numerical solution

The multi-step differential transform method (DTM) adopted from the standard DTM is employed in this case study to solve a model of the transesterification reaction. The DTM is considered in a sequence of time intervals. The accuracy of the proposed method is confirmed by comparing its results with those of the fourth-order RungeKutta (RK4) method. In addition, the experimental results are investigated with the Multi-step DTM to demonstrate the efficiency and effectiveness of these chemical reactions obtained in the laboratory. The present findings confirmed the effectiveness of using the multi-step DTM in validating the chemical models obtained in laboratories.

This paper introduces a new computational method for solving linear and non-linear fractional differential equations (FDEs). Our method essentially consists of the combination of orthonormal Bernoulli polynomials  and the fractional form of the Picard iteration method. We name this method the fractional Bernoulli-Picard iteration method (FBPIM). Unlike the spectral method, the proposed method does not require solving a set of algebraic equations. We also discuss the convergence of the method. Moreover, some numerical examples are included and compared with previously published results to assess both the accuracy and suitability of the developed technique.

The main purpose of this paper is to use the homotopy analytical method in order to solve the linear space-fractional telegraph differential equation. Also, non-standard finite difference method has been used to solve this equation numerically. Next, the concepts of Lie symmetry is established and the symmetries of the equation are calculated. In this article, Matlab software was used for simulation. Numerical results are presented to evaluate the efficiency and usefulness of the proposed method.

In this paper, an implicit second order integro-differential equation governing unsteady motion of a solid particle submerged in a fluid medium and, affected by an arbitrary force field is solved numerically. It is assumed that the particle Reynolds number is quite small to use the well-known Basset kernel for the history force. The implicitness and singularity of the equation are removed by using a hybrid quadrature rule (HQR) and a generalized quadrature rule (GQR), respectively. A recursive plan is used to reduce the required CPU time. Two schemes along with the associated numerical solution algorithms are presented. It is described how the accuracy of the method can be increased in a systematic way. The results obtained by several examples show the effectiveness of the method.

In the present paper, we precisely conduct a quantum calculus method for the numerical solutions of PDEs. A nonlinear Schrödinger equation is considered. Instead of the known classical discretization methods based on the finite difference scheme, Adomian method, and third modified ver-sions, we consider a discretization scheme leading to subdomains according to q-calculus and provide an approximate solution due to a specific value of the parameter q. Error estimates show that q-calculus may produce effi-cient numerical solutions for PDEs. The q-discretization leads effectively to higher orders of convergence provided with faster algorithms. The numer-ical tests are applied to both propagation and interaction of soliton-type solutions.

Our research is about the Sturm-Liouville equation which contains conformable fractional derivatives of order $\alpha \in (0,1]$ in lieu of the ordinary derivatives. First, we present the eigenvalues, eigenfunctions, and nodal points, and the properties of nodal points are used for the reconstruction of an integral equation. Then, the Bernstein technique was utilized to solve the inverse problem, and the approximation of solving this problem was calculated. Finally, the numerical examples were introduced to explain the results. Moreover,  the analogy of this technique is shown in a numerical example with the Chebyshev interpolation technique .

Our work proposes a new numerical method for finding the solution of three-dimensional Volterra-Hammerstein integral equations by using three-dimensional hybrid block-pulse functions and Legendre polynomials. Our integral equation is converted to a system of nonlinear equations. An error bound for the suggested method is established. Eventually, some numerical examples illustrate that our method is feasible and efficient.

In 2010, Alvarez et al. proposed an algorithm for morphological snakes that could detect objects whose edges consist of convex sets and polygonal edges. However, the algorithm may not detect the boundary well if the edges of an object contain a convex set or if there are several separated objects in an image. In this paper, we present two optimal sub-algorithms that are modifications to the Alvarez et al. algorithm. Our algorithms provide optimal edge detection for images and we present examples to demonstrate their effectiveness.

In this article, we use the Haar wavelets (HWs) method to numerically solve the nonlinear Drinfel’d–Sokolov (DS) system. For this purpose, we use an approximation of functions with the help of HWs, and we approximate spatial derivatives using this method. In this regard, to linearize the nonlinear terms of the equations, we use the quasilinearization technique. At the end, to show the effectiveness and accuracy of the method in solving this system one numerical example is provided.

Pricing catastrophe swap as an instrument for insurance companies risk management, has received trivial attention in the previous studies, but in most of them, damage severities caused by the disaster has been considered to be fixed. In this study, through considering jumps for modeling the occurrence of disasters as in Unger [32] and completing it through considering damages caused by natural disasters as stochastic, an integro-differential model was extracted to value catastrophe swap contracts. In determining the swap price changes, the Ito command was followed and to achieve the catastrophic swap model, the generalization of the Black and Scholes modeling method was used. [3]. With regard to the initial and boundary conditions, extracted model does not have an analytical solution; thus, its answer was approximated using the finite difference numerical method and the effect of considering the damage as stochastic on swap value was analyzed. In addition, the model and the extracted numerical solution were separately implemented on the data about the earthquake damage in the United States and Iran. The results showed that prices will experience a regular upward trend until damage growth, damage severities, and occurrence probability of a catastrophe are not so high that the buyer of the swap is forced to pay compensation to the swap’s seller. Of course, the prices will fall sharply as soon as they reach and cross the threshold.

In this work, we consider a collocation method for solving the pantograph-type Volterra Hammerstein integral equations based on the first kind Chebyshev polynomials. We use the Lagrange interpolating polynomial to approximate the solution. The convergence of the presented method has been analyzed by over estimating for error. Finally, some illustrative examples are given to test the accuracy of the method. The presented method is compared with the Legendre Tau method.

In this paper, we will obtain analytical approximate solutions of the HIV infection model of CD4+ T-cells. This model corresponds to a class of nonlinear ordinary differential equation systems. To this end, we combine the Natural transform with the Adomian decomposition method for solving this model. The numerical results obtained by the suggested method are compared with the results obtained by other previous methods. These results indicate that this method agrees with other previous methods.

In this article, two numerical methods based on Touchard and Laguerre polynomials were presented to solve Abel integral (AI) equations. Touchard and Laguerre matrices were utilized to transform Abel integral equations into an algebraic system of linear equations. Solve this system of these equations to obtain Touchard and Laguerre parameters. Four examples are given to demonstrate the presented methods. The solutions were compared with the solutions in the literature.

In this paper we have solved the heat transfer equation by means of the Volterra integral equation and Legendre Wavelets. Since, due to numerical facts, solution of the related partial differential equation is difficult, thus we have applied integral equation model. The integral equation model of this system is a Volterra type of the first kind. These systems are ill posed system, and appropriate method for such systems are wavelets, since wavelets can be generated in the space of solutions. In this work we apply the Legendre wavelets to solve the corresponding integral equation. Numerical implementation of the method is illustrated by benchmark problems originated from heat transfer. The behavior of the initial heat function along with the position axis during the time have been shown through three dimensional plots.

In this work, we investigate fractional version of the Fisher equation and solve it by using an efficient iteration technique based on the Haar wavelet operational matrices. In fact, we convert the nonlinear equation into a Sylvester equation by the Haar wavelet collocation iteration method (HWCIM) to obtain the solution. We provide four numerical examples to illustrate the simplicity and efficiency of the technique.

Algebraic integral equations is a special category of Volterra integral equations system,  that has many applications in physics and engineering. The principal aim of this paper is to serve the numerical solution of an integral algebraic equation by using the Taylor expansion method. In this method, using the Taylor expansion of the unknown function, the algebraic integral equation system becomes a linear equation system of the unknown function and its derivatives. Moreover, the convergence analysis of this method will be shown by preparing some theorems. Several examples are included to illustrate the efficiency and accuracy of the proposed technique and also the results are compared with the different methods.

Many phenomena in various fields of physics are simulated by parabolic partial differential equations with the nonlocal initial conditions, while there are few numerical methods for solving these problems. In this paper, the Ritz–Galerkin method with a new approach is proposed to give the exact and approximate product solution of a parabolic equation with the nonstandard initial conditions. For this purpose, at first, we introduce a function called satisfier function, which satisfies all the initial and boundary conditions. The uniqueness of the satisfier function and its relation to the exact solution are discussed. Then the Ritz–Galerkin method with satisfier function is used to simplify the parabolic partial differential equations to the solution of algebraic equations. Error analysis is worked by using the property of interpolation. The comparisons of the obtained results with the results of other methods show more accuracy in the presented technique.

In this paper, the optimal control of transmission dynamics of hand, foot and mouth disease (HFMD), formulated by a compartmental deterministic SEIPR (Susceptible-Incubation (Exposed)- Infected - Post infection virus shedding - Recovered) model with vaccination and treatment as control parameters is considered. The objective function is based on the combination of minimizing the number of infected individuals and the cost involved in the interventions of vaccination given to the susceptible population and treatment given to the infected population. The existence for the optimal control pair is proved and the characterization of the optimal control pair is obtained by applying the Pontryagin's maximum principle. The variational iteration method is adopted to solve the non-linear Hamilton equations derived from the Pontryagin's maximum principle theory. These equations constitute a two-point boundary value problem. By considering the correction functionals of the Hamilton equations, the Lagrange multipliers are easily identified and practical iteration formulas are derived. An algorithm is developed, based on this formulas, to determine iteratively the solutions of the Hamilton equations with a desired accuracy. With the aid of solutions obtained, the optimal control law can be easily deduced. The results were analyzed and interpreted graphically using Maple.