Approximate Numerical Solutions Research Articles

Control of a nonlinear wave equation with a dynamic boundary condition

Existence, uniqueness, energy decay, and approximate numerical solution for the nonlinear wave equation with dynamic control at the boundary is being studied in this work. The theoretical analysis of the problem will be conducted using the Faedo-Galerkin method and compactness results. To obtain the approximate numerical solution, a combined approach of the finite element method and a finite difference method will be employed, known as the linearized Crank-Nicolson Galerkin method. This method optimizes the calculations and preserves the quadratic order of convergence in time. Finally, numerical experiments are performed, and tables and graphs are presented to illustrate the theoretical convergence rates and demonstrate the consistency between theoretical and numerical results.

Theoretical and numerical study of a Burgers viscous equation type with moving boundary

In this article, we investigate the existence, uniqueness, and numerical aspects of a one‐ and two‐dimensional nonlinear viscous type Burgers problem defined in a noncylindrical domain. In order to obtain the existence and uniqueness of the solution, the problem with a moving ends is transformed into an equivalent problem in a cylindrical through a diffeomorphism between the domains. The numerical simulation for the one‐ and two‐dimensional cases is performed using Lagrange with degrees 1–3 and cubic Hermite polynomials as base functions for applying the linearized Crank–Nicolson–Galerkin method to obtain an approximate numerical solution. Graphs prove the efficiency of the numerical method along with the order of numerical convergence consistent with the degree of the base polynomial.

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.

Operational matrix method approach for fractional partial differential-equations

Abstract Fractional partial differential equations (FPDEs) have become very popular to model and analyze numerous different physical phenomena in recent years. However, it is generally complicated to find the exact solutions of those FPDEs. The objective of this study is to find the approximate numerical solution of FPDEs by introducing a wavelet-based operational matrix technique. In this study we employ Hermite wavelets (HWs) and the operational matrices of the fractional integration for Hermite wavelets. The sparsity of the obtained operational matrices provides fast and efficient computation of the proposed method. The original FPDE equations are converted to Sylvester equations, which then are solved to obtain the final solution. We provide a few numerical examples to demonstrate the versatility and efficiency of the proposed method.

Degenerate Parabolic Equation with Zero-flux Boundary Condition and its Approximations

We study a degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The aim of this paper is to prove convergence of numerical approximate solutions towards the unique entropy solution. We propose an implicit finite volume scheme on admissible mesh. We establish fundamental estimates and prove that the approximate solution converge towards an entropy-process solution. Contrarily to the case of Dirichlet condition, in zero-flux problem unnatural boundary regularity of the flux is required to establish that entropyprocess solution is the unique entropy solution. In the study of well-posedness of the problem, tools of nonlinear semigroup theory (stationary, mild and integral solutions) were used in order to overcome this difficulty. Indeed, in some situations including the one-dimensional setting, solutions of the stationary problem enjoy additional boundary regularity. Here, similar arguments are developed based on the new notion of integral-process solution that we introduce for this purpose.

Numerical search for the effective thermal conductivity of cracked media

Spacecraft parts accumulate damage during operation and defects that are invariably present even in new designs may grow. This leads to changes in the behavior of individual parts of the space vehicle and, consequently, to the risk of fracture. A more accurate assessment of spacecraft safety requires internal defects to be included in the material models under consideration. One of the main hazardous effects on space objects is multiple temperature heating and cooling due to periodic action of solar rays. This paper presents a study of thermal conduction of media containing cracks. It is carried out with the help of a technique developed by the authors to determine the effective thermal conductivity of materials and based on approximate numerical solution of the steady-state thermal conduction problem for a three-dimensional medium with cracks by the boundary element method. This technique allows to obtain the distribution of the temperature field and heat flux density at any point of the body under consideration, as well as to calculate the effective parameters of materials with high accuracy at relatively low calculation time using ordinary personal computers of average power. The basis of the numerical method presented in this paper is the decomposition of the desired solution into a series of some pre-calculated analytical solutions of the heat conduction equations. The dependence of the effective thermal conductivity on the density of thermally insulated cracks was considered. The formula of this dependence is proposed. Verification of the proposed methodology was carried out by comparing the numerical results of a number of problems with the results of other authors.

Approximate Solution of Fuzzy Sedimentary Ocean Basin Boundary Value Problem Using Variational Iteration Method

The problem that is considered in this paper is developed from the migration of the coastline over time in sedimentary ocean basins that may be considered as a moving boundary problem with a variable latent heat transfer equation as a governing equation if one of the boundaries is unknown, with boundary and initial conditions are given. This model discusses the shoreline movement within sedimentary basins. The main objective of the paper is to use fuzzy logic to formulate this problem when modelling sand particles migration in saltwater by utilizing fuzzy numbers whenever replacing some of the problem's parameters. In reality, the proposed model is more general than the nonfuzzy or crisp models. By employing the variational iteration method, the approximate numerical solution of the proposed problem has been found, and computer software written in Mathematica 11 have been used to obtain the results.

Numerical Solution for Conformable Fractional PDEs by Using a New Double Conformable Integral Transform-Decomposition Method

The conformable Sumudu-Elzaki Transform Decomposition Method CSETDM is used in this research to handle the approximate numerical solutions for the regular and singular one-dimensional conformable Fractional Coupled Burger's Equation CFCBsE and the Nonlinear Singular Conformable Pseudohyperbolic equation NSCPE. This method is generalized in the sense of conformable derivatives. Essential results and theorems concerning this method are discussed. Some numerical examples are given to exhibit the proposed technique's viability, applicability, and effortlessness.

Forced vibrations of an elastic triangular plate supported around its perimeter by a unilateral support via the Chebyshev polynomial expansion

The present study introduces static and dynamic analysis of an elastic triangular plate on unilateral edge supports, including forced vibrations due to loading and unloading excitation. The plate is assumed to be subjected to a uniformly distributed load and an eccentrically applied concentrated load. Furthermore, the loads are assumed to display dynamic variations. The governing equation of the problem is derived by considering the static and the dynamic responses of the plate by including inertia forces. The analytical solution is conducted by employing a series of Chebyshev polynomials for the admissible displacement functions and using Lagrange’s equations of motion. Having found the governing equation of the problem, an approximate numerical solution is accomplished using an iterative process due to the non-linear properties of the unilateral edge supports. The static behavior of the plate under a concentrated load is investigated in detail numerically by considering a wide range of parameters of the plate geometry and the stiffness of the support. In deriving the governing equation of the problem and in the numerical solution, special care is taken to use nondimensional parameters. This approach ensures that the analysis and conclusions remain valid across a wide range of parameter values. Dynamic treatment of the problem is carried out in the time domain by assuming a stepwise time variation of the concentrated load and by employing the constant acceleration procedure in the numerical solution of the system of governing differential equations derived from the equation of motion. Time variations of the displacements, the contact points, and the reactions of the plate’s support are presented in figures for various values of the parameters of the plate, as well as those of the supports, particularly focusing on the non-linearity of the problem due to the plate liftoff from the unilateral support. The effects of the parameters and the loading are investigated in detail. The results reveal that the unilateral property of the support stiffness significantly affects the static and dynamic behavior of the triangular plate. The analysis can be extended easily to cover a general type of loading as well. The main issues addressed in this study include: the analysis of a triangular plate with unilateral perimeter support that does not resist tensile forces; the use of Chebyshev polynomials as admissible functions in both directions; the consideration of static and dynamic loads; the determination of lifting areas of the support in both cases and the assessment of the global vertical force balance in the plate, including inertia forces in the dynamic case.

On the existence and uniqueness of the solution to multifractional stochastic delay differential equation

In this paper we study existence and uniqueness of solution stochastic differential equations involving fractional integrals driven by Riemann-Liouville multifractional Brownian motion and a standard Brownian. Then, we obtain approximate numerical solution of our problem and colon cancer chemotherapy effect model are presented to confirm our results. We show that considering time dependent Hurst parameters play an important role to get more realistic results.

The Lorenz system as a gradient-like system

We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for 0⩽ρ⩽12 when σ = 10 and β=8/3 , significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.

The impact of rotation on the onset of cellular convective movement in a casson fluid saturated permeable layer with temperature dependent thermal conductivity and viscosity deviations

In this effort, we examined the impact of rotation on the arrival of cellular convective motion in a Casson fluid saturated permeable layer with temperature dependent thermal conductivity and viscosity deviations. The problem is important to cellular foams prepared from plastics, ceramics, and metallic where radiation conductivity is revealed as a power function of temperature. The altered Darcy model is used to characterize the rheological performance of Casson fluid flow in permeable medium. The approximate analytical solution and numerical solution correct to one decimal place are presented utilizing the Galerkin method. The analysis reveals that the influence of thermal conductivity disparity parameter and the rotation is to delay the convective motion whereas; the viscosity disparity parameter and the Casson parameter have dual impact on the convective motion in the presence of rotation. The range of the convective cell drops with increasing the thermal conductivity disparity parameter, the viscosity disparity parameter, the Casson parameter and rotation parameter. In the absence of rotation, the range of the convective cell does not depend on the Casson parameter and the viscosity disparity parameter. Further, the existing results are compared with the existing literature under the particular case of this study.

Verification of Taylor's theorem

Multivariate function calculus is an important part of mathematical analysis courses, and most conclusions can be found and generalized in univariate calculus. However, the biggest difficulty in teaching multivariate calculus lies in its abstraction, such as Taylors theorem, multiple integral regions drawing, and integral variable transformation. At the same time, ordinary differential equations are also one of the basic courses of the profession, and dynamic systems based on ordinary differential equations have extensive applications in mathematical models of continuity problems and optimal control problems. Software such as Mathematica, Python, Matlab, etc. can solve similar problems. Therefore, this article will use the visualization and computational capabilities of Mathematica to validate important definitions and conclusions in multivariate calculus, and compare the differences among the three software in solving approximate numerical solutions of dynamic systems of ordinary differential equations from different perspectives.

Study the Effect of Suction-Injection parameters Numerically and Analytically for Magneto hydrodynamic Jeffery Hamel Fluid Flow Problem

This paper processes numerically and analytically the magnetohydrodynamic flow among two porous solid plate intersections at an angle or through non-parallel porous walls, which can be interpreted as a mix of Jeffery Hamel fluid flow additives from suction-injection parameters. In fluid mechanics, the equations of governing, which are transferred to the non-linear ordinary differential equation of third order, are modeled instead of the conventional Navier-Stokes equations by Maxwell's electromagnetism. Results obtained from the DTM and the numerical method BVP4c are compared to those resulting from the optimal methods (ODTM), which consist of the collocation differential transform method (CDTM), the sub-domain differential transform method (SDTM), the least squares differential transform method (LSDTM), and the Galerkin differential transform method (GDTM). A comparison is made between the approximate analytical and numerical solutions for different parameters like open angles (ε), Reynolds numbers (Re), injection parameters (S), and Hartmann numbers (Ha). The results demonstrate that the two approaches are very similar.

Flow of nanofluid past a radially stretching disk in an absorbent medium in company of multiple slips, nonlinear radiative heat flux and Arrhenius activation energy

AbstractAn investigation is carried out to analyze the flow, heat and mass transport phenomena of nanofluids over a disk stretched radially in a porous medium. Besides nonlinear radiative heat flux and Arrhenius activation energy, multiple slips are considered in this model. This ensures the novelty of the present research. The main goal of the current investigation is to explore the mutual impacts of nonlinear radiative heat flux, various slip factors and Arrhenius activation energy on flow, heat and mass transfers. The highly nonlinear central equations which are partial differential equations (PDEs) are transformed into nonlinear ordinary ones (ODEs) by employing suitable similarity alterations and then for finding the approximate numerical solutions, shooting technique with 4th order “Runge–Kutta method” are applied. The impact of a variety of parameters on “skin friction coefficient,” “Nusselt number,” “Sherwood number,” velocity of the fluid, heat and mass transfers are computed and presented graphically. It is interesting to observe that fluid velocity diminishes with the growing porosity, suction and velocity slip parameters. Nanofluid's temperature reduces for enhanced values of thermal slip parameter but for rising thermal radiation parameter, temperature is boosted up. Also, nanoparticle concentration diminishes for increasing values of mass slip parameter.

A fresh perspective on solution to fragmentation problems

The population balances offer a continuum framework to model fragmentation of discrete size drops, bubbles, particles, crystals, polymer chains, etc. We have examined the available approaches to solve the resulting population balances and the older discrete kinetic equations. The analytical solutions have enhanced our understanding of breakup processes. We offer in this work new analytical solutions using a fresh and counter-intuitive approach. The approximate numerical solutions approach analytical solutions as the fineness of discretization increases. We looked for scaling patterns in coarse numerical solutions and harnessed the observed scaling to obtain closed-form analytical solutions in the limit of infinitely fine discretization. The approach provides analytical solutions for a family of breakup functions, including evolving populations of infinite number of particles. The success of the new approach is contingent on the discretization step having internal consistency concerning particle number and mass. The approach holds promise for empirical breakage functions.

Honeycomb-configured dissipative nanofluid flow within a squeezed channel with entropy generation: regression and numerical evaluations

PurposeNanosized honeycomb-configured materials are used in modern technology, thermal science and chemical engineering due to their high ultra thermic relevance. This study aims to scrutinize the heat transmission features of magnetohydrodynamic (MHD) honeycomb-structured graphene nanofluid flow within two squeezed parallel plates under Joule dissipation and solar thermal radiation impacts.Design/methodology/approachMass, energy and momentum preservation laws are assumed to find the mathematical model. A set of unified ordinary differential equations with nonlinear behavior is used to express the correlated partial differential equations of the established models, adopting a reasonable similarity adjustment. An approximate convergent numerical solution to these equations is evaluated by the shooting scheme with the Runge–Kutta–Fehlberg (RKF45) technique.FindingsThe impression of pertinent evolving parameters on the temperature, fluid velocity, entropy generation, skin friction coefficients and the heat transference rate is explored. Further, the significance of the irreversibility nature of heat transfer due to evolving flow parameters are evaluated. It is noted that the heat transference rate performance is improved due to the imposition of the allied magnetic field, Joule dissipation, heat absorption, squeezing and thermal buoyancy parameters. The entropy generation upsurges due to rising magnetic field strength while its intensification is declined by enhancing the porosity parameter.Originality/valueThe uniqueness of this research work is the numerical evaluation of MHD honeycomb-structured graphene nanofluid flow within two squeezed parallel plates under Joule dissipation and solar thermal radiation impacts. Furthermore, regression models are devised to forecast the correlation between the rate of thermal heat transmission and persistent flow parameters.

Variable order fractional diabetes models: numerical treatment

ABSTRACT In this paper, we present two numerical approximation solutions for the updated nonlinear variable order fractional differential equations of diabetes disease. The variable order fractional differential operator is based on the Caputo fractional derivative concept. The first used method is developed using the nonstandard finite difference process, such that a new quick approximation is applied to tackle the nonlinear terms. By applying some reasonable assumptions to the given data, we can prove that this scheme maintains the positivity and boundedness of the discretized solutions. The second method is based on approximation with shifted Jacobi polynomials. The properties of Jacobi polynomials, together with the shifted Jacobi-Gauss-Lobatto nodes were utilized to reduce the proposed system into a set of algebraic nonlinear equations. Furthermore, some numerical experiments are conducted to compare the performance of the introduced schemes. Our numerical simulations show that the nonstandard finite difference approach requires less computational time compared to the Jacobi-Gauss-Lobatto spectral collocation. While the last one is more accurate when used for solving the variable order fractional diabetes model.

Efficient decomposition shooting method for tackling two-point boundary value models

AbstractThis research aims to present an efficient numerical method to find approximate numerical solutions for ordinary differential equations (ODEs), specifically, the class of two-point boundary value models. This approach is predicated on coupling the shooting method and the decomposition method, which the method is named as Efficient Decomposition Shooting Method (EDSM). Thus, we provide the complete outline of EDSM, and thereafter, apply it to some test models, comprising linear and nonlinear models. Further, the results are then validated by comparing them to the exact solution and other computational methods. Thus, the proposed method performs excellently over its peers, and is found to converge rapidly to the available exact solution with a good precision.

A new method for determining crack propagation

This paper presents a new method for obtaining approximate closed-form solutions for crack propagation problems. In general, it is not possible to obtain explicit solutions of the parameters and elementary functions of these problems, due to their terms complexity, present in their definitions. In this way, we used numerical methods to obtain approximate numerical solutions. The method presented in this article provides a proposal, which, when defined by differential equations, is applicable and extensible to other problems. The approximate solutions are obtained as an exact solution of a modified initial value problem. The performance is evaluated in relation to problems with known exact solution.