An Iterative Technique for Solving Time Fractional Systems with Variable Coefficients

Abstract

This paper presents a comprehensive study on constructing exact and approximate solutions to Cauchy problems for time-fractional systems with variable coefficients. An innovative iterative approach is developed for solving functional equations with initial conditions, combining rigorous mathematical foundations with practical computational efficiency. The proposed technique effectively handles the nonlocal nature of fractional operators through a carefully designed iterative scheme that maintains simplicity while achieving high accuracy. It demonstrates particular strength in solving nonlinear systems with well-defined conditions and variable coefficients, where traditional methods often fail. Through systematic theoretical analysis and numerical validation, we establish the method’s convergence properties and computational advantages, showing its capability to generate both exact closed-form solutions, when available, and high-precision approximations otherwise. The approach remains computationally tractable even for complex cases where variable coefficients and memory effects of fractional systems present significant challenges to conventional solution approaches.

1. Introduction

This paper presents a systematic investigation of the initial value problems arising in time-fractional partial differential equations, where the nonlocal nature of fractional derivatives introduces unique mathematical challenges that fundamentally distinguish them from their integer-order counterparts. The inherent memory effects and historical dependence of fractional operators render the initial value problem particularly intricate yet fascinating from both theoretical and computational perspectives. Our comprehensive study develops and analyzes a suite of analytical and numerical methodologies specifically designed to overcome these challenges, with the dual objectives of establishing rigorous mathematical foundations and providing practical computational tools for understanding the temporal evolution of solutions in such systems. The research builds upon and significantly extends existing work in this rapidly developing field, as documented in [1,2,3,4,5,6,7,8] and related references, while introducing novel approaches to address the distinctive features of fractional-order initial value problems.

The historical development of fractional calculus provides crucial context for understanding its modern applications. Our exploration begins with the seminal correspondence between L’Hôpital and Leibniz in 1695, a foundational moment in mathematical history documented in [9]. In this historic exchange, L’Hôpital posed the now-famous question to Leibniz:

“What would be the result of ${\displaystyle \frac{d^{n}y}{d{x}^n}}$ if $n={\displaystyle \frac{1}{2}}$?”

Leibniz’s insightful reply demonstrated remarkable foresight:

“$d^{1/2}x$ will be equal to $x\sqrt{dx:x}$. This is an apparent paradox, from which, one day useful consequences will be drawn.”

This early conceptual breakthrough, however, remained largely undeveloped for nearly two centuries. The mathematical community of the Enlightenment era faced several barriers to advancing fractional calculus: The lack of a rigorous theoretical framework for non-integer derivatives; The absence of clear physical interpretations; Significant computational challenges; The dominance of integer-order calculus in solving contemporary problems. This historical evolution underscores how theoretical mathematics often precedes practical applications, with Leibniz’s initial insight eventually finding numerous applications across science and engineering disciplines.

The fundamental nature of fractional operators distinguishes them significantly from traditional differential operators. Researchers have established that these operators are intrinsically non-local, typically expressed through integral formulations with singular kernels. This essential characteristic gives fractional calculus its unique capacity to model memory effects and hereditary properties in physical systems, where the current state depends on the complete historical evolution rather than just instantaneous conditions.

A particularly noteworthy application emerges in electrochemical systems, where fractional-order operators have proven remarkably effective: They provide superior modeling of diffusion processes in electrochemical cells compared to classical approaches; They more accurately characterize electrode impedance behavior; They offer enhanced precision in describing charge transfer mechanisms.

The theoretical framework of fractional calculus has consequently found extensive utility across numerous scientific and technological domains, including: modeling anomalous diffusion and viscoelastic material properties, analyzing complex reaction-diffusion processes, understanding neuronal dynamics and biomechanical tissue behavior, characterizing long-range dependent processes in financial markets and so on.

These diverse applications, meticulously documented in [10] and extensively developed in subsequent research, vividly illustrate the remarkable transformation of fractional calculus from a purely theoretical construct to an indispensable analytical framework in modern science and engineering. The continually expanding scope of its applications provides compelling evidence for the fundamental importance of these non-local operators, with demonstrated successes ranging from nanoscale phenomena to large-scale economic systems. Their unique capacity to capture long-range dependencies and memory effects has enabled breakthroughs in modeling complex systems that defy description by classical approaches, particularly in situations involving power-law behaviors and anomalous dynamics. Recent advances have particularly highlighted how fractional operators offer superior fidelity in modeling multiscale phenomena with coupled temporal and spatial dependencies, while also providing natural frameworks for systems exhibiting complex hierarchical or fractal-like structures. The operational success of fractional calculus across such disparate domains not only underscores its profound theoretical significance but has also driven important innovations in computational methods and analytical techniques. This dual impact—simultaneously deepening our fundamental understanding while enhancing practical problem-solving capabilities—continues to motivate vigorous research efforts at the frontiers of applied mathematics and interdisciplinary science, promising further discoveries and applications in emerging fields.

Fractional calculus, representing a significant generalization of classical differential and integral operators to non-integer orders, has emerged as a vibrant research field attracting considerable attention from both theoretical and applied perspectives. This mathematical framework extends the traditional notions of differentiation and integration to fractional orders, thereby enabling the description of complex phenomena that exhibit non-local behavior and memory effects. Throughout its historical development, numerous distinct definitions of fractional derivatives and integrals have been proposed, each offering unique advantages for specific applications and mathematical contexts.

The absence of a unified definition for fractional derivatives has led to a rich diversity of operational formulations, including several prominent variants that have proven particularly useful in practice. Among these, the Grünwald–Letnikov approach provides a discrete interpretation based on finite differences, while the Riemann–Liouville formulation offers a continuous integral-based perspective. The Caputo derivative has gained prominence for its advantageous treatment of initial conditions in differential equations. Additional important formulations include the Riesz potential for symmetric fractional operations, the conformable fractional derivative for maintaining certain classical properties, and various local fractional derivatives designed for specific applications.

Within this diverse landscape, the Riemann–Liouville fractional integral and Caputo fractional derivative have emerged as particularly influential, owing to their mathematical tractability and physical interpretability. These operators have found extensive application across numerous scientific and engineering disciplines, including advanced control systems where they enable more robust controller designs, the modeling of anomalous diffusion processes in complex media, and the analysis of fractal structures and multi-scale systems. The continued development and refinement of these fractional operators, as documented in [4,7,8,11,12,13,14] and related works, reflects their growing importance in addressing challenging problems that lie beyond the scope of classical calculus.

Among the various formulations of fractional operators, the Grünwald–Letnikov fractional derivative (see Appendix A.2) occupies a special historical and conceptual position as one of the most intuitive and computationally accessible approaches. This operator, rooted in the generalization of finite difference methods, provides a natural discrete framework for fractional differentiation that closely parallels the classical limit definition of integer-order derivatives. The Grünwald–Letnikov approach is particularly valuable in numerical analysis and scientific computing, where its discrete nature allows for efficient implementation in algorithmic solutions to fractional differential equations. Its computational advantages have made it indispensable for simulations of complex systems with memory effects, as extensively documented in [15] and subsequent computational studies.

In contrast, the Riemann–Liouville fractional derivative represents a continuous formulation defined through a convolution operation with a power-law kernel. This definition enjoys several theoretical advantages, including a rigorous mathematical foundation in functional analysis and a natural connection to the theory of integral equations. The Riemann–Liouville operator has proven particularly powerful in analytical approaches to fractional differential equations arising in mathematical physics, where its properties facilitate the development of existence theorems and solution representations. The complementary strengths of these two approaches—with Grünwald–Letnikov favoring computational implementations and Riemann–Liouville supporting theoretical analysis—serve to illustrate the rich diversity of tools available in fractional calculus, as discussed in detail in [16] and related mathematical physics literature.

In contrast, the Caputo fractional derivative was introduced to address this issue, offering a more physically intuitive way to incorporate initial conditions in models. It is particularly popular in applied settings such as control systems. The Riesz potential, often employed in potential theory and the fractional Laplacian, generalizes the idea of a potential field to fractional dimensions and is crucial in studying processes like anomalous diffusion and Lévy flights; see [17] and the references therein.

In contrast to the Riemann–Liouville formulation, the Caputo fractional derivative was specifically developed to overcome certain mathematical challenges in modeling physical systems. This alternative definition provides several crucial advantages for practical applications, most notably its ability to handle initial conditions in a manner that aligns more naturally with physical interpretations. The Caputo operator’s formulation ensures that initial value problems can be specified using standard function values rather than fractional derivatives of the initial state, making it particularly valuable for modeling real-world phenomena. This property has led to its widespread adoption in applied mathematics and engineering disciplines, especially in control theory, where it enables more natural formulations of dynamical systems with memory effects. The Caputo derivative’s compatibility with conventional physical interpretations has made it a preferred choice for modeling viscoelastic materials, thermal systems, and other processes where the initial state is typically specified in terms of classical variables.

Similarly important in fractional analysis is the Riesz potential, which extends classical potential theory to fractional dimensions through an elegant symmetric formulation. This operator plays a fundamental role in the theory of the fractional Laplacians and has become indispensable for studying non-local phenomena. The Riesz transform provides a powerful mathematical framework for analyzing complex diffusion processes, particularly those exhibiting anomalous behavior that deviates from classical Brownian motion. Its applications span diverse areas, including the modeling of Lévy flights in statistical physics, the analysis of non-local interactions in quantum mechanics, and the study of long-range correlations in complex systems. The theoretical foundations and practical applications of the Riesz potential are thoroughly examined in [17], along with its connections to other fractional operators and its growing importance in modern mathematical physics.

The rich diversity of fractional derivative definitions, while reflecting the wide-ranging applications of fractional calculus across scientific disciplines, presents significant challenges for researchers working to select, compare, and apply these various operators. This proliferation of definitions—each with distinct properties and suited for different applications—can create confusion and complicate theoretical comparisons between results obtained using different formulations. Recognizing this challenge, the mathematical community has increasingly turned to axiomatization approaches that seek to unify these diverse operators within a coherent theoretical framework.

In [12], the authors adopt such an axiomatic perspective, examining several fractional derivative definitions in a systematic way. By identifying the core properties that these operators share, they developed a general framework that facilitates a deeper understanding of their similarities and differences. For example, the axioms might include the following several items:

• (Linearity) The operator is linear.
• (Identity) The zero order derivative of a function returns the function itself.
• (Backward compatibility) When the order is integer, the fractional derivative gives the same result as the classical derivative.
• (Index law) For any $\alpha <0$ and $\beta <0$, we have

  $$D^{\alpha }{D}^{\beta }f\left(t\right)=D^{\alpha +\beta }f\left(t\right).$$
• (Generalized Leibniz rule)

  $$D^{\alpha }\left[f\left(t\right)g\left(t\right)\right]=\sum _{i=0}^{\infty }\left(\begin{array}{c}\alpha \\ i\end{array}\right)D^{i}f\left(t\right)D^{\alpha -i}g\left(t\right).$$

  Remark 1.

  If $\alpha \in \mathbb{N}$, this is exactly the classical Leibniz rule. By the way, another generalization of this rule, such as

  $$D^{\alpha }\left[f\left(t\right)g\left(t\right)\right]=\sum _{i=-infty}^{\infty }\left(\begin{array}{c}\alpha \\ i+\beta \end{array}\right)D^{\beta +i}f\left(t\right)D^{\alpha -\beta -i}g\left(t\right).$$

This approach not only highlights the underlying mathematical structure common to various fractional operators but also provides a way to compare and contrast their behaviors. By formalizing fractional calculus through axiomatization, it becomes easier to extend these operators to new contexts, such as multi-dimensional systems, stochastic processes, or even quantum mechanics, where fractional derivatives are increasingly finding applications; see [5,18,19,20] and the references therein.

The axiomatization of fractional operators has profound implications for both the theoretical development of fractional calculus and its practical applications. On the theoretical side, it provides a clearer framework for exploring the connections between different types of fractional derivatives and integrals, potentially leading to the discovery of new operators or generalizations that encompass existing ones. It also helps to clarify which definitions are most suitable for specific problems, depending on the required properties of the derivative, such as memory effects, non-locality, or boundary behavior; see [21,22] and the references therein.

From a practical standpoint, this unifying approach simplifies the application of fractional derivatives in diverse fields, from engineering to biology. For instance, in modeling materials with memory effects, the choice of a fractional derivative is crucial, and an axiomatic perspective can help guide that choice. Similarly, in complex systems where fractal geometry or anomalous diffusion is present, such as in finance, geology, or network theory, an axiomatized framework for fractional derivatives ensures that the correct operator is used to capture the necessary dynamics.

In this paper, we mainly explore the construction of exact solutions or approximate solutions to the initial value problems of time fractional partial differential equations. This is a very direct problem in the study of the theory of differential equations. However, it is not easy and even rather difficult. In addition, solving fractional systems is usually more difficult than classical systems, for their operator is defined by an integral (a non-local operator with a singular kernel). Luckily, there are some different effective methods that have been developed to construct the exact solutions or the approximate solutions of fractional systems with initial conditions. Such as the iterative method [23,24,25,26,27], the homotopy analysis method [28,29,30], the Cole-Hopf transformation method [31], the residual power series method [32,33,34,35,36], the Laplace transform method [37], and so on.

Outline of Paper

This manuscript presents a systematic investigation of iterative techniques for solving initial value problems in time-fractional partial differential equations. The paper is organized as follows:

In Section 2, we establish the fundamental mathematical framework by introducing essential concepts from fractional calculus, including the various definitions of fractional derivatives and their key properties. This section provides the necessary theoretical foundation by presenting important lemmas regarding fractional operators, particularly focusing on their non-local characteristics and memory effects. We carefully define all mathematical notations to ensure clarity throughout subsequent developments, paying special attention to the function spaces and operator norms relevant to our analysis of fractional differential equations.

Section 3 develops an iterative approach for solving functional equations with initial conditions, with particular emphasis on applications to fractional-order systems. We present a comprehensive convergence analysis of the proposed iterative scheme, establishing sufficient conditions under which the method guarantees convergence to the true solution. The section includes detailed proofs of convergence theorems and examines the rate of convergence under various regularity assumptions. Furthermore, we discuss the computational advantages of this approach compared to traditional methods, highlighting its efficiency in handling the non-local nature of fractional operators.

Section 4 demonstrates the practical implementation of our iterative technique through an in-depth study of the one-dimensional time-fractional Tricomi equation. We derive both exact solutions (where obtainable) and high-precision approximate solutions, comparing our results with known analytical solutions to validate the method’s accuracy. The section includes detailed case studies with different fractional orders and initial conditions, accompanied by rigorous error analysis and convergence rate verification. Special attention is given to the physical interpretation of solutions and their relationship to the fractional order parameter.

The analysis is extended in Section 5 to multidimensional time-fractional Tricomi equations, where we address the additional complexities introduced by higher spatial dimensions. We develop dimensionally-adaptive versions of our iterative scheme and examine their performance for various initial boundary value problems. The section presents numerical results for two- and three-dimensional cases, including visualization of solution surfaces and comparison with alternative numerical methods. We particularly focus on the method’s scalability and its ability to handle different types of initial conditions, from smooth analytic functions to discontinuous data.

Finally, Section 6 synthesizes our findings into a comprehensive conclusion, summarizing the key advantages and limitations of the proposed iterative technique. We discuss broader implications for the field of fractional calculus and suggest several directions for future research, including potential applications to nonlinear fractional systems and coupled fractional-order equations. The section concludes with practical recommendations for implementing the method and its possible extensions to other classes of fractional differential equations.

2. Fractional Calculus

This subsection introduces the fundamental theoretical framework necessary for our investigation of fractional systems. We begin with the essential special functions that form the backbone of fractional calculus. The Gamma function, which extends the factorial concept to continuous values, plays an indispensable role in defining fractional operators due to its appearance in their integral kernels. Equally crucial is the Mittag–Leffler function, a sophisticated generalization of the exponential function that naturally describes processes with memory effects and power-law relaxation.

The mathematical landscape of fractional calculus features a diverse collection of definitions for fractional integrals and derivatives, each developed to serve specific theoretical or applied purposes. Among these various formulations, several have emerged as particularly significant. The Riemann–Liouville fractional integral represents the most direct extension of classical integration theory to non-integer orders, while the Caputo fractional derivative has become preferred for physical modeling due to its more intuitive treatment of initial conditions. Other important variants include the Riesz potential, which provides symmetric fractional operations valuable for certain boundary value problems, and the Weyl derivative, especially useful for analyzing periodic phenomena. The comparative advantages and specific applications of these different definitions are thoroughly examined in the comprehensive references [7,8,11,12,13,14].

Within this spectrum of fractional operators, the Riemann–Liouville integral and Caputo derivative have gained particular prominence in both theoretical and applied contexts. The Riemann–Liouville approach provides the most natural mathematical foundation for fractional integration, while the Caputo formulation offers distinct practical advantages for solving initial value problems that model physical systems. This preference stems from the Caputo derivative’s compatibility with conventional initial conditions specified in terms of standard derivatives, making it more suitable for applications in engineering and physics.

Our subsequent analysis will build upon several key lemmas concerning the operational properties of these fractional operators and the characteristic behaviors of Mittag–Leffler functions. These fundamental results enable rigorous investigation of existence, uniqueness, and qualitative properties for solutions to fractional differential equations. The interplay between these various concepts creates a robust framework for examining the nonlinear fractional systems that constitute the primary focus of this work, while also providing the necessary tools for developing effective solution methods.

2.1. The Gamma Function and the Mittag–Leffler Function

This subsection introduces the definitions for the Gamma function and the Mittag–Leffler function, which will be used in the definitions of the fractional integral and derivative.

Definition 1

([38]). The Gamma function is defined by

$$\mathrm{\Gamma }\left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt\mathrm{Re}\left(x\right)>0,$$

where Re$\left(x\right)$ denotes the real part of the complex number x.

Proposition 1

([38]). Some main properties of the Gamma function:

• $\mathrm{\Gamma }(x+1)=x\mathrm{\Gamma }\left(x\right)$ for all x with Re$x>0$.
• $\mathrm{\Gamma }\left(1\right)=1$.
• $\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}}\right)=\sqrt{\pi }$.

Definition 2

([39]). The two-parameters Mittag–Leffler function is defined by letting

$$E_{\alpha ,\beta }\left(x\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{x^k}{\mathrm{\Gamma }(\alpha k+\beta )}}\alpha ,\beta >0.$$

And the one-parameter Mittag–Leffler function is defined by letting

$$E_{\alpha }\left(x\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{x^k}{\mathrm{\Gamma }(\alpha k+1)}}\alpha >0.$$

Remark 2.

The two-parameter Mittag–Leffler function is a generalization of the one-parameter Mittag–Leffler function, that is, $E_{\alpha ,1}\left(x\right)=E_{\alpha }\left(x\right)$. The one-parameter Mittag–Leffler function is a generalization of the exponential function, or rather, $E_1\left(x\right)=e^x$.

2.2. The Riemann–Liouville Fractional Integral and the Caputo Derivative

This subsection recalls the definition of the Riemann–Liouville fractional integral and Caputon derivative, see [4,5,6,13,40] and the references therein.

Definition 3.

Let $\theta :[0,T]\to \mathbb{R}$ is a continuous function. Then, the fractional-order integral of θ in the Riemann–Liouville sense is defined by

$$J^{\alpha }\theta \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}\theta \left(s\right)ds,\alpha >0,t\ge 0,$$

(1)

where $\mathrm{\Gamma }\left(\alpha \right)={\int }_0^{\infty }{e}^{-u}{u}^{\alpha -1}du.$

Definition 4.

Let $\theta :[0,T]\to \mathbb{R}$ is a continuous function. Then, the Caputo fractional-order derivative of θ is given by

$${}^{c}D^{\alpha }\theta \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}{\int }_0^t{(t-s)}^{n-\alpha -1}{\theta }^{\left(m\right)}\left(s\right)ds,$$

where $m-1<\alpha \le m;m\in \mathbb{N}.$

Lemma 1

([27]). For any $m-1<\beta \le m\in {\mathbb{N}}^{\ast }$, one has

$${\mathrm{I}}^{\beta }{\mathrm{D}}^{\beta }u(x,t)=u(x,t)-\sum _{k=0}^{m-1}{\partial }^{k}u(x,0){\displaystyle \frac{t^k}{k!}}.$$

Lemma 2.

For any $\beta ,\gamma$, one has

$${\mathrm{I}}^{\beta }{t}^{\gamma }={\displaystyle \frac{\mathrm{\Gamma }(\gamma +1)}{\mathrm{\Gamma }(\gamma +\beta +1)}}{t}^{\gamma +\beta }.$$

3. An Iterative Technique for Solving a Class of Nonlinear Functional Equations

In this section, we present a fundamental lemma that plays a pivotal role in our subsequent mathematical analysis. It provides the essential theoretical foundation for establishing the key propositions and theorems developed throughout this paper. It has been chosen for its particular relevance to our investigation of fractional differential systems, offering crucial insights into the behavior of solutions and the properties of fractional operators. Its proper application will be carefully demonstrated in subsequent sections where we develop our main results.

Lemma 3.

Consider a class of abstract functional equations of the following form

$$u(x,t)=f(x,t)+\mathcal{L}\left(u(x,t)\right)+\mathcal{N}\left(u(x,t)\right),$$

(2)

where $(t,x)\in D=\{(t,x):t\in \mathbb{R},x\in {\mathbb{R}}^n,n\in \mathbb{N}\}$, $u(x,t)$ is the unknown function, f is a given known function, $\mathcal{L}$ is a linear operator and $\mathcal{N}$ is a nonlinear operator with respect to u from a Banach space Dom$\left(\mathcal{L}\right)\cap$Dom$\left(\mathcal{N}\right)$ to itself (Dom$\left(\mathcal{L}\right)\cap$Dom$\left(\mathcal{N}\right)$ stands for the domain of $\mathcal{L}$ and $\mathcal{N}$ respectively).

For abstract function Equation (2), consider the series solutions of the following form

$$u(t,x)=\sum _{n=0}^{\infty }{u}_n(t,x)=u_0+u_1+u_2+\cdots ,$$

where the iterative sequence $\left\{u_n\right\}$ can be obtained by the following recurrence equations

$$\begin{array}{cc}\hfill u_0& =f,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill u_1& =\mathcal{L}\left(u_0\right)+\mathcal{N}\left(u_0\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill u_{m+1}& =\mathcal{L}\left(u_m\right)+\left\{\mathcal{N}\left(\sum _{n=0}^m{u}_n\right)-\mathcal{N}\left(\sum _{n=0}^{m-1}{u}_n\right)\right\},m=1,2\cdots .\hfill \end{array}$$

(5)

Furthermore, if the operators $\mathcal{L}$ and $\mathcal{N}$ are contractive, then the series solution ${\displaystyle \sum _{n=0}^{\infty }}{u}_n(t,x)$ converges absolutely and uniformly.

Proof. 

Obviously, the nonlinear operator $\mathcal{N}$ can be decomposed as

$$\begin{array}{cc}\hfill \mathcal{N}\left(u\right(t,x\left)\right)& =\mathcal{N}\left(\sum _{n=0}^{\infty }{u}_n(t,x)\right)\hfill \\ \hfill & =\mathcal{N}\left(u_0\right)+\sum _{n=1}^{\infty }\left\{\mathcal{N}\left(\sum _{j=0}^n{u}_j\right)-\mathcal{N}\left(\sum _{j=0}^{n-1}{u}_j\right)\right\}.\hfill \end{array}$$

Similarly, the linear operator $\mathcal{L}$ can also be decomposed as

$$\begin{array}{cc}\hfill \mathcal{L}\left(u\right(t,x\left)\right)& =\mathcal{L}\left(\sum _{n=0}^{\infty }{u}_n(t,x)\right)\hfill \\ \hfill & =\mathcal{L}\left(u_0\right)+\sum _{n=1}^{\infty }\left\{\mathcal{L}\left(\sum _{j=0}^n{u}_j\right)-\mathcal{L}\left(\sum _{j=0}^{n-1}{u}_j\right)\right\}.\hfill \end{array}$$

Denote by

$$\mathcal{M}\left(u(t,x)\right)=\mathcal{L}\left(u(t,x)\right)+\mathcal{N}\left(u(t,x)\right).$$

then one has

$$\mathcal{M}\left(u(t,x)\right)=\mathcal{M}\left(u_0\right)+\sum _{n=1}^{\infty }\left\{\mathcal{M}\left(\sum _{j=0}^n{u}_j\right)-\mathcal{M}\left(\sum _{j=0}^{n-1}{u}_j\right)\right\}.$$

Define the following recurrence equations

$$\begin{array}{cc}\hfill u_0& =f,\hfill \\ \hfill u_1& =\mathcal{M}\left(u_0\right),\hfill \\ \hfill u_{m+1}& =\mathcal{M}\left(\sum _{n=0}^m{u}_n\right)-\mathcal{M}\left(\sum _{n=0}^{m-1}{u}_n\right),m=1,2\cdots .\hfill \end{array}$$

Since operators $\mathcal{L}$ and $\mathcal{N}$ are contractive, then $\mathcal{M}$ is also contractive, i.e., there exists a constant $0<K<1$, such that

$$\parallel \mathcal{M}\left(v_i\right)-\mathcal{M}\left(v_j\right)\parallel \le K\parallel v_i-v_j\parallel ,\forall v_i,v_j\in \mathrm{Dom}\left(\mathcal{L}\right)\cap Dom\left(\mathcal{N}\right),$$

where $\parallel ·\parallel$ denotes the norm on Banach space Dom$\left(\mathcal{L}\right)\cap$Dom$\left(\mathcal{N}\right)$. Furthermore, for $u_{m+1}$, one can obtain

$$\begin{array}{c}\parallel u_{m+1}\parallel =∥\mathcal{M}\left({\displaystyle \sum _{n=0}^m}{u}_n\right)-\mathcal{M}\left({\displaystyle \sum _{n=0}^{m-1}}{u}_n\right)∥\le K\parallel u_m\parallel \le K^{m+1}\parallel u_0\parallel .\hfill \end{array}$$

Since $0<K<1$, the series ${\displaystyle \sum _{m=1}^{\infty }}{K}^{m+1}\Vert u_0\Vert$ converges absolutely as well as uniformly.

According to the Weierstrass M-test, one can obtain that the series ${\displaystyle \sum _{i=0}^{\infty }}{u}_i$ converges absolutely as well as uniformly. □

Some more specific details about this type of iterative method could be found in [23,24,25,26,27] and the references therein.

4. One-Dimensional Time Fractional Tricomi Equation with Variable Coefficients

Consider the following one-dimensional time-fractional Tricomi equation of order $\alpha$, as studied in [41]:

$$\begin{array}{c}\hfill {\mathrm{D}}_t^{\alpha }u+\xi \left(t\right){\partial }_x^{2}u=f(x,t),\end{array}$$

(6)

subject to the initial conditions

$$\begin{array}{c}\hfill u(x,0)={\omega }_1\left(x\right),{\mathrm{D}}_{t}u(x,0)={\omega }_2\left(x\right),\end{array}$$

(7)

where $1<\alpha \le 2$ represents the fractional order of the time derivative, $t\in [0,T]$ denotes the temporal variable, and $x\in \mathbb{R}$ is the spatial variable.

The special case when $\alpha =2$ reduces (6) to the classical Tricomi equation, first introduced by Francesco Tricomi in his seminal 1923 work [42]. This equation occupies a particularly important position in the theory of partial differential equations due to its remarkable property of exhibiting mixed-type characteristics. Specifically, the nature of the equation undergoes a fundamental change depending on the sign of the coefficient function $\xi \left(t\right)$: when $\xi \left(t\right)>0$, the equation demonstrates elliptic behavior, while for $\xi \left(t\right)<0$, it becomes hyperbolic. This dual character makes the Tricomi equation a paradigmatic example of mixed-type partial differential equations.

The physical significance of this equation was further elucidated by Frankl [43], who established its crucial connection to transonic gas dynamics. More precisely, the Tricomi equation provides a mathematical model describing the transition between subsonic and supersonic flow regimes in gas dynamics. The elliptic region ($\xi \left(t\right)>0$) corresponds to subsonic flow, where disturbances propagate in all directions, while the hyperbolic region ($\xi \left(t\right)<0$) represents supersonic flow, characterized by directional wave propagation and the formation of shock waves. This interpretation underscores the equation’s importance in aerodynamics and fluid mechanics.

In the following analysis, we will develop an iterative solution technique to solve the initial value problem (6) and (7) for the time-fractional generalization of this important equation. Our approach will carefully account for both the fractional temporal derivative and the mixed-type spatial operator, while maintaining the essential physical interpretation of the solutions.

By integrating both sides of Equation (6) with respect to the temporal variable from 0 to t, we derive the equivalent integral formulation:

$$\begin{array}{c}\hfill u(x,t)=u(x,0)+{\mathrm{D}}_{t}u(x,0)t-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\partial }_x^{2}u\right)+{\mathrm{I}}_t^{\alpha }f(x,t).\end{array}$$

(8)

Incorporating the initial conditions (7) into the integral Equation (8) yields:

$$u(x,t)={\omega }_1\left(x\right)+{\omega }_2\left(x\right)t+{\mathrm{I}}_t^{\alpha }f(x,t)-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\partial }_x^{2}u\right).$$

From the perspective of iterative methods, we can interpret this equation as follows: The terms ${\omega }_1\left(x\right)+{\omega }_2\left(x\right)+{\mathrm{I}}_t^{\alpha }f(x,t)$ constitute the initial iterative data, incorporating both the given initial conditions and the source term $f(x,t)$. The remaining term $-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\partial }_x^{2}u\right)$ acts as a nonlinear feedback operator that propagates the solution through successive iterations.

We propose a series solution of the form:

$$u(x,t)=\sum _{n=0}^{\infty }{u}_n(x,t)=u_0+u_1+u_2+\cdots ,$$

where the iterative sequence is determined by the following recursive scheme:

$$\begin{array}{cc}\hfill u_0& ={\omega }_1\left(x\right)+{\omega }_2\left(x\right)t+{\mathrm{I}}_t^{\alpha }f(x,t),\hfill \\ \hfill u_1& =-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\partial }_x^2{u}_0\right),\hfill \\ \hfill u_2& =-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\partial }_x^2{u}_1\right),\hfill \\ \hfill & ⋮\hfill \\ \hfill u_{m+1}& =-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\partial }_x^2{u}_m\right).\hfill \end{array}$$

To demonstrate the effectiveness of this iterative approach, we now examine several specific cases with carefully chosen initial conditions, source terms, and coefficient functions.

Proposition 2

(Trivial Solution Case). For the simplest case where ${\omega }_1\left(x\right)\equiv 0$, ${\omega }_2\left(x\right)\equiv 0$, and $f(x,t)\equiv 0$, we observe that for any well-behaved coefficient function $\xi \left(t\right)$, the solution identically vanishes:

$$u(x,t)\equiv 0.$$

This trivial solution serves as an important consistency check for our method.

Proposition 3

(Exponential Source Term with Constant Coefficient). Consider the case with zero initial conditions ${\omega }_1\left(x\right)\equiv 0$, ${\omega }_2\left(x\right)\equiv 0$, an exponential source term $f(x,t)\equiv e^x$, and constant coefficient $\xi \left(t\right)\equiv -1$. The iterative scheme yields:

$$u_n=e^x{\displaystyle \frac{t^{(n+1)\alpha }}{\mathrm{\Gamma }\left(\right(n+1)\alpha +1)}}\forall n\in \mathbb{N}.$$

The complete solution then takes the elegant form:

$$\begin{array}{cc}\hfill u(x,t)& =\sum _{n=0}^{\infty }{u}_n=e^x\sum _{n=0}^{\infty }{\displaystyle \frac{t^{(n+1)\alpha }}{\mathrm{\Gamma }\left(\right(n+1)\alpha +1)}}\hfill \\ \hfill & =e^x\left(E_{\alpha }\left(t^{\alpha }\right)-1\right),\hfill \end{array}$$

where $E_{\alpha }(·)$ denotes the Mittag–Leffler function, which naturally emerges in solutions to fractional differential equations.

Proposition 4

(Modified Exponential Source Term). For the case with ${\omega }_1\left(x\right)\equiv 0$, ${\omega }_2\left(x\right)\equiv 0$, $f(x,t)\equiv e^x{\displaystyle \frac{t^{\alpha }}{\mathrm{\Gamma }(-\alpha +1)}}$, and $\xi \left(t\right)\equiv -1$, we obtain:

$$u_n=e^x{\displaystyle \frac{t^{n\alpha }}{\mathrm{\Gamma }(n\alpha +1)}}\forall n\in \mathbb{N}.$$

The solution series converges to:

$$u(x,t)=e^x{E}_{\alpha }\left(t^{\alpha }\right),$$

demonstrating how different source terms can lead to distinct Mittag–Leffler function representations.

Proposition 5

(Non-constant Coefficient Case). Consider the problem with

$${\omega }_1\left(x\right)=e^x,{\omega }_2\left(x\right)=0,f(x,t)=0,\xi \left(t\right)=t^{\alpha }.$$

Then the iterative scheme generates the sequence

$$\begin{array}{cc}\hfill u_0& =e^x,\hfill \\ \hfill u_1& =-e^x{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)}}{t}^{2\alpha },\hfill \\ \hfill u_2& =e^x{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }(3\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)\mathrm{\Gamma }(4\alpha +1)}}{t}^{4\alpha }.\hfill \end{array}$$

By the induction of mathematics, we establish the general term:

$$u_n={(-1)}^n{e}^x{\displaystyle \frac{{\displaystyle \prod _{k=1}^n\mathrm{\Gamma }((2k-1)\alpha +1)}}{{\displaystyle \prod _{k=1}^n\mathrm{\Gamma }(2k\alpha +1)}}}{t}^{2n\alpha },n\ge 0.$$

The complete solution is expressed as the infinite series:

$$u(x,t)=e^x\sum _{n=0}^{\infty }{(-1)}^n{\displaystyle \frac{{\displaystyle \prod _{k=1}^n\mathrm{\Gamma }((2k-1)\alpha +1)}}{{\displaystyle \prod _{k=1}^n\mathrm{\Gamma }(2k\alpha +1)}}}{t}^{2n\alpha },$$

revealing the intricate relationship between the solution’s temporal behavior and the fractional order α.

Proof. 

Step 1: Iterative structure.

Under the given assumptions, the iteration takes the form

$$u_{n+1}(x,t)=-{\mathrm{I}}_t^{\alpha }\left(t^{\alpha }{u}_n(x,t)\right),$$

where ${\mathcal{I}}_t^{\alpha }$ denotes the fractional integral operator.

Because the initial data is separable and $u_0=e^x$, we observe that each iterate preserves the spatial factor $e^x$. Hence we may write

$$u_n(x,t)=e^x{a}_n\left(t\right),$$

reducing the problem to determining the scalar sequence $a_n\left(t\right)$.

Step 2: First iterates.

For $n=0$,

$$a_0\left(t\right)=1.$$

Using the fractional integral formula

$${\mathrm{I}}_t^{\alpha }{t}^{\beta }={\displaystyle \frac{\mathrm{\Gamma }(\beta +1)}{\mathrm{\Gamma }(\beta +\alpha +1)}}{t}^{\beta +\alpha },$$

we obtain

$$a_1\left(t\right)=-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)}}{t}^{2\alpha }.$$

Repeating the same computation yields

$$a_2\left(t\right)={\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }(3\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)\mathrm{\Gamma }(4\alpha +1)}}{t}^{4\alpha }.$$

Step 3: Inductive hypothesis.

Assume that for some $n\ge 1$,

$$u_n={(-1)}^n{e}^x{\displaystyle \frac{{\prod }_{k=1}^n\mathrm{\Gamma }((2k-1)\alpha +1)}{{\prod }_{k=1}^n\mathrm{\Gamma }(2k\alpha +1)}}{t}^{2n\alpha }.$$

Step 4: Inductive step.

Applying the iteration formula,

$$u_{n+1}=-{\mathrm{I}}_t^{\alpha }\left(t^{\alpha }{u}_n\right).$$

Substituting the inductive expression gives

$$t^{\alpha }{u}_n={(-1)}^n{e}^x{\displaystyle \frac{{\prod }_{k=1}^n\mathrm{\Gamma }((2k-1)\alpha +1)}{{\prod }_{k=1}^n\mathrm{\Gamma }(2k\alpha +1)}}{t}^{(2n+1)\alpha }.$$

Applying the fractional integral formula once more,

$${\mathrm{I}}_t^{\alpha }{t}^{(2n+1)\alpha }={\displaystyle \frac{\mathrm{\Gamma }\left(\right(2n+1)\alpha +1)}{\mathrm{\Gamma }\left(\right(2n+2)\alpha +1)}}{t}^{(2n+2)\alpha }.$$

Therefore,

$$u_{n+1}={(-1)}^{n+1}{e}^x{\displaystyle \frac{{\prod }_{k=1}^{n+1}\mathrm{\Gamma }((2k-1)\alpha +1)}{{\prod }_{k=1}^{n+1}\mathrm{\Gamma }(2k\alpha +1)}}{t}^{2(n+1)\alpha }.$$

This completes the induction.

Step 5: Series representation.

Summing the iterates yields the solution as the convergent power-type series

$$u(x,t)=\sum _{n=0}^{\infty }{u}_n(x,t),$$

which gives the stated expansion. □

5. Higher-Dimensional Time Fractional Tricomi Equation with Variable Coefficients

Consider the following higher-dimensional time fractional Tricomi equation

$$\begin{array}{c}\hfill {\mathrm{D}}_t^{\alpha }u+\xi \left(t\right){\Delta }_{x}u=f(x,t)\end{array}$$

(9)

with initial conditions

$$\begin{array}{c}\hfill u(x,0)={\omega }_1\left(x\right),{\mathrm{D}}_{t}u(x,0)={\omega }_2\left(x\right).\end{array}$$

(10)

where $1<\alpha \le 2,t\in [0,T],x\in {\mathbb{R}}^n$, and ${\Delta }_x$ denotes the standard Laplace operator and it is defined by ${\Delta }_{x}u(x,t)={\sum }_{i=1}^n{\partial }_{x_i}^{2}u(x,t)$.

Similarly, one can obtain the equivalent integral Equations (9) and (10) as follows

$$u(x,t)={\omega }_1\left(x\right)+{\omega }_2\left(x\right)t+{\mathrm{I}}_{\mathrm{t}}^{\alpha }f(x,t)-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\Delta }_{x}u(x,t)\right).$$

Suppose that the unknown function u has the series form

$$u(x,t)=\sum _{n=0}^{\infty }{u}_n(x,t)=u_0+u_1+u_2+\cdots ,$$

where the sequence $\left\{u_n\right\}$ can be obtained by the following iterative process

$$\begin{array}{cc}\hfill u_0& ={\omega }_1\left(x\right)+{\omega }_2\left(x\right)t+{\mathrm{I}}_t^{\alpha }f(x,t)\hfill \\ \hfill u_1& =-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\Delta }_x{u}_0\right)\hfill \\ \hfill u_2& =-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\Delta }_x{u}_1\right)\hfill \\ \hfill ⋮\\ \hfill u_{m+1}& =-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\Delta }_x{u}_m\right).\hfill \end{array}$$

Proposition 6.

If ${\omega }_1\left(x\right)\equiv 0,{\omega }_2\equiv 0,f(x,t)\equiv 0$, then for any reasonable $\xi \left(t\right)$, $u(x,t)\equiv 0$.

Proposition 7.

Assume ${\omega }_1\left(x\right)\equiv 0$, ${\omega }_2\left(x\right)\equiv 0$, $f(x,t)\equiv e^{x_1+x_2+\cdots +x_n}$ and $\xi \left(t\right)\equiv -1$.

• Iterative construction. Since the initial data vanish, the first term of the iteration is obtained directly from the source term:

$$u_0=e^{x_1+\cdots +x_n}{\displaystyle \frac{t^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}.$$

Applying the iterative scheme once more and using

$${\mathcal{I}}_t^{\alpha }{t}^{\beta }={\displaystyle \frac{\mathrm{\Gamma }(\beta +1)}{\mathrm{\Gamma }(\beta +\alpha +1)}}{t}^{\beta +\alpha },$$

we obtain

$$u_1=e^{x_1+\cdots +x_n}{\displaystyle \frac{t^{2\alpha }}{\mathrm{\Gamma }(2\alpha +1)}}.$$

Proceeding inductively, one easily verifies that

$$u_n=e^{x_1+\cdots +x_n}{\displaystyle \frac{t^{(n+1)\alpha }}{\mathrm{\Gamma }\left(\right(n+1)\alpha +1)}},\forall n\in \mathbb{N}.$$

Series solution. Summing all iterates,

$$\begin{array}{cc}\hfill u(x,t)& =\sum _{n=0}^{\infty }{u}_n\hfill \\ \hfill & =e^{x_1+\cdots +x_n}\sum _{n=0}^{\infty }{\displaystyle \frac{t^{(n+1)\alpha }}{\mathrm{\Gamma }\left(\right(n+1)\alpha +1)}}\hfill \\ \hfill & =e^{x_1+\cdots +x_n}\sum _{n=1}^{\infty }{\displaystyle \frac{t^{n\alpha }}{\mathrm{\Gamma }(n\alpha +1)}}.\hfill \end{array}$$

Rewriting the series,

$$u(x,t)=e^{x_1+\cdots +x_n}\left(\sum _{n=0}^{\infty }{\displaystyle \frac{t^{n\alpha }}{\mathrm{\Gamma }(n\alpha +1)}}-1\right).$$

Hence,

$$u(x,t)=e^{x_1+\cdots +x_n}\left(E_{\alpha }\left(t^{\alpha }\right)-1\right).$$

Proposition 8.

Assume ${\omega }_1\left(x\right)\equiv 0$, ${\omega }_2\left(x\right)\equiv 0$,

$$f(x,t)\equiv e^{x_1+x_2+\cdots +x_n}{\displaystyle \frac{t^{\alpha }}{\mathrm{\Gamma }(1-\alpha )}},\xi \left(t\right)\equiv -1.$$

Iterative construction. Since the initial data vanish and the source term already contains $t^{\alpha }$, the first iterate reduces to

$$u_0=e^{x_1+\cdots +x_n}.$$

Applying the iterative scheme once,

$$u_1=e^{x_1+\cdots +x_n}{\displaystyle \frac{t^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}.$$

Continuing in the same manner, one verifies inductively that

$$u_n=e^{x_1+\cdots +x_n}{\displaystyle \frac{t^{n\alpha }}{\mathrm{\Gamma }(n\alpha +1)}},\forall n\in \mathbb{N}.$$

Series solution. Summing the iterates gives

$$\begin{array}{cc}\hfill u(x,t)& =\sum _{n=0}^{\infty }{u}_n\hfill \\ \hfill & =e^{x_1+\cdots +x_n}\sum _{n=0}^{\infty }{\displaystyle \frac{t^{n\alpha }}{\mathrm{\Gamma }(n\alpha +1)}}.\hfill \end{array}$$

Recognizing the Mittag–Leffler series,

$$E_{\alpha }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{z^n}{\mathrm{\Gamma }(n\alpha +1)}},$$

we conclude that

$$u(x,t)=e^{x_1+\cdots +x_n}{E}_{\alpha }\left(t^{\alpha }\right).$$

Proposition 9.

Assume ${\omega }_1\left(x\right)\equiv 0$, ${\omega }_2\left(x\right)\equiv e^x$, $f(x,t)\equiv 0$, and

$$\xi \left(t\right)\equiv -{\displaystyle \frac{t^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}.$$

Iterative construction. Since the source term vanishes and the initial velocity is $e^x$, the first term of the iteration is

$$u_0=e^x.$$

Proceeding with the iterative scheme, one obtains successively terms of increasing order in $t^{\alpha }$. By induction it follows that

$$u_n=e^x{\displaystyle \frac{t^{n\alpha }}{\mathrm{\Gamma }(n\alpha +1)}},\forall n\in \mathbb{N}.$$

Series solution. Summing the iterates,

$$\begin{array}{cc}\hfill u(x,t)& =\sum _{n=0}^{\infty }{u}_n\hfill \\ \hfill & =e^x\sum _{n=0}^{\infty }{\displaystyle \frac{t^{n\alpha }}{\mathrm{\Gamma }(n\alpha +1)}}.\hfill \end{array}$$

Recognizing the Mittag–Leffler expansion, we conclude that

$$u(x,t)=e^x{E}_{\alpha }\left(t^{\alpha }\right).$$

Proposition 10.

Assume

$${\omega }_1\left(x\right)\equiv e^{x_1+\cdots +x_n},{\omega }_2\left(x\right)\equiv 0,f(x,t)\equiv 0,\xi \left(t\right)\equiv t^{\alpha }.$$

First iterates. The initial term is

$$u_0=e^{x_1+\cdots +x_n}.$$

Since ${\partial }_x^2{e}^{x_1+\cdots +x_n}=e^{x_1+\cdots +x_n}$, the first iteration gives

$$\begin{array}{cc}\hfill u_1& =-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\partial }_x^2{u}_0\right)\hfill \\ \hfill & =-e^{x_1+\cdots +x_n}{\mathrm{I}}_t^{\alpha }{t}^{\alpha }\hfill \\ \hfill & =-e^{x_1+\cdots +x_n}{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)}}{t}^{2\alpha }.\hfill \end{array}$$

Applying the scheme once more,

$$\begin{array}{cc}\hfill u_2& =-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\partial }_x^2{u}_1\right)\hfill \\ \hfill & =e^{x_1+\cdots +x_n}{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)}}{\mathrm{I}}_t^{\alpha }{t}^{3\alpha }\hfill \\ \hfill & =e^{x_1+\cdots +x_n}{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }(3\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)\mathrm{\Gamma }(4\alpha +1)}}{t}^{4\alpha }.\hfill \end{array}$$

Inductive step. Assume that for some $k\ge 1$,

$$u_k={(-1)}^{k+1}{e}^{x_1+\cdots +x_n}{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }(3\alpha +1)\cdots \mathrm{\Gamma }\left(\right(2k-1)\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)\mathrm{\Gamma }(4\alpha +1)\cdots \mathrm{\Gamma }(2k\alpha +1)}}{t}^{2k\alpha }.$$

Then

$$\begin{array}{cc}\hfill u_{k+1}& =-{\mathrm{I}}_t^{\alpha }\left(\xi \left(t\right){\partial }_x^2{u}_k\right)\hfill \\ \hfill & ={(-1)}^{k+2}{e}^{x_1+\cdots +x_n}{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)\cdots \mathrm{\Gamma }\left(\right(2k-1)\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)\cdots \mathrm{\Gamma }(2k\alpha +1)}}{\mathrm{I}}_t^{\alpha }{t}^{(2k+1)\alpha }.\hfill \end{array}$$

Using

$${\mathrm{I}}_t^{\alpha }{t}^{\beta }={\displaystyle \frac{\mathrm{\Gamma }(\beta +1)}{\mathrm{\Gamma }(\beta +\alpha +1)}}{t}^{\beta +\alpha },$$

we obtain

$$u_{k+1}={(-1)}^{k+2}{e}^{x_1+\cdots +x_n}{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)\cdots \mathrm{\Gamma }\left(\right(2k+1)\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)\cdots \mathrm{\Gamma }\left(\right(2k+2)\alpha +1)}}{t}^{(2k+2)\alpha }.$$

Hence, by induction, for all $n\in \mathbb{N}$,

$$u_n={(-1)}^{n+1}{e}^{x_1+\cdots +x_n}{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }(3\alpha +1)\cdots \mathrm{\Gamma }\left(\right(2n-1)\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)\mathrm{\Gamma }(4\alpha +1)\cdots \mathrm{\Gamma }(2n\alpha +1)}}{t}^{2n\alpha }.$$

Series representation. Summing the iterates,

$$u(x,t)=\sum _{n=0}^{\infty }{u}_n,$$

that is,

$$u(x,t)=e^{x_1+\cdots +x_n}\sum _{n=0}^{\infty }{(-1)}^{n+1}{\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }(3\alpha +1)\cdots \mathrm{\Gamma }\left(\right(2n-1)\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)\mathrm{\Gamma }(4\alpha +1)\cdots \mathrm{\Gamma }(2n\alpha +1)}}{t}^{2n\alpha }.$$

6. Concluding Remarks

In this paper, we systematically investigate an innovative iterative technique for solving a broad class of functional equations that frequently arise in mathematical analysis and applied mathematics. Building upon established theoretical foundations, we develop and rigorously analyze this computational method, demonstrating its effectiveness through both theoretical considerations and practical applications.

The primary focus of our study lies in the application of this methodology to initial value problems associated with time fractional partial differential equations. Through careful implementation of our approach, we are able to derive either exact analytical solutions or highly accurate approximate solutions, depending on the specific characteristics of each problem under consideration.

A particularly noteworthy aspect of our research reveals that this iterative technique offers several significant advantages: (1) conceptual simplicity that facilitates straightforward implementation, (2) computational efficiency that enables practical solutions to complex problems, and (3) remarkable adaptability to various types of nonlinear systems. Most importantly, we establish that our method maintains these advantageous properties even when applied to challenging nonlinear systems of differential equations with initial conditions, where many traditional approaches often encounter difficulties.

The robustness and versatility of our proposed method are thoroughly examined through multiple carefully selected examples, which serve to illustrate both the theoretical framework and practical applications. Our results suggest that this approach may provide a valuable addition to the existing toolkit for analyzing fractional differential equations and related functional equations in applied mathematics.