Abstract
The study of the existence and uniqueness of solutions to 2D systems utilizing the generalized proportional fractional derivative operator is the focus of this work. We also derive a finite difference scheme in order to numerically approximate such an operator, and we prove that the method we propose is convergent. Several tests are performed at the end to illustrate the robustness of our algorithm.
1. Introduction
Fractional calculus is a branch of mathematics that allows for the differentiation and integration of noninteger orders, extending the scope of traditional calculus. It involves the study of derivatives and integrals of arbitrary order. This allows for a more general approach to calculus, offering greater flexibility and accuracy. Fractional calculus can be used to model various physical phenomena, including turbulent fluid flow and the diffusion of gases. It is also used in signal processing applications, such as in the design of filters, and in the analysis of fractal images. The goal of fractional calculus is to provide a more comprehensive understanding of phenomena by allowing for the study of derivatives and integrals of arbitrary order. By allowing for the differentiation and integration of noninteger orders, fractional calculus is able to offer greater accuracy and flexibility in the study of various phenomena and has a wide range of applications (see [1–4]). Ordinary and partial fractional differential equations have developed significantly in recent years (see [5–9]).
In [10–12], authors have generalized the Caputo proportional fractional derivatives with respect to another function involving exponential functions. The Caputo derivatives are used to represent the fractional order of derivatives with respect to initial conditions. The exponential functions in the kernels of the derivatives provide a description of the time dynamics of the system. This type of fractional derivatives allows for a more precise description of complex nonlinear systems and provides a better representation of fractional order dynamics (see [13–17]).
The Darboux problem for partial hyperbolic differential equations is an important topic of research in mathematics. It concerns the existence and uniqueness of solutions to certain partial differential equations (PDEs). In recent years, there have been numerous articles discussing the Darboux problem. For instance, readers can refer to [5, 6, 18–22] for further information.
This paper extends [18] to the case of the GPF-Caputo derivative of order with respect to . The main contribution of this article is to investigate the existence, uniqueness, and numerical solutions of the following fractional partial differential systems:where , , and ; is the GPF-Caputo derivative of order with respect to and ; and , , and are given continuous functions.
2. Preliminarily
This section presents the definitions, lemmas, and propositions necessary for our findings. In the rest of this paper, we take .
Definition 1. Let and be positive strictly increasing functions such that for all and . The GPF-integral of order of with respect to is defined aswhere , and and are positive real numbers.
Definition 2. Let and be positive strictly increasing functions such that for all and . The GPF-Riemann–Liouville derivative of order of with respect to is defined aswhere and
Definition 3. Let and be positive strictly increasing functions such that for all and . The GPF-Caputo derivative of order of with respect to is defined aswhere , and .
Lemma 4. Let , , and be positive strictly increasing functions such that for all and . Then, we obtain the following relation:where , and are positive real numbers.
Proof. We haveBy using Fubini’s theorem, we obtainOn the other hand, we haveBy using the change of variables and and by using the fact that and , we get
Lemma 5. Let , , and be positive strictly increasing functions such that for all and . Then, we havewhere , and .
Proof. From Definition 2 and Lemma 4, we getUsing integration by parts, we obtain the following.
Lemma 6. Let , , and be positive strictly increasing functions such that for all and . Then, we have where , and .
As a consequence, we obtain the following.
Lemma 7. Let , , and be positive strictly increasing functions such that for all and . Then, we havewhere , and .
Proposition 8. Let and be positive strictly increasing functions such that for all and . The GPF-Caputo derivative of order of with respect to is given bywhere , and .
Definition 9. (see [23]). Let , , such that for . The generalized Mittag–Leffler function is defined bywhereWe have the following particular case ( and ):
3. Existence Results
We first require the following lemma in order to demonstrate the existence and uniqueness results.
Lemma 10. satisfies (1)-(2) if and only if
Proof. Assume that satisfies (20). By applying and using Lemma 5, we get which satisfies (1). We have the integral as zero when or ; therefore, the initial conditions in (2) are satisfied. Then, satisfies (1)-(2). Conversely, suppose satisfies (1)-(2) and letBy applying to (21), we findApplying the operator to this last equation, we obtain
Theorem 11. Let , , and . Define the function as follows: where
Then, problem (1)-(2) has at least one solution
Proof. If , then for all . Then, the function with satisfies (1)-(2). For define the set byclearly that is nonempty, closed, and convex subset of . We define the operator on this set byWe shall show that satisfies the assumption of Schauder’s fixed point theorem. We have as continuous. Now, we prove that is defined to into itself; let and thenThus, we have if . Now, we show that is relatively compact. Firstly, we show that is uniformly bounded. Indeed, from the previous step, we getSecondly, we show that is equicontinuous. Let such that and and . Then,As and , the right-hand side of the abovementioned inequality tends to zero. Hence, is equicontinuous, since is uniformly continuous in . As a consequence of Arzela–Ascoli theorem and Schauder’s fixed point theorem, we deduce that has a fixed point . This fixed point is a solution of problem (1)-(2).
3.1. Uniqueness of Solution
In this subsection, we discuss the uniqueness of solution of problem (1)-(2).
Lemma 12. Assume that there exists a constant such thatfor all and , then we have
Proof. Let and , we haveIt implies that
Theorem 13. Suppose that the assumptions of Theorem 11 are satisfied, and suppose that the function satisfies the Lipschitz condition with respect to the third variable with the Lipschitz constant . Also, let and . Then,
Proof. For , the inequality (36) holds. Suppose that (36) is true for , then for all and , we haveIt implies thatHence, the proof is complete.
Theorem 14. Let and be the same in Theorem 11. Suppose that satisfies a Lipschitz condition with respect to the third variable with the Lipschitz constant , where the set is defined as in Theorem 11. Then, there exists a unique solution of problem (1)-(2).
Proof. It follows from Theorem 11 that (1)-(2) has a solution. To show the uniqueness, we adapt Theorem 13. We use the operator as defined in 3.5, the function as defined in 3.3, and the set as defined in 3.4. We will use Weissinger’s fixed point theorem to show that has a unique fixed point. From 3.6, we getLet . Then, we haveso the series converges. This completes the proof.
4. Numerical Method for the Approximation of the 2D Generalized Proportional Fractional Derivative
We derive and investigate in this section a finite difference method to approximate the solution of system (1)-(2).
4.1. The Finite Difference Scheme
Let and be positive integers and let be a partition of defined by and , where and with and . We introduce the approximation of the solution with and as follows: and with and . In the sequel, we will assume that for simplicity.
Proposition 15. If , then scheme (41)-(42) is consistent with order (at least) one.
Proof. Let and . We define the truncation error at a grid node bywhere . We obtain by Taylor expansions of in .with , yielding by (1).Hence, we get for any and .with . Moreover, we have It followsNow, using the estimate,for any value of , with is a constant that do not depend on (see [24]), we deducewith . The same estimate holds for the second term in the right-hand side of (49). Finally, the result follows from (49) and (51).
Theorem 16. Assume is -Lipschitz with respect to its third variable. Then, scheme (41)-(42) is convergent.
Proof. All we need is to prove that scheme (41)-(42) is stable with respect to perturbations. Let be the solution of (41)-(42) and let be a solution of system (41)-(42) with additional perturbations: for all and ,where denotes the perturbations. Define the error term by , and then (41) and (52) yieldIn addition, we have for all ,and for all ,where and (with ), , and is defined by ). Denoting , thenUsing some index changes, one may findwith Now, we need the following results.
Lemma 17. Let and , . Then, we haveand .
Proof. The first inequality is a direct consequence of the estimate for any . Using the identity for any and , we obtain for ,We obtain the following from (54) and (58) and Lemma 17:yieldingwith . Let us notice that , , , and if is sufficiently small (a sufficient condition is ). Thus, we deduce using the discrete Gronwall inequality (see Lemma 4.4 in [2]).In addition, there exists a constant that depends on and only but not on such thatUsing Lemma 17 once more, we deducewith is a constant that do not depend on . This achieves the proof.
4.2. Numerical Tests
4.2.1. Example 1
We consider system (1)-(2) with the data as follows:and . According to Lemma 10, the solution of system (1)-(2) is given by
Figure 1 shows the numerical solution obtained by using the FD scheme (41)-(42) versus the exact solution. The parameters used are , , and . Clearly, the approximated solution fits very well with the theoretical one. Table 1 reports the difference of the solutions, as well as the orders of convergence for various values of . Notice that these errors go to zero when the path tends to zero, which is in total agreement with our theoretical study.
4.2.2. Example 2
Consider system (1)-(2) with the data:with , , , , and . One can check that the solution of (1)-(2) is given by . We plotted in Figure 2 the exact versus the numerical solutions obtained by using the FD scheme (41)-(42) for various values of . The fractional order is set to . One can notice that all the solutions are in good agreement. We also compute the errors in the norm between the solutions in Table 2. The obtained results clearly confirm the robustness of the proposed scheme.
(a)
(b)
5. Conclusion
The existence and uniqueness of solutions to 2D systems using the proportional fractional derivative operator with respect to other functions are introduced and studied in this work. We also use the finite differences method to study the numerical approximation of such operator, and we establish the convergence of the numerical scheme.
Data Availability
No underlying data were collected or produced in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was funded by “Researchers Supporting Project number (RSPD2023R683), King Saud University, Riyadh, Saudi Arabia.”