saedeh foadian

In this paper, a numerical method is proposed for the numerical solution of a linear wave equation with initial and boundary conditions by using the cubic B-spline method to determine the unknown boundary condition. We apply the cubic B-spline for the spatial variable and the derivatives, which generate an ill-posed linear system of equations. In this regard, to overcome, this drawback, we employ the Tikhonov regularization (TR) method for solving the resulting linear system. It is proved that the proposed method has the order of convergence O((Δt)2+h2). Also, the conditional stability by using the Von-Neumann method is established under suitable assumptions. Finally, some numerical experiments are reported to show the efficiency and capability of the proposed method for solving inverse problems.

Although designing and developing a mathematical model is extremely important in the mathematics but finding solution for designing model is essential as well. Thus one cannot propose a model without offering its solutions. In the mathematical modeling, there are many models based on nonlinear partial differential equations. In such models, there is no general method for solving any problem. However, numerical methods, approximate methods or analytical methods are available for some problems. It is clear that among the methods for solving a model based on partial differential equations, analytical methods are preferred, but for all problems, it is not possible to provide an exact solution. In this case, some methods can provide a class of solutions. In such methods, techniques that lead to more solutions are more important, but the use of different methods can provide a wide class of solutions. For this reason, various methods are used to find the possible solution of nonlinear partial differential equations. One of these methods is the expansion method. Since one of the well-known equations with wide application in genetics and gene mutation is the Fisher Kolmogorov-Petrovskii-Piskunov (Fisher KPP) equation, we applied expansion method, for finding exact traveling wave solutions which are based on the solutions of Bernoulli ordinary differential equation.

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.

The Newell-Whitehead-Segel (NWS) equation is an important model arising in fluid mechanics. Various researchers worked on approximate solutions to this model by using different methods. In this paper, the Sine-Cosine wavelets method is applied for solving numerically the NWS equation. The Sine-Cosine wavelet operational matrix of integration is obtained and used to transform the equations into a system of algebraic equations. To demonstrate the effectiveness and applicability of this method, two numerical examples are included.

‎In this paper‎, ‎the Tau method based on shifted Legendre polynomials is proposed for solving a class of fractional stochastic integro-differential equations‎. ‎For this purpose‎, ‎shifted Legendre polynomials and their properties are introduced‎. ‎By using the operational matrices of integration and stochastic Ito-integration we transform the problem into the corresponding linear system of algebraic equations‎. ‎Finally the efficiency of the proposed method is confirmed by some examples‎. ‎The results show that this method is very accurate and efficient‎.

‎In this paper‎, ‎the stochastic weakly singular integro-differential equation is discussed‎. ‎The shifted Legendre Tau method is introduced for finding the unknown function‎. ‎For this purpose‎, ‎shifted Legendre polynomials and their properties are introduced‎. ‎The proposed method is based on expanding the approximate solution as the elements of shifted Legendre polynomials‎. ‎We reduce the problem to set of algebraic equations by using operational matrices‎. ‎Also the convergence analysis of shifted Legendre polynomials and error estimation for this method have been discussed‎. ‎Finally‎, ‎several numerical examples are given to demonstrate the high accuracy of the method‎.

Glucose tolerance test is advised to detection of pre-diabetics and mild diabetics in the medical practice. Several mathematical models such as glucose model, were proposed for mimicking the blood glucose-insulin interaction. To predict accurate insulin requirement for each glucose concentration, we need to solve optimal control problems. In this model, constraints are linear and nonlinear forms of cost function. Although ordinary methods can be used in glucose model, but non-negative natures of medical variables made them difficult to use. To finding a new approximate solution of glucose model, parameterization method with non-negative coefficients in polynomial was used. In this state parameterization method, we use polynomials that ensure that the control variable is non-negative in this model. And increases the time for the model solution to be non-negative compared to conventional methods. The simplicity, lower requirement for mathematical calculations and more compatibility with variables are positive aspects of our parameterization method.