Construction of operational matrices based on linear cardinal B-spline functions for solving fractional stochastic integro-differential equation

Application of flatlet oblique multiwavelets to solve the fractional stochastic integro‐differential equation using Galerkin method

“…In Table 3, the proposed method is compared with cardinal B-spline method [10] and two different methods based on radial basis functions in [57] for 𝜎 = 0, 𝜐 = 0.75, and various values of N. This table illustrates L ∞ , RMS errors, and CPU times for all four compared methods. Considering the results of this table, it is entirely clear that the proposed method [10,57] for Example 1 in cases 𝜎 = 0, 𝜐 = 0.75.…”

“…Example Consider the following FSI‐DE [10, 55, 56]: $$$$ {}_0^C{D}_x^{\upsilon }V(x)=-\frac{x^5{e}^x}{5}+\frac{6{x}^{3-\upsilon }}{\Gamma \left(4-\upsilon \right)}+{\int}_0^x{e}^x sV(s) ds+\sigma {\int}_0^x{e}^x sV(s)d\mathcal{B}(s),s,x\in \left[0,1\right], $$$$ with the initial condition $$$ V(0)=0 $$$ that has not an analytical solution. But, when $$$ \sigma =0 $$$, the problem () has the exact solution as $$$ V(x)={x}^3 $$$.…”

“…In Figure 2, $$$ {\log}_{10}\left(\mathrm{absolute}\ \mathrm{error}\right) $$$ of the proposed method is compared with cardinal B‐spline method [10], the Bernstein [56], and the cubic B‐spline methods [55] for $$$ \upsilon =0.75 $$$ in case $$$ \sigma =0 $$$. The results show that the new scheme for almost the entire interval $$$ \left[0,1\right] $$$ has better performance than the mentioned methods.…”

“…Comparison absolute errors of the proposed method with presented methods in [10, 55, 56] for Example 1 in cases $$$ \sigma =0,\upsilon =0.75 $$$. [Colour figure can be viewed at wileyonlinelibrary.com]…”

“…In recent years, there have been several methods introduced for solving SI-DEs and FSI-DEs. These methods include Bernoulli functions [7], homotopy perturbation method [8], meshless method [9], operational matrices method [10][11][12], and other methods [13][14][15][16][17][18][19][20][21]. FSI-DEs are essential for modeling and analyzing complex systems with memory and randomness.…”