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Due to the non-locality of fractional derivative, the analytical solution and good approximate solution of fractional partial differential equations are usually difficult to get. Reproducing kernel space is a perfect space in studying this type of equations, however the numerical results of equations by using the traditional reproducing kernel method (RKM) isn't very good. Based on this problem, we present the piecewise technique in the reproducing kernel space to solve this type of equations. The focus of this paper is to verify the stability and high accuracy of the present method by comparing the absolute error with traditional RKM and study the effect on absolute error for different values of α. Furthermore, we can study the distribution of entire space at a particular time period. Three numerical experiments are provided to verify the efficiency and stability of the proposed method. Meanwhile, it is tested by experiments that the change of the value of α has little effect on its accuracy. [ABSTRACT FROM AUTHOR]

The key of the reproducing kernel method (RKM) to solve the initial boundary value problem is to construct the reproducing kernel meeting the homogenous initial boundary conditions of the considered problems. The usual method is that the initial boundary conditions must be homogeneous and put them into space. Another common method is to put homogeneous or non-homogeneous conditions directly into the operator. In addition, we give a new numerical method of RKM for dealing with initial boundary value problems, homogeneous conditions are put into space, and for nonhomogeneous conditions, we put them into operators. The focus of this paper is to further verify the reliability and accuracy of the latter two methods. Through solving three numerical examples of integral-differential equations and comparing with other methods, we find that the two methods are useful. [ABSTRACT FROM AUTHOR]

Today, most of the real physical world problems can be best modelled with fractional telegraph equation. Besides modelling, the solution techniques and their reliability are the most important. Therefore, high accuracy solutions are always needed. As we all know, reproducing kernel method (RKM) has been successfully presented for solving various ordinary differential equations. However, the numerical results are not perfectly satisfactory when we directly use the traditional RKM for solving fractional partial differential equation. The aim of this paper is to fill this gap. In this paper, a new method is provided for solving fractional telegraph equation in the reproducing kernel space by piecewise technique, which can obtain more accurate solution than traditional method. Three experiments are given to demonstrate the effectiveness of the present method. [ABSTRACT FROM AUTHOR]

Recently, the reproducing kernel Hilbert space methods(RKHSM) (see Wang et al (2011) [2]; Lin and Lin (2010) [3]; Wu and Li (2011) [4]; Zhou et al. (2009) [6]; Jiang and Chen (2014) [7]; Wang et al. (2010) [8]; Du and Cui (2008) [9]; Akram et al. (2013) [10]; Lü and Cui (2010) [11]; Wang et al. (2008) [12]; Yao and Lin (2009) [13]; Geng et al. (2014) [14] ; Arqub et al. (2013) [15]) emerged one after the other. But, a lot of difficult work should be done to deal with multi-point boundary value problems(BVPs). Our work is aimed at giving a new reproducing kernel method for multi-point BVPs. We do not put the homogenization conditions into the reproducing kernel space which can avoid to compute the reproducing kernel satisfying boundary conditions and the orthogonal system. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate that new algorithm has the following advantages: small computational work, fast convergence speed and high precision. [ABSTRACT FROM AUTHOR]