Remarks on a fractional nonlinear partial integro-differential equation via the new generalized multivariate Mittag-Leffler function

Introducing a new generalized multivariate Mittag-Leffler function which is a generalization of the multivariate Mittag-Leffler function, we derive a sufficient condition for the uniqueness of solutions to a brand new boundary value problem of the fractional nonlinear partial integro-differential equation using Banach’s fixed point theorem and Babenko’s technique. This has many potential applications since uniqueness is an important topic in many scientific areas, and the method used clearly opens directions for studying other types of equations and corresponding initial or boundary value problems. In addition, we use Python which is a high-level programming language efficiently dealing with the summation of multi-indices to compute approximate values of the generalized Mittag-Leffler function (it seems impossible to do so by any existing integral representation of the Mittag-Leffler function), and provide an example showing applications of key results derived.

Let a(x) ∈ C[0, T], g : [0, T] × R → R, and f : C[0, T] → R. Very recently, Li [14] studied the uniqueness of solutions for the following nonlinear integro-differential equation with a nonlocal boundary condition and variable coefficients for l < α ≤ l + 1: where λ is a constant.In particular for l = 1, equation (1.1) turns out to be This paper aims to study the uniqueness of solutions for the following new equation with 0 < α ≤ 1 and m = 1, 2, . . ., in the space S([0, 1] 2 ): where all λ i are constants, ψ(x) is a continuous function on [0, 1], and v : [0, 1] 2 × R → R is a function which satisfies conditions to be given.Equation (1.2) with its initial condition is new and, to the best of our knowledge, has never been investigated earlier.
The remainder of the paper is organized in the following manor.Section 3 studies the uniqueness of solutions for equation (1.2) by the newly introduced generalized multivariate Mittag-Leffler function and Banach's fixed point theory.Section 4 presents a demonstrative example which illuminates applications of the key results based on the value of a generalized multivariate Mittag-Leffler function calculated by our Python code.Finally, in Sect.5, we provide a summary of the work.
Definition 1 A generalized multivariate Mittag-Leffler function is defined by the following series: where α i , , δ > 0, In particular, E (0,...,0),1 (α 1 ,...,α m ), (ζ 1 , . . ., which is the multivariate Mittag-Leffler function given in [19] since which is the well-known two-parameter Mittag-Leffler function. Babenko's approach [20] is a highly effective method that can be employed to solve various integral and differential equations [9,17] by treating a bounded integral operator as a "normal" variable and using the inverse operator to deduce solutions.The method itself is similar to the Laplace transform while working on differential and integral equations with constant coefficients, but it can be applied to equations with continuous and bounded variable coefficients.To show this approach, we will consider the following equation in the space C[0, 1] (the space of all continuous functions on [0, 1]) for constants a and b: Obviously, Applying I α 0 to equation (2.1), we have Considering the inverse operator of (1 + aI + bI β+α 0 ), we informally get by Babenko's technique which gives that the series solution

Uniqueness of solutions
Theorem 2 Let ψ ∈ C[0, 1], λ i be real constants, β i , γ i ≥ 0 for all i = 1, 2, . . ., m, and v : [0, 1] 2 × R → R be a continuous and bounded function.In addition, we assume that 0 < α ≤ 1 and Then u(t, x) is a solution to equation (1.2) if and only if it satisfies the following integral equation in the space S([0, 1] 2 ): Proof Applying I α t to equation (1.2), we get This implies that x, u(t, x) , and This further implies that Using Babenko's method, we deduce that by the multinomial theorem.We will now show that u ∈ S([0, 1] 2 ).Indeed, . This marks the completion of the proof.
Then there is a unique solution in the space S([0, 1] 2 ) to equation (1.2).
Proof Let T be the mapping defined on the space S([0, 1] 2 ) by From the proof of Theorem 2, we claim that T u ∈ S([0, 1] 2 ).We will prove that T is contractive.In fact, for u 1 , u 2 ∈ S([0, 1] 2 ), we get from Theorem 2 that Since W < 1, there is a unique solution to equation (1.2) in the space S([0, 1] 2 ) by Banach's fixed point theorem.Hence Theorem 3 follows.

Example
Example 4 Consider the following equation with a boundary condition: ∂t 0.5 u(t, x) + 1 15 I 1.5 t I 0.5 x u(t, x) + 1 21 I 2.5 t I 1.1 x u(t, x) = 1  18 cos(txu) Then there is a unique solution in the space S([0, 1] 2 ) to equation (4.1).
Remark 5 The Python language is quite useful when computing the values of the multivariate Mittag-Leffler function or the newly introduced generalized multivariate Mittag-Leffler function.These functions appear often in many fields and play an important role in studying integral or differential equations with various conditions, as well as in finding approximate solutions, such as for equation (2.1) as an example.

Conclusion
We have obtained a sufficient condition for uniqueness of solution to the new boundary value problem (1.2) involving double integral operators by using the new generalized multivariate Mittag-Leffler function, Babenko's approach, as well as by applying Banach's fixed point theorem.Moreover, we made use of the Python language to aid in finding the approximate value of a generalized Mittag-Leffler function, which currently seems unfeasible to do so by any existing integral representations of the Mittag-Leffler function.
Finally, we presented an example that applies the results of the key theorems derived.The technique used certainly works for different types of PDE and corresponding initial or boundary value problems.