rayleigh–Taylor instability, two-phase still water and air bubble rising in water have been conducted, by which the capability of accurately capturing the physics of multiphase flows is verified. The results of these simulations are in a good agreement with analytical and previous numerical solutions. Finally, the simulation of the breakup process of liquid jet into surrounding air is accomplished. The whole numerical solutions are accomplished for both Wendland and cubic spline kernel functions and Wendland kernel function gave more accurate results. Length of liquid breakup in Rayleigh regime is calculated for various flow conditions such as different Reynolds and Weber numbers. The results of breakup length demonstrate in satisfactory agreement with the experimental correlation. Finally, impinging distance and breakup length of a simple multijet setup are analyzed. The two-jet multijet has a longer breakup length than a three-jet one.

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Dutykh, Denys. "Numerical Simulation of Feller’s Diffusion Equation." Mathematics 7, no. 11 (2019): 1067. http://dx.doi.org/10.3390/math7111067.

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This article is devoted to Feller’s diffusion equation, which arises naturally in probability and physics (e.g., wave turbulence theory). If discretized naively, this equation may represent serious numerical difficulties since the diffusion coefficient is practically unbounded and most of its solutions are weakly divergent at the origin. In order to overcome these difficulties, we reformulate this equation using some ideas from the Lagrangian fluid mechanics. This allows us to obtain a numerical scheme with a rather generous stability condition. Finally, the algorithm admits an elegant implementation, and the corresponding Matlab code is provided with this article under an open source license.

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Singh, Jashanpreet, and Chanpreet Singh. "Numerical analysis of heat dissipation from a heated vertical cylinder by natural convection." Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 231, no. 3 (2015): 405–13. http://dx.doi.org/10.1177/0954408915600109.

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Natural convection heat transfer from a hot vertical hollow brass cylinder has been studied experimentally and numerically. The governing equations of continuity, momentum and energy are discretised by using an implicit finite difference technique. The velocity and temperature profiles, boundary layer thickness, local and average heat transfer coefficient are obtained using the numerical simulation. The predictions of the numerical simulation are compared with the experiments conducted on a laboratory-scale apparatus and with the results obtained from analytical solutions available in literature. The numerical simulation results are obtained for two fluids; air and water vapour whereas the experiments are conducted for air only. The induced flow is laminar in both the simulation and the experiments. The dependence of boundary layer thickness on Prandtl number is discussed. The numerically obtained Nusselt number is found quite close to the analytical one. The results show the heat dissipation from the cylinder to surrounding fluid is higher for air than for water vapour. The various factors that affect the comparison of the experimental results with the numerical simulation are discussed.

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Portavoce, Alain, Ivan Blum, Lee Chow, Jean Bernardini, and Dominique Mangelinck. "Numerical Simulation Support for Diffusion Coefficient Measurements in Polycrystalline Thin Films." Defect and Diffusion Forum 309-310 (March 2011): 63–72. http://dx.doi.org/10.4028/www.scientific.net/ddf.309-310.63.

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The measurement of diffusion coefficients in today’s materials is complicated by the down scaling of the studied structures (nanometric effects in thin films, nano-crystalline layers, etc.) and by the complex production process conditions of industrial samples or structures (temperature variations, complex solute and point defect distributions, stress gradients, etc.). Often diffusion measurements have to be performed in samples for which initial experimental conditions do not offer the possibility of using conventional diffusion analytical solutions. Furthermore, phenomena involved with diffusion are sometimes so numerous and complex (stress, matrix composition inhomogeneities, time dependence of point defect generation sources, electrical effects, clustering effects, etc…) that the use of analytical solutions to solve the observed diffusion behavior is difficult. However, simulations can be of use in these cases. They are time consuming compared to the use of analytical solutions, but are more flexible regarding initial conditions and problem complexity. The use of simulations in order to model physical phenomena is quite common nowadays, and highly complex models have been developed. However, two types of simulations have to be considered: i) simulations aiming to understand and predict phenomena, and ii) simulations for measurement purposes, aiming to extract the (average) value of a physical parameter from experimental data. These two cases have different constrains. In the second case, that is the subject of this article, one of the most important stress is that the simulation has to precisely scale the experiment (sample size, experiment duration, etc.), sometimes preventing the measurement due to computational time consumption. Furthermore, the simpler the model (small number of parameters) used in the simulation, the more relevant the measurement (minimum error). In this paper, examples of recent works using two- and three-dimensional finite element simulations for diffusion coefficient measurements in thin polycrystalline films and nano-crystalline layers are presented. The possible use of simulations for diffusion coefficient measurements considering GB migration, GB segregation, or triple junctions is also discussed.

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Lu, Chun, Xiaohua Ding, and Mingzhu Liu. "Numerical simulation of periodic solutions for a class of numerical discretization neural networks." Mathematical and Computer Modelling 52, no. 1-2 (2010): 386–96. http://dx.doi.org/10.1016/j.mcm.2010.03.005.

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Galybin, A. N. "Numerical solutions for polygonal cracks." International Journal of Fracture 131, no. 2 (2005): L15—L20. http://dx.doi.org/10.1007/s10704-005-2595-x.

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Azis, Mohammad Ivan. "NUMERICAL SOLUTIONS FOR 2D UNSTEADY LAPLACE-TYPE PROBLEMS OF ANISOTROPIC FUNCTIONALLY GRADED MATERIALS." Mathematical Modelling and Analysis 27, no. 2 (2022): 303–21. http://dx.doi.org/10.3846/mma.2022.14463.

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The time-dependent Laplace-type equation of variable coefficients for anisotropic inhomogeneous media is discussed in this paper. Numerical solutions to problems which are governed by the equation are sought by using a combined Laplace transform and boundary element method. The variable coefficients equation is transformed to a constant coefficients equation. The constant coefficients equation after being Laplace transformed is then written in a boundary-only integral equation involving a time-free fundamental solution. The boundary integral equation is therefore employed to find the numerical solutions using a standard boundary element method. Finally the numerical results are inversely transformed numerically using the Stehfest formula to obtain solutions in the time variable. Some problems of anisotropic functionally graded media are considered. The results show that the combined Laplace transform and boundary element method is accurate and easy to implement.

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Grbčić, Luka, Lado Kranjčević, Ivan Filiplić, and Kristijan Mavrić. "Numerical Simulation of River Inflows in Rijeka Bay Coastal Area." Journal of Maritime & Transportation Science 3, no. 3 (2020): 117–24. http://dx.doi.org/10.18048/2020.00.08.

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In this paper, a model of water flow in the Porto Baroš has been developed, which is the part of the Rijeka coastal area, for the purpose of its renovation and conversion. For numerical simulation purposes, the depth of the seabed of Port was previously performed, based on which the geometry and numerical domain of Port were made. By conducting the flow simulation, the analysis was carried out, after which the analyses of the conceptual solutions with the introduction of the pipe discharge were performed with the aim of reducing the water pollution of the Porto Baroš area. Port geometry will be made in commercial SMS software and numerical domains and simulations in OpenFOAM open-source software.

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Raisal, Abu Yazid, Rosynanda Nur Fauziah, and Heru Kuswanto. "SIMULATION OF FREE ENERGY OF MIXING FOR A POLYMER SOLUTION USING A SPREADSHEET FOR LEARNING ACTIVITIES." Jurnal Pendidikan Fisika 12, no. 2 (2023): 165. http://dx.doi.org/10.24114/jpf.v12i2.52810.

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Teaching physics is sometimes difficult to convey without using props or visualizations. Simulations can help teachers create and demonstrate real conditions in front of the class. In this article, we describe the usage of spreadsheets in simulating Gibbs free energy in mixing polymer solutions. We have created an model for the simulation consisting of a main spreadsheet and several secondary spreadsheets. A spreadsheet was chosen to simulate Gibbs free energy because spreadsheets can perform numerical representations in tables. This simple simulation can be used when discussing the topic of polymer thermodynamics. Teachers can start by deriving mathematical equations and then show simulations to visualize the equations. Another option is for students to be asked to create their simulations after deriving a mathematical equation. Using simulation in learning can make learning more interactive and help students understand material physics subjects more easily. Spreadsheets can be an alternative for teachers when explaining abstract material to students. Furthermore, simulations with spreadsheets can also support physics learning remotely.

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Dashkov A.S., Kostromin N. A., Babichev A. V., Goray L.I., and Egorov A. Yu. "Simulation of the energy-band structure of superlattice of quaternary alloys of diluted nitrides." Semiconductors 57, no. 3 (2023): 203. http://dx.doi.org/10.21883/sc.2023.03.56237.4163.

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The paper describes an algorithm for computing the interband transition energy for superlattices of quaternary solid solutions of diluted nitrides. Using the described method, the authors have conducted several numerical simulations of test structures with InGaAsN quantum wells for the method verification using experimental data and comparison with other approaches. Simulation results showed the validity of the used approach. The hybridization parameter estimation method for Indium mole-fraction below 30% is presented. Based on simulation results, the authors propose InGaAs/GaAsN superlattices' parameters for the implementation of the source emitting in the 1.3 μm spectral range Keywords: superlattices, diluted nitrides, interband transitions, numerical simulations, hybridization parameter.

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Phan, Thành Trung. "Mô phỏng giải pháp bảo vệ kết cấu bê tông cốt thép dưới tác dụng của tải trọng nổ tiếp xúc". Vietnam Institute for Building Science and Technology 2023, vi.vol3 (2023): 20–26. http://dx.doi.org/10.59382/j-ibst.2023.vi.vol3-3.

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The paper aim to assess the fracture failure mode of reinforced concrete components under contact blast loading using both on site experiment and numerical simulation. From there, we propose a solution to protect reinforced concrete components under explosive load contact. Based on the results, the selection of computational models, constitutive laws of the material in the simulation of the structure under the impact of blast loading in the ABAQUS program has been evaluated.

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Lu, Chun. "Periodic Solutions for a Numerical Discretization Neural Network." Discrete Dynamics in Nature and Society 2010 (2010): 1–17. http://dx.doi.org/10.1155/2010/239787.

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The existence and global exponential stability of periodic solutions for a class of numerical discretization neural networks are considered. Using coincidence degree theory and Lyapunov method, sufficient conditions for the existence and global exponential stability of periodic solutions are obtained. Numerical simulations are given to illustrate the results.

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Wang, Qi, and Jiechang Wen. "Oscillations of Numerical Solutions for Nonlinear Delay Differential Equations in the Control of Erythropoiesis." Discrete Dynamics in Nature and Society 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/936351.

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We consider the oscillations of numerical solutions for the nonlinear delay differential equations in the control of erythropoiesis. The exponentialθ-method is constructed and some conditions under which the numerical solutions oscillate are presented. Moreover, it is proven that every nonoscillatory numerical solution tends to the equilibrium point of the continuous system. Numerical examples are given to illustrate the main results.

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ZHANG, WEN, and IAN GLADWELL. "A CATALYTIC SURFACE REACTION MODEL: ANALYSIS AND NUMERICAL SIMULATION." International Journal of Bifurcation and Chaos 02, no. 03 (1992): 577–96. http://dx.doi.org/10.1142/s0218127492000719.

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We investigate a reaction-diffusion-convection model describing a catalyzed surface reaction. The model consists of parabolic partial differential equations involving non-diffusive and non-convective variables where the steady state is the solution of a system of ordinary differential algebraic equations. We analyze solution behavior through perturbation methods and exhibit a linear instability induced by the algebraic equations. Also we find multiple front solutions resulting from a jump discontinuity in the algebraic components. Numerical simulations are performed on the original system as well as the reduced system resulting from the singular perturbation technique. In addition to the stationary front solutions and multiple steady states, we find oscillations which closely relate to the stability at the equilibrium of the algebraic manifold of solutions. Experimentally we observe a regular oscillation when there is one Hopf point on the algebraic manifold, a period-doubling when two Hopf points bifurcate from one Hopf point and a chaotic oscillation when two Hopf points separate (in this last case an unstable segment occurs between the Hopf points).

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Cheneke, Kumama Regassa. "Fractional Derivative Model for Analysis of HIV and Malaria Transmission Dynamics." Discrete Dynamics in Nature and Society 2023 (October 16, 2023): 1–17. http://dx.doi.org/10.1155/2023/5894459.

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In this study, a Caputo fractional derivative is employed to develop a model of malaria and HIV transmission dynamics with optimal control. Also, the model’s basic properties are shown, and the basic reproduction number is computed using the next-generation matrix method. Additionally, the order of fractional derivative analysis shows that the infected group decreases at the beginning for the higher-order of fractional derivative. Moreover, the early activation of memory effects through public health education reduces the impact of malaria and HIV infections on further progression and transmission. On the other hand, effective optimal controls reduce the occurrence and prevalence of HIV and malaria infections from the beginning to the end of the investigation. Finally, the numerical simulations are done for the justification of analytical solutions with numerical solutions of the model. Moreover, the MATLAB platform is incorporated for numerical simulation of the solutions.

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DAVIDSON, B. D., and D. E. STEWART. "A NUMERICAL HOMOTOPY METHOD AND INVESTIGATIONS OF A SPRING-MASS SYSTEM." Mathematical Models and Methods in Applied Sciences 03, no. 03 (1993): 395–416. http://dx.doi.org/10.1142/s0218202593000217.

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A numerical technique is developed to determine the behavior of periodic solutions to highly nonlinear non-autonomous systems of ordinary differential equations. The method is based on shooting in conjunction with a probability one homotopy method and an implementation of the topological index. It is shown that solutions may be characterized a priori in terms of an index and this is developed into a powerful numerical and investigative tool. This method is used to investigate the periodic solutions of a nonlinear fourth order system of differential equations. These equations describe the motion of a forced mechanical oscillator and are extremely difficult to evaluate numerically. Solutions are presented which could not be found using local methods. These include flip, saddle node and symmetry breaking pitchfork bifurcations.

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Ratas, Mart, Andrus Salupere, and Jüri Majak. "SOLVING NONLINEAR PDES USING THE HIGHER ORDER HAAR WAVELET METHOD ON NONUNIFORM AND ADAPTIVE GRIDS." Mathematical Modelling and Analysis 26, no. 1 (2021): 147–69. http://dx.doi.org/10.3846/mma.2021.12920.

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The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers’ equation, the Korteweg–de Vries equation, the modified Korteweg–de Vries equation and the sine–Gordon equation are used as model equations. Adaptive as well as nonadaptive nonuniform grids are developed and used to solve the model equations numerically. The numerical results are compared to the known analytical solutions as well as to the numerical solutions obtained by application of the HOHWM on a uniform grid. The proposed methods of using nonuniform grid are shown to significantly increase the accuracy of the HOHWM at the same number of grid points.

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Merdan, Mehmet, Ahmet Gökdoğan, Ahmet Yıldırım, and Syed Tauseef Mohyud-Din. "Numerical Simulation of Fractional Fornberg-Whitham Equation by Differential Transformation Method." Abstract and Applied Analysis 2012 (2012): 1–8. http://dx.doi.org/10.1155/2012/965367.

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An approximate analytical solution of fractional Fornberg-Whitham equation was obtained with the help of the two-dimensional differential transformation method (DTM). It is indicated that the solutions obtained by the two-dimensional DTM are reliable and present an effective method for strongly nonlinear partial equations. Exact solutions can also be obtained from the known forms of the series solutions.

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Kadyirov, Aidar, Rustem Gataullin, and Julia Karaeva. "Numerical Simulation of Polymer Solutions in a Single-Screw Extruder." Applied Sciences 9, no. 24 (2019): 5423. http://dx.doi.org/10.3390/app9245423.

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Single-screw extruders are the most common equipment used for polymer extrusion. The study of the hydrodynamics of a polymer melts flow in the extruder channel is the basis for modeling and understanding the extrusion process. In general form, the extruder includes a straight section with a screw installed in it. In this study, the three-dimensional mathematical modeling of the polymer solutions flow in the metering zone of a single-screw extruder is performed. The influences of the screw geometry (L/D2 = 1…3) on the flow structure and the pressure drop are analyzed under a speed rotation up to 60 rpm. Aqueous solutions of 0.5% polyacrylamide (0.5% PAA) and 1.5% sodium salt of carboxymethyl cellulose (1.5% CMC) are considered as the working fluid.

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Klymenko, O. V., and I. B. Svir. "Modelling Complex Chemical Processes in Homogeneous Solutions: Automatic Numerical Simulation." Nonlinear Analysis: Modelling and Control 11, no. 3 (2006): 247–61. http://dx.doi.org/10.15388/na.2006.11.3.14746.

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Two algorithms for the determination of the necessary limit of local error for the numerical solution of ordinary differential equation (ODE) systems describing homogeneous chemical and biochemical processes, and for the evaluation of their stiffness are developed. The approach for finding the necessary limit of local error of a numerical ODE solver is justified by the proof of the corresponding theorems. The application of the new algorithms implemented in version 2.1 of KinFitSim software to the simulation of real chemical systems is considered on the example of Belousov-Zhabotinsky reaction.

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Potapov, V. V., I. A. Kashutina, and E. V. Shunina. "Numerical simulation for polycondensation of orthosilicic acid in hydrothermal solutions." Journal of Volcanology and Seismology 10, no. 5 (2016): 320–31. http://dx.doi.org/10.1134/s0742046316050067.

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Journal articles on the topic 'Numerical simulation of solutions'

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