A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-di ff erential equations

: A spectral collocation method is proposed to solve variable order fractional stochastic Volterra integro-di ff erential equations. The new technique relies on shifted fractional order Legendre orthogonal functions outputted by Legendre polynomials. The original equations are approximated using the shifted fractional order Legendre-Gauss-Radau collocation technique. The function describing the Brownian motion is discretized by means of Lagrange interpolation. The integral components are interpolated using Legendre-Gauss-Lobatto quadrature. The approach reveals superiority over other classical techniques, especially when treating problems with non-smooth solutions.


Introduction
Fractional calculus [1] generalizes the standard calculus and has been applied in many fields of engineering and science [2]. Often, finding the exact solutions of fractional order differential equations is not possible, and accurate numerical methods have to be developed.
Stochastic differential equations (SDE) are utilised to simulate a variety of phenomena, including volatile stock prices or systems sensitive to thermal variations. SDEs comprise a variable that corresponds to random white noise, computed as the derivative of a Wiener process or Brownian motion. Other forms of random behaviour, such as jump processes, are also feasible. Stochastic

Fractional calculus
There are various definitions of fractional integration and differentiation, but the most reliable and extensively used are the famous Caputo and Riemann-Liouville formulations. There exist fractional derivatives of fixed, variable, distributed, and tempered orders for both definitions. In this paper, we will look at VOFSV-IDEs with the Caputo formulation.
Definition 2.1. The Caputo fractional derivative of variable order ρ(x) is: where s = ⌈ρ(x)⌉ and the Gamma function Γ(.) is given by e −t t n−1 dt, n > 0.
The Legendre polynomials Ω m (t), m = 0, 1 . . . , obey to the Rodrigues formula: Moreover, Ω m (t) corresponds to a Legendre polynomial [45,46] of degree m and we can get the pth derivative of Ω m (t) as: .
We obtain the orthogonality by: ( where ω(t) = 1, h m = 2 2m+1 . The integrals given above were evaluated efficiently using the Legendre-Gauss-Lobatto quadrature. For ψ ∈ S 2N−1 [−1, 1], we have: (2.5) Let the discrete inner product be given as: In the case of smooth solutions, the employment of classical polynomials in spectral techniques, such as Legendre and Jacobi, suffices to find approximate solutions with great accuracy. On the contrary, in the case of non-smooth solutions, this precision degrades, prompting us to utilize polynomials of fractional orders to prevent this issue, see [26,47,48] for more details.
Definition 2.2. Let SFOLOF, the inferred function from the Legendre polynomial, be provided by [26,47,48] L,r : 0 ≤ r ≤ M}, be the fractional-polynomial space of finite dimensions. Along with the orthogonal characteristic (2.8), the function Θ(ξ) ∈ L 2 Theorem 2.3. The relevant nodes and Christoffel numbers of the shifted fractional Legendre-Gauss (Gauss-Radau or Gauss-Lobatto) interpolation in the interval [0, L] may be obtained from [26,47,48] x (ε) L,K,s = L . (2.9) In addition, the exponential error convergence of fractional Legendre (as a special case of fractional Jacobi) has already been gained, see [47,48]. In the case of a smooth solution, the errors drop exponentially as N → ∞.
2) For 0 ≤ s < t ≤ L the random variable given by the increment ω(t) − ω(s) is normally distributed with zero mean and variance t − s, which means ω(t) − ω(s) ∼ √ t − s N(0, 1), where N(0, 1) denotes a normally distributed random variable with zero mean and unit variance.
A stochastic integral is the integral of some function ψ(t) over some interval [0, L], but with respect to a Brownian motion ω(t) as

VOFSV-IDEs with initial condition
We solve the VOFSV-IDEs: with initial condition: where X(x), H(x, X), Q 1 (ζ, x) and Q 2 (ζ, x), with x ∈ [0, L], denote stochastic processes defined on the probability space (Ω, , P), and X is unknown. Moreover, Therefore, using integration by parts, Eq (3.1) yields: with The solution of Eq (3.2) can be approximated by: Despite the location of the nodes is optional, we take x ε L,Q, j as SFL-GR nodes. The Ω (ε) L, j (x) is given analytically by: By means of Eq (2.1), we find: and, thus, we have: Therefore, we get: (3.7) Using the quadrature, we have: where λ L,R r are the shifted Legendre-Gauss-Lobatto nodes. As a result, the integral terms can be written as: The residual of (3.2) can thus be computed as: which is complied to be zero at M points: where r = 1, 2, . . . , M. Hence, for M + 1 unknowns, we have M algebraic equations. One additional equation may be constructed from the beginning condition (3.2) as follows: Finally, from Eqs (3.11) and (3.12), we have a system of algebraic equations. Consider the discretized Brownian motion for computational purposes, in which ω(x) is given at x discrete values and ω(x) is evaluated using Lagrange interpolation. We have: where ω 0 = 0 with probability 1, and △ω i is a random variable such that △x ε L,Q,i N(0, 1).

Numerical results and assessment
We present numerical results obtained with the new algorithm, illustrating its effectiveness and high accuracy. Therefore, three problems are solved. The algorthim code is run via MATHEMATICA version 12.2.

Problem I
We solve the VOFSV-IDEs: where, for σ = 0, we get X(x) = x 3 . Table 1 shows the maximum absolute error between the approximate and exact solutions, given σ = 0 and ρ(x) = x 2 . We verify that the exact solution is obtained when εM = p, with p denoting the power exponent in the exact solution. Table 2 lists the absolute error for σ = 1 2 and ρ(x) = x 2 , while Figure 1 depicts the numerical and exact solutions for σ = 0, ε = 1 5 and M = Q = 12. The variation of the absolute errors are shown in Figure 2. Additionally, we can see the error degradation in Figure 3. When adopting σ = 1 2 , ρ(x) = x 2 , we get the approximate solution of Problem I as:     Figure 3. M E convergence of problem (4.1), for ε = 1 5 .

Conclusions
A collocation strategy was proposed to solve VOFSV-IDEs. Numerical examples proved the new method's accuracy and applicability. Indeed, the results showed that good precision is achieved even with a small number of points. Other methods need a larger number of points to achieve identical results and, thus, involve a higher computational burden. For example, the Euler-Maruyama approximation arrives at 10 −4 as the best solution with step size 1 512 . It is important to note that the proposed approach also thrives in situations with non-smooth solutions, where the accuracy of many other methods is affected.