Cubic B-spline Least-square Method Combine with a Quadratic Weight Function for Solving Integro-Differential Equations

In this article, a numerical scheme was implemented for solving the integro-differential equations (IDEs) with the weakly singular kernel by using a new scheme depend on the cubic B-spline least-square method and a quadratic B-spline as a weight function. The numerical results are in suitable agreement with the exact solutions via calculating and norms errors. Theoretically, we discussed the stability evaluation of the current method using the Von-Neumann method, which explained that this technique is unconditionally stable.


Introduction
The integro-differential equations appear in a wide range of disciplines including physics, chemistry and engineering.
Consider the following IDE with a weakly singular kernel: where − = − , 0 < ! < 1 101 and the cubic B-spline -. , = −1 1 / + 1 , at the knots . which form a basis over the solution domain , , is defined as [13] The set of splines -% , -, … , -D , -D9% forms a basis for functions defined over , . Consider the approximate solution E D , to the exact solution E , given by where H G are unknown time-dependent parameters to be determined from the boundary and weighted residual conditions. We will use the following local coordinate transformation ℎJ = − . , 0 ≤ J ≤ 1, a cubic B-spline shape functions in terms of η over the element [ . , .9% ] that can be defined as all splines apart from -. % , -. , -.9% and -.9 are zero over the element [ . , .9% ] on each time interval [ K , K9% ], ∆ = K9% − K is a local coordinate M, where By using the transformations (7) and (9) in equation (4) The integral equation takes its minimum value with the variation in over each element The least square method turns into a Petrov-Galerkin method with then weight function, the variation of over the element . , .9% defined by Write the weight function as Using the expansion (12) so that Substituting (11) in equation (17), integration with respect to ζ and integration by parts as required leads to the following matrix system of equations for each individual element The equation (18) can be written in a matrix form as follows: where H = H , H % , … , H D q is a global element.
After some simplifications, get |Ϋ| < 1, so cubic B-spline least square method with quadratic weight function for PIDE is unconditionally stable.

Numerical Examples
In this section, we will apply the scheme described in Section 3 to test two examples to demonstrate the efficiency, accuracy, and applicability of the present scheme. Results obtained by this scheme are compared with the analytical solution of each example by computing the maximum norm error and norm error . The exact solution is:

Conclusions
In this paper, we introduced a new numerical scheme to solving the integrodifferential equations with the weakly singular kernel by using the cubic B-spline leastsquare method with quadratic B-spline as a weight function. The method was performed when taking values / = 100, 150, 200, 250 and 300 with ∆ = 0.00001 with a different M, which presented in Tables 1-2. From Figures 1-2, the numerical and the exact solutions are very harmonic which signalizes the numerical solutions effectively. We calculated and norms errors varied to test the accuracy of the proposed method, also, the numerical results are in good agreement with the exact solutions. The proposed method is an effective and unconditionally stable method.