Method For Approximate Solution Research Articles

ECCPA: Calculation of classical and quantum cross sections for elastic collisions of charged particles with atoms

ECCPA: Calculation of classical and quantum cross sections for elastic collisions of charged particles with atoms

The Extended Galerkin Method for Approximate Solutions of Nonlinear Vibration Equations

An extension has been made to the popular Galerkin method by integrating the weighted equation of motion over the time of one period of vibrations to eliminate the harmonics from thee deformation function. A set of successive equations of coupled higher-order vibration amplitudes is resulted, and a nonlinear eigenvalue problem is obtained for the frequency-amplitude dependence of nonlinear vibrations with successive displacements. The subsequent solutions of vibration frequencies and deformation are consistent with other successive approximate methods, such as the harmonics balance method. This is an extension of the Galerkin method which has broad applications for asymptotic solutions, particularly for problems in solid mechanics. This extended Galerkin method can also be utilized for the analysis of free and forced nonlinear vibrations of structures as a new technique with significant advantages in calculations.

Approximate solutions of aggregation and breakage population balance equations

The work aims to present a semi-analytical approach based on the homotopy analysis method for finding closed-form solutions and approximate solutions to aggregation and breakage population balance equations. The population balance equations are specific integro-partial differential equations. In this work, we first transform both the aggregation and breakage models into integral equations. Then the resulting integral equations are solved by the homotopy analysis method (HAM) to get the series solution, which in particular cases eventually converges to the exact solution. The semi-analytical solutions for various benchmark aggregation kernels like the Ruckenstein/Pulvermacher kernel and breakage kernels such as the Austin kernel are provided using HAM.

Numerical Method for Approximate Solution of Fisher's Equation

In this paper, Fisher's reaction diffusion equation has been solved numerically by Strang splitting technique depending on collocation method with cubic B-spline. For our purpose, the initial and boundary value problem consisting of Fisher's equation is split into two sub-problems to be one linear and the other nonlinear such that each one contains the derivative in terms of time. Then, the whole problem is reduced to the algebraic equation system using finite element collocation method combined with the cubic B-spline for spatial discretization and the convenient classical finite difference approaches for time discretization. The effective and efficiency of the newly given method have been shown on the four examples. In addition, the newly obtained numerical results are shown in formats graphical profiles and tables to compare with studies available in the literature.

Hermite Method of Approximate Particular Solutions for Solving Time-Dependent Convection-Diffusion-Reaction Problems

This article describes the development of the Hermite method of approximate particular solutions (MAPS) to solve time-dependent convection-diffusion-reaction problems. Using the Crank-Nicholson or the Adams-Moulton method, the time-dependent convection-diffusion-reaction problem is converted into time-independent convection-diffusion-reaction problems for consequent time steps. At each time step, the source term of the time-independent convection-diffusion-reaction problem is approximated by the multiquadric (MQ) particular solution of the biharmonic operator. This is inspired by the Hermite radial basis function collocation method (RBFCM) and traditional MAPS. Therefore, the resultant system matrix is symmetric. Comparisons are made for the solutions of the traditional/Hermite MAPS and RBFCM. The results demonstrate that the Hermite MAPS is the most accurate and stable one for the shape parameter. Finally, the proposed method is applied for solving a nonlinear time-dependent convection-diffusion-reaction problem.

A moving boundary model for oxygen diffusion in a sick cell

In this paper we use an approximate analytical method for numerical solution of one dimensional moving boundary problem. We consider the oxygen diffusion problem where the oxygen is allowed to diffuse into a sick cell which absorbs and immobilizes oxygen at a constant rate. Our main problem consists in tracking the moving boundary that represents the oxygen penetration depth inside the sick cell. We can find an accurate solution which is obtained by a polynomial of third and fourth degree and we show some mistakes in the paper published by Seval Çatal in (App.Math.Comput 145:361–369, 2003).

Scheduling in a flexible flow shop with unrelated parallel machines and machine-dependent process stages: Trade-off between Makespan and production costs

This paper aims to tackle bi-objective scheduling problem in a flexible flow shop containing unrelated parallel machines in the first stage. Due to the different technology levels of the parallel machines, their process speeds and production costs vary to each other. Therefore, minimizing the maximum completion time (Makespan) and the total production cost are considered as two objective functions. In addition, setup times are considered sequence-dependent and this system considers machine-dependent process steps and the process steps of an order depend on the assigned machine in the first stage. First, the problem is described and formulated as a bi-objective mathematical model. Since the problem is known to be strongly NP-hard, an approximate solution method is introduced based on the Non-dominated Sorting Genetic Algorithm (NSGA-II) to provide proper solutions for decision makers. The performance of the proposed solution method is investigated in comparison to another powerful multi-objective algorithm (SPEA 2) by solving different test problems. The computational results using various metrics such as Error Ratio (ER) and Generational Distance (GD) show the effectiveness of the proposed method in terms of optimality. The other indices such as Spacing (S), Diversification (D), and Mean Ideal Distance (MID) emphasize the superiority of the proposed algorithm compared to the rival algorithm in solving medium and large instances. In addition, supplementary analysis provided proper trade-off between two objectives for managers to select the best solution based on their preferences.

Optimal Feedback Control of Cancer Chemotherapy Using Hamilton–Jacobi–Bellman Equation

Cancer chemotherapy has been the most common cancer treatment. However, it has side effects that kill both tumor cells and immune cells, which can ravage the patient’s immune system. Chemotherapy should be administered depending on the patient’s immunity as well as the level of cancer cells. Thus, we need to design an efficient treatment protocol. In this work, we study a feedback control problem of tumor‐immune system to design an optimal chemotherapy strategy. For this, we first propose a mathematical model of tumor‐immune interactions and conduct stability analysis of two equilibria. Next, the feedback control is found by solving the Hamilton–Jacobi–Bellman (HJB) equation. Here, we use an upwind finite‐difference method for a numerical approximate solution of the HJB equation. Numerical simulations show that the feedback control can help determine the treatment protocol of chemotherapy for tumor and immune cells depending on the side effects.

Numerical method for approximate solution of problems of modeling physical characteristics of radio-absorbing materials

Numerical method for approximate solution of problems of modeling physical characteristics of radio-absorbing materials

The method of approximate solution of a system of differential equations from the Ramsey-Kass-Koopmans model, based on the solution in quadratures of one subclass of similar systems

The method of approximate solution of a system of differential equations from the Ramsey-Kass-Koopmans model, based on the solution in quadratures of one subclass of similar systems

Variable Piecewise Interpolation Solution of the Transport Equation

In this paper, we construct a piecewise interpolation method of approximate solution of the transport equation based on the Newton interpolation polynomial of two variables. We transform the polynomial to the algebraic form with numerical coefficients; this leads us to a sequence of iterations, which improves the accuracy of the approximation. The method is implemented in software and numerical experiments are performed. The possibility of generalizations to systems of partial differential equations and integro-differential equations is discussed.

Comparison of cooling strategies in simulated annealing algorithms for flow-shop scheduling

Flow-shop scheduling is considered. The order of operations must be the same for each job to minimize the maximum completion time. The Simulated Annealing algorithm is a standard approximate solution method in scheduling and optimization in general. Since the algorithm depends on cooling as a heuristic to generate better approx- imations, choosing the strategy with which the temperature decreases can affect the final result.

Optimization of Renewable Energy Project Portfolio Using a Developed Approximate Solution Method, an International Case Study

Optimization of Renewable Energy Project Portfolio Using a Developed Approximate Solution Method, an International Case Study

Prediction of Breakdown Field Strength for Large-area and Multi-layer Film Dielectrics

The polymer film dielectric has been widely used in high voltage equipment for bulk insulation owing to the excellent insulation performance. Martin&#x2019;s empirical formula for bulk breakdown is frequently used to predict the breakdown field strength of multilayer thin film with different electrode areas. In this article, based on the academic consensus that the cumulative breakdown probability of single-layer film obeys the Weibull distribution, the relationship between the cumulative breakdown probability of film dielectric under different electrode areas and the applied electrical field is deduced. Moreover, the relationship between the cumulative breakdown probability of film dielectric with various numbers of film layers and the applied electrical field is obtained and simplified. On these bases, combined with the experimental results of the average breakdown field of film dielectric with smaller electrode areas and fewer layers, a prediction model of the average breakdown field strength of film dielectric with different electrode areas and film layers is established. To obtain the solution of the prediction model concerning the number of film layer, an effective approximate solution method is proposed to solve the calculation problem with ultralarge and ultrasmall data, which significantly reduces the complexity of the solution process. A coaxial film capacitor is developed and tested to verify the prediction model with the &#x201C;Chenguang&#x201D; accelerator. The experimental breakdown field strength of 54 layers of 15 <inline-formula> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> biaxially oriented polypropylene (BOPP) film with an electrode area of 218.1 cm² is 355.6 V/<inline-formula> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula>, which is comparable to the predicted result of 374 V/<inline-formula> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula>. The prediction model presented in this article is of certain reference significance for the insulation design of film dielectrics.

New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method

Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor’s series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implemented to other TSFDEs applied in the mathematical modeling of a real-world aspect.

Meshless method of approximate particular solution for an initial and boundary value problem of the Korteweg–de Vries type equation and eventual periodicity

In this work, we approximate the solution of an IBVP of the Korteweg–de Vries type equation with periodic forcing at the boundary using the meshless method of approximate particular solution (MMAPS). This method is based on radial basis functions (RBFs). Further, it is shown that the solution exhibits the eventual periodicity and method approximates the solution accurately and efficiently. To test the accuracy of the method the conserved quantities are also computed for the KdV equation. The accuracy of the method depends on the shape parameter and the optimal value of the shape parameter is also approximated.

Bernstein operator method for approximate solution of singularly perturbed Volterra integral equations

An approximate solution of integral equations takes an active role in the numerical analysis. This paper presents and tests an algorithm for the approximate solution of singularly perturbed Volterra integral equations via the Bernstein approximation technique. The method of computing the numerical approximation of the solution is properly demonstrated and exemplified in the matrix notation. Besides, the error bound and convergence associated with the numerical scheme are constituted. Finally, particular examples indicate the dependability and numerical capability of the introduced scheme in comparison with other numerical techniques.

Fractal-fractional Anthroponotic Cutaneous Leishmania model study in sense of Caputo derivative

The current paper is the analysis of an Anthroponotic Cutaneous Leishmania infection caused by a parasite known as Leishmania Tropica under the Caputo fractal-fractional operator. The problem has seven compartmental agents having two categories of human and vector population. By using the fixed point theory we show the existence and uniqueness of solutions of the system. Applying the methods of basic theorems of fractal-fractional calculus and the iterative numerical techniques of fractional Adams–Bashforth method for approximate solution. For the simulation of the model, we have considered different values for fractional-order ϱ and fractal dimension χ and compare the results with integer order for real data. The fractal-fractional calculus technique is applied as a beneficial technique to know about the real-world problem and also to controls the whole world situation of the aforesaid pandemic in the different continents and territories of the world. Finally, numerical simulations are performed to elaborate on the significance of the arbitrary order derivative. Our analysis indicates that the fractal-fractional order derivatives are more informative about the complexity of the dynamics of the proposed Leishmania model.

Galerkin Approximations for the Solution of Fredholm Volterra Integral Equation of Second Kind

In this research, we have introduced Galerkin method for finding approximate solutions of Fredholm Volterra Integral Equation (FVIE) of 2nd kind, and this method shows the result in respect of the linear combinations of basis polynomials. Here, BF (product of Bernstein and Fibonacci polynomials), CH (product of Chebyshev and Hermite polynomials), CL (product of Chebyshev and Laguerre polynomials), FL (product of Fibonacci and Laguerre polynomials) and LLE (product of Legendre and Laguerre polynomials) polynomials are established and considered as basis function in Galerkin method. Also, we have tried to observe the behavior of all these approximate solutions finding from Galerkin method for different problems and then a comparison is shown using some standard error estimations. In addition, we observe the error graphs of numerical solutions in Galerkin method for different problems of FVIE of second kind. GANITJ. Bangladesh Math. Soc.41.1 (2021) 1–14

Solution of linear and non-linear partial differential equation of fractional order

We know that the solution of partial differential equations by analytical method is better than the solution by approximate or series solution method. In this paper, we discuss the solution of linear and non-linear fractional partial differential equations involving derivatives with respect to time or space variables by converting them into the partial differential equations of integer order. Also we develop an analytical formulation to solve such fractional partial differential equations. Moreover, we discuss the method to solve the fractional partial differential equations in space as well as time variables simultaneously with the help of some examples.