modern research physics

---

This paper presents an appropriate numerical method to solve nonlinear Fredholm integro-differential equations with time delay. Its approach is based on the Taylor expansion. This method converts the integro-differential equation and the given conditions into the matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Taylor expansion coefficients, so that the solution of this system yields the Taylor expansion coefficients of the solution function. Then, the performance of the method is evaluated with some examples

---

In this paper, we use operational matrices of piecewise constant orthog-
onal functions on the interval [0,1] to solve fractional differential , integral
and integro-differential equations without solving any system. We first ob-
tain Laplace transform of the problem and then we find numerical inversion
of Laplace transform by operational matrices. Numerical examples show
that the approximate solutions have a good degree of accuracy.

---

In this paper, a collocation method based on the Bessel polynomials is used for the solution of nonlinear Fredholm-Volterra-Hammerstein integro-differential equations (FVHIDEs) under mixed condition. This method of estimating the solution, transforms the nonlinear (FVHIDEs) to matrix equations with the help of Bessel polynomials of the first kind and collocation points. The matrix equations correspond to a system of nonlinear algebraic equations with the unknown Bessel coefficients. Present results and comparisons demonstrate that our estimate has good degree of accuracy and this method is more valid and useful than other methods.In this paper, a collocation method based on the Bessel polynomials is used for the solution of nonlinear Fredholm-Volterra-Hammerstein integro-differential equations (FVHIDEs) under mixed condition. This method of estimating the solution, transforms the nonlinear (FVHIDEs) to matrix equations with the help of Bessel polynomials of the first kind and collocation points. The matrix equations correspond to a system of nonlinear algebraic equations with the unknown Bessel coefficients. Present results and comparisons demonstrate that our estimate has good degree of accuracy and this method is more valid and useful than other methods.

---

Diffusion Processes such as Brownian motions and Ornstein-Uhlenbeck processes are the classes of stochastic processes that have been under considerations of the researchers in various scientific disciplines including biological sciences. It is usually assumed that the outcomes of these processes
are lied on the Euclidean spaces. However, some data are appeared in physical, chemical and biological phenomena that cannot be considered as the observations in Euclidean spaces due to various features
such as the periodicity of the data. Hence, we cannot analysis them using the common mathematical methods available in Euclidean spaces. In addition, studying and analyzing them using common linear statistics are not possible. One of these typical data is the dihedral angles that are utilized to identifying, modeling and predicting the proteins backbones. Because these angles are representatives of points on the surface of torus, it seems that proper statistical modeling of diffusion processes on the torus could be of a great help for the research activities on dynamic molecular simulations in predicting the proteins backbones. In this article, using the Riemannian distance on the torus, the stochastic differential equations to describe the Brownian motions and Ornstein-Uhlenbeck processes on this geometrical objects will be derived. Then, in order to evaluate the proposed models, the statistical simulations will be performed using the equilibrium distributions of aforementioned stochastic processes. Moreover, the link between the gained results with the available concepts in the non-linear statistics will be highlighted.

---

Fractional derivatives and integrals are new concepts of derivatives and integrals of arbitrary order. Partial differential equations whose derivatives can be of fractional order, are called fractional partial differential equations (FPDEs). Recently, these equations have been under special attentions due to their most practical usages. In this paper, we survey a rather general case of FPDE, to obtain a numerical scheme, the fractional derivatives in the equation are replaced by common definitions such as Grundwald-Letnikov, Riemann-Liouville and Caputo, to improve the numerical solution, partial derivatives inside the equation are discrete using non-standard finite difference scheme. Then, we survey the stability of numerical scheme and prove that the proposed method is unconditionally stable. Eventually, in order to approve the theoretical results, we use presented technique to solve wave equation with fractional-order that is very practical and widely used in physics and its branches. Numerical results confirm the findings of the theory and show that this technique is effective.

---

Hybrid of rationalized Haar functions are developed to approximate the solution of the differential equations. The properties of hybrid functions which are the combinations of block-pulse functions and rationalized Haar functions are first presented. These properties together with the Newton-Cotes nodes are then utilized to reduce the differential equations to the solution of algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.