Employing a Fractional Basis Set to Solve Nonlinear Multidimensional Fractional Differential Equations

Abstract

Fractional-order partial differential equations have gained significant attention due to their wide range of applications in various fields. This paper employed a novel technique for solving nonlinear multidimensional fractional differential equations by means of a modified version of the Bernstein polynomials called the Bhatti-fractional polynomials basis set. The method involved approximating the desired solution and treated the resulting equation as a matrix equation. All fractional derivatives are considered in the Caputo sense. The resulting operational matrix was inverted, and the desired solution was obtained. The effectiveness of the method was demonstrated by solving two specific types of nonlinear multidimensional fractional differential equations. The results showed higher accuracy, with absolute errors ranging from ${10}^{-12}$ to ${10}^{-6}$ when compared with exact solutions. The proposed technique offered computational efficiency that could be implemented in various programming languages. The examples of two partial fractional differential equations were solved using Mathematica symbolic programming language, and the method showed potential for efficient resolution of fractional differential equations.

1. Introduction

Fractional-order partial differential equations are gaining popularity due to their increasing applications in various fields, such as physics and engineering. These types of equations were first discovered in the 18th century by Euler, Liouville, Lacroix and Abel [1,2,3,4,5], but at that time their significance was not understood. However, as the 20th century progressed, researchers realized that many real-world phenomena are governed by fractional-order differential and integral equations. Fractional-order equations can provide a deeper understanding of different physical and conceptual properties in fields such as engineering, chemistry, physics, optics, biology, and sociology [6,7,8,9,10,11,12,13,14]. Fractional-order calculus has been used to analyze and solve various phenomena, including Quantum mechanics [15,16], fluid-dynamics, traffic [17,18,19,20,21], colored noise [22], shock waves [18], underwater acoustics [23], statistical mechanics [24], soliton dynamics [25], earthquake science [26], materials properties [16,27], semiconductor [27], super-fluids [16], particle and plasma physics [6], cryptography [28], steganography [28], etc. The field of fractional calculus is being applied in physics, chemistry, mathematics, engineering, and biosciences [29,30,31,32,33,34,35,36,37]. As a result, there has been a growing need to find solutions for these types of equations.

Many research studies have been conducted to approximate solutions for fractional differential equations. Different authors have used a range of analytical and numerical methods, such as the Adomian Decomposition Method [38], Laplace Transform [39], Galerkin technique [40], Bernstein method [41], finite difference schemes method [42], Monotone iterative technique [43], wavelet method [44], and many others [39,45]. These methods vary in their level of complexity, but most of them are analytical methods, with only a few of them being numerical methods. Numerical methods have been found to be more manageable when applied to fractional differential equations and can be implemented using symbolic coding for an efficient solution [41]. In recent studies, the use of modified Bhatti-fractional polynomials has been explored as a means to solve nonlinear partial fractional differential equations. Many authors have successfully solved linear and nonlinear fractional differential equations using this method, as cited in their works [46,47,48,49,50,51,52,53,54,55].

This paper presents the application of a novel technique for solving multidimensional Nonlinear Fractional Differential Equations (NFDE) using a modified version of the Bernstein polynomial known as the Bhatti-fractional polynomials. In previous research articles, we assessed our method by applying it to linear and nonlinear one-dimensional fractional equations [48,51,52,53,54]. In those studies, we conducted comparisons between our method and existing approaches, and the results consistently demonstrated that our method outperforms the others in terms of both accuracy and computational efficiency. Motivated by the facts, in this article we demonstrate the effectiveness of our method in the case of multidimensional NFDE by applying it to two specific nonlinear partial fractional differential equations. Our approach involves, first, approximating a desired solution with initial conditions and second, substituting it into the nonlinear equation. The resulting equation is rearranged and treated as a matrix equation. This matrix represents the operational matrix, which incorporates the initial and boundary conditions. The operational matrix equation is inverted and solved to obtain an estimated solution.

The paper is organized in six Sections. In Section 2, we briefly describe the basic preliminaries of Bhatti-fractional polys and Caputo fractional differentiation rules to be used in the calculation of the solutions. We also present a rule for taking fractional derivatives of fractional polynomials. A linear combination of fractional B-polys is taken to approximate the desired solution of a partial fractional differential equation. Section 3 consists of application of the method to two fractional partial differential equations. An absolute error is determined by taking the absolute difference between exact and numerical solutions. Our results are compared with the known numerical results of other authors. In Section 4, we apply the method to two NFDEs and presents results to compare with the existing methods. Section 5 discusses the possible error analysis for the current method for a particular example considered in the article. Finally, we present conclusions about the current method and its limitation in terms of the accuracy of the results. A review of the comparison is also described in this section.

2. Basic Definition of B-polys and Fractional Differentiation Rules

This study considers Bhatti-fractional polynomials as a modified version of the Bernstein polynomials for partial fractional polynomials. Here we present a brief review of the Bhatti-fractional polynomials. In terms of a single variable x over the range [0, R], fractional B-polys are defined as follows [49,50,51,52,53]:

$$B_{i,n}\left(\alpha ,x\right)={\displaystyle \sum _{k=0}^n}{\beta }_{i,k}{\left(\frac{x}{R}\right)}^{\alpha k}$$

(1)

An $(n+1)$ fractional-order B-polynomial basis set is provided by Equation (1), where the fractional-order parameter $\alpha$ reflects the fractional B-polys and n represents the number of polynomials with n-degree. When $\alpha$ is replaced by 1, Equation (1) will represent Bernstein polynomials. The factor ${\beta }_{i,k}$ is described in Equation (1) as follows:

$${\beta }_{i,k}={\left(-1\right)}^{i-k}\left(\begin{array}{c}n\\ k\end{array}\right)\left(\begin{array}{c}k\\ i\end{array}\right)$$

(2)

The following are expressions for the fractional Caputo differentiation [51],

$$\begin{array}{c}{{}_a{}^{C}D}_t^{\gamma }f\left(t\right)=\frac{1}{\Gamma \left(n-\gamma \right)}{\int }_a^t\frac{f^{\left(n\right)}\left(\tau \right)}{{\left(t-\tau \right)}^{\gamma +1-n}}d\tau .\\ \mathrm{for}n-1<\gamma <n,n\in N,t>0,f\in C^n.\end{array}$$

(3)

In this scheme, any constant has zero Caputo’s fractional derivative, $D^{\gamma }C=0$. Let us consider polynomial $f\left(t\right)=t^{\alpha }$, and $a=0$ [56]. Therefore, the fractional derivative of a fractional polynomial is defined as follows:

$$D^{\gamma }{x}^{\alpha }=\left\{\begin{array}{cc}0\hfill & \hfill \mathrm{for}\alpha \in N_0\mathit{and}\alpha <\gamma \\ & \\ \frac{\Gamma \left(\alpha +1\right)x^{\alpha -\gamma }}{\Gamma \left(\alpha +1-\gamma \right)}\hfill & \hfill \mathrm{otherwise}.\end{array}\right.$$

(4)

In this scenario, the order of the fractional derivative is represented by $\gamma$ and the degree of the fractional poly is denoted by $\alpha$.

3. Approximate Solution with Initial Condition

The desired approximate solution to the nonlinear fractional differential equation may be considered as a linear combination of fractional B-polys,

$$U\left(x,t\right)={\displaystyle \sum _{i=0}^n}{a}_i\left(t\right)B_{i,n}\left(\alpha ,x\right).$$

(5)

where $a_i\left(t\right)$ is represented by a set of polynomials as a function of another variable t, and n represents the number of polynomials.

$$a_i\left(\alpha ,t\right)={\displaystyle \sum _{j=0}^n}{b}_j^i{B}_{j,n}(\alpha ,t).$$

(6)

With respect to the nonlinear fractional differential equation we are exploring, the desired solution with initial condition is as follows:

$$U\left(x,t\right)={\displaystyle \sum _{i=0}^n}{a}_i(\alpha ,t)B_{i,n}(\alpha ,x)+f\left(x\right).$$

(7)

where $f\left(x\right)$ is the initial condition at t = 0 in the interval [0, T]. In the following section, we are going to apply our multidimensional technique to two examples of NFDEs and the results will be compared with exact solutions. Also, for simplicity, we will ignore indices $\alpha$ and n-degree on the fractional-order B-polys ($B_{i,n}(\alpha ,x)$ = $B_i\left(t\right)$) in the following sections.

4. Application of the Technique

Example 1: Let us consider the seeking of a solution to a multidimensional nonlinear fractional differential equation, which is given by [57],

$$\frac{{\partial }^{\gamma }U}{\partial t^{\gamma }}+U\frac{\partial U}{\partial x}=\left(x+x{t}^2\right).$$

(8)

The exact analytic solution of this Equation (8) is well known for $\gamma =1$, and it is provided by $U_{exact}\left(x,t\right)=xt$. We utilize the initial condition $U\left(x,0\right)=f\left(x\right)=0$ and the numerical solution of this equation is sought out in the intervals $0\le t\le 1$ and $0\le x\le 1$. The above equation’s approximate solution can be expressed in terms of the fractional B-poly expansion $U\left(x,t\right)={\sum }_{i=0}^n{a}_i\left(t\right)B_i\left(x\right)+f\left(x\right)$, where $a_i\left(t\right)={\sum }_{j=0}^n{b_j^{i}B}_j\left(t\right)$. After inserting these fractional B-poly expansions into Equation (8) and using the Galerkin method [56] for minimization, we obtain the following expression,

$${\displaystyle \sum _{i,j=0}^n}{b}_j^i[⟨B_i\left(x\right)|B_n\left(x\right)⟩⟨B_j^{\gamma }\left(t\right)|B_m\left(t\right)⟩+{\displaystyle \sum _{k,l=0}^n}{b}_l^k{g}_{ikn}^{\prime }{g}_{jlm}]=⟨\left(x+x{t}^2\right)|B_n\left(x\right)B_m\left(t\right)⟩.$$

(9)

where, in Equation (9), $g_{ikn}^{\prime }$ and $g_{jlm}$ are the nonlinear terms that display nonlinearity through the unknown coefficients $b_l^k$. We can arrange this equation in terms of matrices, so that,

$$\Rightarrow B\left[A_{mn}+D_{mn}\right]=W_{mn},$$

(10)

where matrices A, D and W are represented by the matrix elements,

$$\begin{array}{c}{A}_{mn}={\displaystyle \sum _{i,j=0}^n}⟨B_i\left(x\right)|B_n\left(x\right)⟩⟨B_j^{\gamma }\left(t\right)|B_m\left(t\right)⟩={\displaystyle \sum _{i,j=0}^n}\underset{0}{\overset{R}{{\displaystyle \int }}}{B}_i\left(x\right)B_n\left(x\right)dx\underset{0}{\overset{T}{{\displaystyle \int }}}{D^{\gamma }(B}_j\left(t\right))B_m\left(t\right)dt\\ D_{mn}={\displaystyle \sum _{k,l=0}^n}{b}_l^k{g}_{ikn}^{\prime }{g}_{jlm}={\displaystyle \sum _{k,l}^n}{b}_l^k{{\displaystyle \int }}_0^R{B}_i^{\prime }\left(x\right)B_k\left(x\right)B_n\left(x\right)dx{{\displaystyle \int }}_0^T{B}_j\left(t\right)B_l\left(t\right)B_m\left(t\right)dt,\\ W_{mn}=⟨\left(x+x{t}^2\right)|B_n\left(x\right)B_m\left(t\right)⟩=\underset{0}{\overset{R,T}{{\displaystyle \iint }}}\left(x+x{t}^2\right)B_n\left(x\right)B_m\left(t\right)dxdt.\end{array}$$

(11)

In this equation, $D$ stands for the nonlinear terms, $B^{\prime }\left(x\right)$ represents the ordinary first-order derivative and $D^{\gamma }$ represents the fractional-order derivative. Without considering its nonlinear component in first iteration, Equation (10) can be written,

$$\Rightarrow B\left[A_{mn}\right]=W_{mn}.$$

(12)

This formula results in an $\left(n+1\right)$ by $\left(n+1\right)$ system of equations $XB=W_{mn}$, in unknown variables $B=\{b_1^1,b_2^1,b_3^1,\dots ,b_1^2,b_2^2,b_3^2,\dots ,\}$, which comprise the matrix B of initial guess and $X=\left[A_{mn}+D_{mn}\right]$ or $\left[A_{mn}\right]$. Equations (10) and (12) gives the matrices $X\&W$ in terms of the inner products of fractional B-polys which are evaluated using a computer code. The value of the unknown variable coefficients of B is determined in the linear equation using Equation (12). As an initial guess, the coefficient values of B matrix are substituted back into the nonlinear Equation (10) to obtain a refined guess. This iterative procedure is repeated to obtain converged values of the unknown coefficients. The operational matrix is $X$, whose inverse is multiplied by the column matrix $W_{mn}$ to obtain values of the unknown coefficients $b_j^i$, which are generated from the nonlinear partial differential equation by solving the equation $B=X^{-1}{W}_{mn}$. Initial conditions are enforced by modifying the rows and the corresponding columns of Equation (10) before building the operational matrix equation $XB=W_{mn}$, for the result to vanish at $t=0$ and $x=0$. The subsequent approximate solution is obtained using the B-poly basis set and the converged coefficients. When the iterations were finished, the coefficient $b_j^i$ values converged to values $\{0,0,0,1\}$ for example 1. The solution for Equation (8) is obtained by using this procedure, and the converged solution is given for the closed intervals $t\in \left[0,1\right]$ and $x\in \left[0,1\right]$,

$$U\left(x,t\right)=t(0.\times {10}^{-30}+1.0x)\approx xt.$$

(13)

The above result is highly accurate and matches the exact solution $U\left(x,t\right)=xt.$ For the case $\gamma =1$, the B-polys with $n=1$ have been applied in both $x$ and $t$ variables in order to solve the Nonlinear Fractional Partial Differential (NFPD) Equation (8). For the two B-polys basis sets employed $\{1-t,t\}$ and $\{1-x,x\}$, multiplying both sets produced a 4 × 4 operational matrix. The perfect agreement between the two results, exact and numerical solutions, at the level of machine accuracy is demonstrated in Figure 1. We find that, when $t=x$ is substituted in Equation (13), the order of accuracy in the solution can be reached at the level of ${10}^{-12}$, which shows the excellent quality of the numerical solution in one-dimension $x$. For $\gamma =1$, a graph of both solutions, numerical and the solution from Ref. [57], is shown in Figure 1 (left). Both solutions match very well.

For further comparison of our solution with that in Ref. [57], we have created

Table 1. For $\gamma =1$ and x = 0.25, the comparison of the B-poly method to the other method is presented [57]. The values of x = 0.25 and t vary. It is shown that both methods agree except at the last point in the table.

Value of x = 0.25 and t Varies t	$\mathbf{Value}\mathbf{of}\mathit{U}(\mathit{x},\mathit{t})\mathbf{When}\mathit{\gamma }=1$ from Current Method	$\mathbf{Value}\mathbf{of}\mathit{U}(\mathit{x},\mathit{t})\mathbf{for}\mathit{\gamma }=1$ from the Reference [57], h = −0.75
0	$0$	0
0.2	$0.050000$	$0.050038$
0.4	$0.100000$	$0.100216$
0.6	$0.150000$	$0.150346$
0.8	$0.200000$	$0.200173$
1	$0.250000$	$0.251506$

for both solutions for $\gamma =1$, x = 0.25 and variable t = 0, 0.2, 0.4, 0.6, 0.8, 1. We find that both solutions agree, but they slightly disagree at the end of the intervals. This is because the solution from Ref. [57] depends on an adjuatable parameter h = −0.75. Our solutions do not depend on a fixed parameter.

Next, we are going to campare values of both solutions for various fractional-orders of $\gamma$ below: take $\gamma =0.8$ and x = 0.25, the solutions are plotted as shown in Figure 2. It is shown that both graphs match, except for the last point in the graph. This happens because the solution of the same example 1 in Ref. [57] depends on a parameter h = −0.75 (see

Table 2. For $\gamma =0.8$ and x = 0.25, the comparison of the B-poly method to the other method in Ref. [57] is presented for various values of t variable. The values disagree near the end of the interval [0, 1].

Value of x = 0.25 and t Varies t	$\mathbf{Value}\mathbf{of}\mathbf{U}(\mathit{x},\mathit{t})\mathbf{When}\mathit{\gamma }=0.8$ Using Current Method	$\mathbf{Value}\mathbf{of}\mathbf{U}(\mathit{x},\mathit{t})\mathbf{for}\mathit{\gamma }=0.8$ from [57], h = −0.75
0	0	0
0.2	$0.072434$	$0.072556$
0.4	$0.122903$	$0.123100$
0.6	$0.167311$	$0.166983$
0.8	$0.209886$	$0.209714$
1	$0.252794$	$0.262224$

).

For further comparion with the solution in Ref. [57], we have created Table 2 for both solutions for $\gamma =0.8$, x = 0.25 and variable t = 0, 0.2, 0.4, 0.6, 0.8, 1. It is shown that both solutions agree, but they slightly disagree at the end of the intervals.

Now we comapre the graphs for $\gamma =0.5$ and x = 0.25. The solutions are plotted as shown in Figure 3. It is shown that both graphs match, except the last few point in the graph. This happens because the solution of the same example in Ref. [57] depends on a parameter h = −0.75, as its value can be adjusted to exactly match with the B-poly method. Our method does not depend on any external ajustable parameter. It is believed that the results from fractioanl B-polys are accurate.

It is shown that as fractional-order $\gamma$ takes a smaller value than 1, the graphs for the soltuions differ quite significantly. As a matter of fact, the solution from Ref. [57] diverges rapidly for $\gamma$ less than 1. However, the solutions obtained from the current method do not diverge near the end of the intervals. A comaprison of the solutions for $\gamma =0.5$ is shown in

Table 3. For $\gamma =0.5$, the comparison of the B-poly method to the other method [57] is presented. The value of x = 0.25 is fixed and t varies.

Value of x = 0.25 and t Varies t	$\mathbf{Value}\mathbf{of}\mathbf{U}(\mathit{x},\mathit{t})\mathbf{When}\mathit{\gamma }=0.5$ Using Current Method	$\mathbf{Value}\mathbf{of}\mathbf{U}(\mathit{x},\mathit{t})\mathbf{for}\mathit{\gamma }=0.5$ from Ref. [57] and h = −0.75
0	$0$	0
0.2	$0.111518$	$0.111237$
0.4	$0.150163$	$0.149572$
0.6	$0.189551$	$0.182115$
0.8	$0.250308$	$0.215547$
1	$0.377037$	$0.252536$

. It is shown in Table 3 that values for the solutions differ from each other near the end of the t-interval.

Example 2: We consider another multidimensional nonlinear fractional differential equation taken from [58],

$$\frac{{\partial }^{\gamma }U}{\partial t^{\gamma }}+U\frac{\partial U}{\partial x}=1+x+t.$$

(14)

when fractional order $\gamma =1$, the exact solution to Equation (14) is given by $U_{exact}\left(x,t\right)=x+t$. We use the initial condition $U\left(x,0\right)=f\left(x\right)=x$ and calculate the numerical solution to this problem in the intervals $0\le t\le 1$ and $0\le x\le 1$. As before in the first example, this equation’s estimated solution can be expressed in terms of fractional B-polys, $U\left(x,t\right)={\sum }_{i=0}^n{a}_i\left(t\right)B_i\left(x\right)+f\left(x\right)$, where t dependent coefficients are given by $a_i\left(t\right)={\sum }_{j=0}^n{b}_j^i{B}_j\left(t\right)$. After we put this approximate solution into Equation (14) and after modifying this equation in the Galerkin approximation, we arrive at an equation resembling Equation (15),

$$\begin{array}{c}{\displaystyle \sum _{i,j=0}^n}{b}_j^i[⟨B_i\left(x\right)|B_n\left(x\right)⟩⟨B_j^{\gamma }\left(t\right)|B_m\left(t\right)⟩+⟨{xB}_i^{\prime }\left(x\right)|B_n\left(x\right)⟩⟨B_j\left(t\right)|B_m\left(t\right)⟩\\ +⟨B_i^{\prime }\left(x\right)|B_n\left(x\right)⟩⟨B_j\left(t\right)|B_m\left(t\right)⟩+{\displaystyle \sum _{k,l=0}^n}{b}_l^k{g}_{ikn}^{\prime }{g}_{jlm}]=⟨\left(1+t\right)|B_n\left(x\right)B_m\left(t\right)⟩.\end{array}$$

(15)

where in Equation (15) $g_{ikn}^{\prime }$ and $g_{jlm}$ are the nonlinear terms that display nonlinearity through the unknown coefficients $b_l^k$. We organize this equation as follows,

$$\Rightarrow B\left[A_{mn}+C_{mn}+D_{mn}+E_{mn}\right]=W_{mn},$$

(16)

where the matrix elements of these matrices are given as follows,

$$\begin{array}{c}{A}_{mn}={\displaystyle \sum _{i,j=0}^n}⟨B_i\left(x\right)|B_n\left(x\right)⟩⟨B_j^{\gamma }\left(t\right)|B_m\left(t\right)⟩={\displaystyle \sum _{i,j=0}^n}\underset{0}{\overset{R}{{\displaystyle \int }}}{B}_i\left(x\right)B_n\left(x\right)dx\underset{0}{\overset{T}{{\displaystyle \int }}}{D^{\gamma }(B}_j\left(t\right))B_m\left(t\right)dt\\ C_{mn}={\displaystyle \sum _{i,j=0}^n}⟨{xB}_i^{\prime }\left(x\right)|B_n\left(x\right)⟩⟨B_j\left(t\right)|B_m\left(t\right)⟩={\displaystyle \sum _{i,j=0}^n}\underset{0}{\overset{R}{{\displaystyle \int }}}{xB}_i^{\prime }\left(x\right)B_n\left(x\right)dx\underset{0}{\overset{T}{{\displaystyle \int }}}{B}_j\left(t\right)B_m\left(t\right)dt\\ D_{mn}={\displaystyle \sum _{i,j=0}^n}⟨B_i^{\prime }|B_n\left(x\right)⟩⟨B_j\left(t\right)|B_m\left(t\right)⟩={\displaystyle \sum _{i,j=0}^n}\underset{0}{\overset{R}{{\displaystyle \int }}}{B}_i^{\prime }\left(x\right)B_n\left(x\right)dx\underset{0}{\overset{T}{{\displaystyle \int }}}{B}_j\left(t\right)B_m\left(t\right)dt\\ E_{mn}={\displaystyle \sum _{k,l=0}^n}{b}_l^k{g}_{ikn}^{\prime }{g}_{jlm}={\displaystyle \sum _{k,l}^n}{b}_l^k{{\displaystyle \int }}_0^R{B}_i^{\prime }\left(x\right)B_k\left(x\right)B_n\left(x\right)dx{{\displaystyle \int }}_0^T{B}_j\left(t\right)B_l\left(t\right)B_m\left(t\right)dt,\\ W_{mn}=⟨\left(1+t\right)|B_n\left(x\right)B_m\left(t\right)⟩=\underset{0}{\overset{R,T}{{\displaystyle \iint }}}\left(1+t\right)B_n\left(x\right)B_m\left(t\right)dxdt.\end{array}$$

(17)

This linear version of the equation can be written as follows, without considering the nonlinear component (matrix E = 0) of equation (16),

$$\Rightarrow B\left[A_{mn}+C_{mn}+D_{mn}\right]=W_{mn}.$$

(18)

This algorithm leads to an $\left(n+1\right)$ by $\left(n+1\right)$ system of equations $XB=W_{mn}$, in unknown variables $B=\{b_1^1,b_2^1,b_3^1,\dots ,b_1^2,b_2^2,b_3^2,\dots ,\}$, which comprise the matrix B and $X=\left[A_{mn}+C_{mn}+D_{mn}+E_{mn}\right],$ or without the nonlinear term $X=\left[A_{mn}+C_{mn}+D_{mn}\right]$. The matrices $XandW$ are based on the inner products of fractional-order B-polys given in Equations (16) and (17). The initial guess of the unknown value of B may be determined from the linear Equation (18). Next, these values for the B matrix are substituted into Equation (16). The column matrix $W_{mn}$ is multiplied by the inverse of the operational matrix X created from Equation (16) to produce the values of the unknown coefficients, $b_j^i$ or $B,$ utilizing the formula $B=X^{-1}{W}_{mn}$. Initial conditions are imposed by removing the rows and corresponding columns from Equation (16), causing the solution to vanish at $t=0$ and $x=0$. The fractional-order B-poly basis set is used to derive the final approximate solution using Equation (7). The converged coefficient values of $b_j^i=\left\{0,0,0,1\right\}$ were reached after seven iterations. Using this method, the approximate result for Equation (14), for closed intervals $t\in \left[0,1\right]$ and $x\in \left[0,1\right]$, is achieved,

$$U\left(x,t\right)=t\left(1.0+0.\times {10}^{-30}x\right)+x\approx x+t.$$

(19)

From the above result, the outcome is exceptionally accurate solutions for Equation (14). For the special case $\gamma =1$, the B-polys of degree n = 1 have been used in both variables x and t to solve the nonlinear partial differential Equation (14). A 4 × 4 operational matrix was produced by multiplying the two B-poly basis sets employed, $\{1-t,t\}$ and $\{1-x,x\}$. For the sake of comparison, we have provided plots of the exact and numerical results of Equation (19). Figure 4 illustrates the precise agreement between the two solutions at the level of machine accuracy. We find that, when $t=x$ is substituted in Equation (19), the error of the order ${10}^{-16}$ is detected, which shows excellent resolution in one-dimensional space. The plot of both solutions for $\gamma =1$, and x = 0.25, the numerical solution and exact solution, match as shown in Figure 4.

The nonlinear fractional differential Equation (14) has an exact general solution provided in Ref. [58] that can be compared with the present solutions. As a special case when $\gamma =1/2$, the exact solution is given [58],

$$U\left(x,t\right)=x+t+2-2e^{t}erfc\left(\sqrt{t}\right)-2\sqrt{\frac{t}{\pi }}$$

(20)

At a fixed value of x = 0.25, the solutions are compared for various values of t and $\gamma =1/2$ in

Table 4. For $\gamma =1/2$, the comparison of the fractional B-poly method to the general solution from the reference is presented [58].

Value of x = 0.25 and t Varies t	$\mathbf{Value}\mathbf{of}\mathbf{U}(\mathit{x},\mathbf{t})\mathbf{When}\mathit{\gamma }=\frac{1}{2}$ from Current Method	$\mathbf{Value}\mathbf{of}\mathbf{U}(\mathit{x},\mathbf{t})\mathbf{for}\mathit{\gamma }=\frac{1}{2}$ from the General Solution Ref. [58]
0	$0.25$	$0.25$
0.2	$0.657794$	$0.657797$
0.4	$0.829134$	$0.829138$
0.6	$0.979911$	$0.979912$
0.8	$1.124255$	$1.124255$
1.0	$1.266451$	$1.266454$

. It is noted that both exact solution and converged numerical solution matched after using n = 6 fractional B-polys and seven iterations. The comparison of values of the solutions is provided in Table 4. A graph of both solutions for $\gamma =\frac{1}{2}$ is also presented in Figure 5 for the purpose of comparison. In this case, again both graphs overlap, showing excellent agreement of solutions. As mentioned earlier, the dotted line represents the B-poly solution method, and the dashed line represents the solution from Ref. [58].

For $\gamma =0.8$, both solutions are compared at a fixed value of x = 0.25 for various values of t shown in

Table 5. For $\gamma =0.8$, the comparison of the fractional B-poly solution method to the general solution from the reference is presented [58].

Value of x = 0.25 and t Varies t	$\mathbf{Value}\mathbf{of}\mathbf{U}(\mathit{x},\mathbf{t})\mathbf{When}\mathit{\gamma }=0.8$ from Current Method	$\mathbf{Value}\mathbf{of}\mathbf{U}(\mathit{x},\mathbf{t})\mathbf{for}\mathit{\gamma }=0.8$ from the General Solution Ref. [58]
0	$0.25$	$0.25$
0.2	$0.528553$	$0.528579$
0.4	$0.729216$	$0.729242$
0.6	$0.913309$	$0.913316$
0.8	$1.090647$	$1.090665$
1.0	$1.265139$	$1.265150$

and Figure 6. For this solution, n = 5 and six iterations were used to obtain the solutions compared with the solution from Ref. [58]. As we increase the number of fractional B-polys, the computational time increases. A graph of both solutions for $\gamma =0.8$ is also presented in Figure 6 for the purpose of comparison. In this case, again both graphs overlap, showing excellent agreement of solutions. As mentioned earlier, the dotted line represents the B-poly method, and the dashed line represents the solution from Ref. [58]. For the purpose of comparison, we have created Table 5 for values for both solutions. For $\gamma =0.8$, x = 0.25 and various values of t, the Table is provided below:

In the following section, we would like to present error analysis of example 2 in more depth for $\gamma =1/2$ as the number of fractional B-Polys are changed. It is shown that the solutions for the fractional B-Poly method are very accurate and are comparable with the available solutions for Refs. [57,58].

5. Error Analysis

We solved two examples of nonlinear fractional partial differential equations with one initial condition using fractional Bhatti-polynomials, defined over the intervals $x\in [0,1]$ and $t\in [0,1]$ (see Section 2). Our results depend on the chosen parameter $n$ which represents the number of fractional Bhatti-polynomials in the basis set used in the calculations. The error analysis is assessed based on the accuracy of the method as $n$ increases for a basis set. In Example 2, $n$ = 3 was chosen to give us 4 B-polys in t-variable and n = 1 was chosen to give 2 B-polys in x-variable. These two sets of polynomials were multiplied to give a total of 8 B-polys in a basis set. A basis set of eight fractional B-polys was used in the calculations which resulted in an 8 × 8 operational matrix to be inverted to find the solution. In the case of 4 B-polys in t-variable, we detected an absolute error on the order of ${10}^{-3}$ between the exact and numerical solutions. The computational time was about 15 min with seven iterations. The plot of both solutions is displayed in Figure 7. The absolute difference between the two solutions was further explored by increasing the number of fractional polynomials in the t-variable and keeping the same number of B-polys in the x-variable. Increasing the number of fractional B-polys in the t-variable to 6 gave us a set of 7 × 2 = 14 B-poly basis elements, and the calculations resulted in a 14 × 14-dimensional operational matrix that was solved to find the solution. This time the results were more accurate, and the absolute error was reduced to ${10}^{-6}$ orders of magnitude. At the cost of desired accuracy, the computational complexity increased. Beyond this point, the accurate matrix inversion became more problematic and time consuming. The computational time for these calculations was about 1 h and 10 min. A plot of these results is depicted in Figure 8.

Graphical representations (Figure 7 and Figure 8) illustrate how the absolute error decreases as the fractional B-poly basis set increases in size. The method, due to the analytical nature of fractional B-polynomials, does not require grid representations during integration and differentiation. The graphs (Figure 7 and Figure 8) depict the systematic error reduction with larger basis sets, ultimately converging to a solution comparable to the exact solutions. It is worth noting that larger basis sets lead to longer CPU times for computation.

6. Conclusions

Previous successful applications of the current method have motivated us to test the method on various nonlinear partial fractional differential equations—see Refs. [48,49,50,51,52,53,54,59]. This technique allowed us to successfully solve the complex problem of multidimensional nonlinear fractional differential equations. Additionally, the technique effectively resolved two specific types of nonlinear fractional partial differential equations with a desired degree of accuracy. The absolute difference between exact and numerical solution is called absolute error. The absolute error ranged from ${10}^{-15}$ to ${10}^{-17}$ between exact and our semi analytical solutions for integrals values of $\gamma$. For fractional values of $\gamma$, the accuracy changes as shown in the graphs. It was also determined that the accuracy of the solutions depends on the number of fractional B-polys used in the calculations. The larger the set of fractional B-polys, the higher the accuracy of the solutions found, and more iterations were required to obtain converged solutions, raising the computational time and resulting in higher computational cost. Our method is characterized by its simplicity and user-friendliness, as it can be readily employed in any symbolic programming language. Our method utilized the fractional polynomial basis set that did not require a grid representation and it resulted in fewer computational iterations to reach converged values of the expansion coefficients—see Equation (1). This user-friendly approach was implemented in the symbolic programing language, Mathematica [60].

The method enabled a quick resolution of nonlinear partial fractional differential (NPFD) equations, which took only 10 min to 1.5 h for each problem to solve, whereas integer order nonlinear partial differential equations were resolved in just 3–5 min. Therefore, the computational cost was at the bare minimum using the four-node laptop computer. The proposed method is sensitive to variations in input parameters or initial conditions. A small mistake or miscoding can lead to a wrong converged answer. Many fractional differential equations could be computed efficiently in a shorter amount of CPU time. All the examples provided have been solved using Mathematica symbolic programing version 13.2 [60] and compared with existing solutions. Thus, the method can be employed to solve many linear and nonlinear partial fractional differential equations in applied science. It is determined that the fractional B-poly method is highly accurate, as the results were compared with the existing results of Refs. [57,58] in this article.