# Constructing and Predicting Solutions for Different Families of Partial Differential Equations: A Reliable Algorithm

Authors: Mubashir Qayyum; Amna Khan
Journal: Journal of Mathematics
 (2022)

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## Abstract

The construction of mathematical models for different phenomena, and developing their solutions, are critical issues in science and engineering. Among many, the Buckmaster and Korteweg-de Vries (KdV) models are very important due to their ability of capturing different physical situations such as thin film flows and waves on shallow water surfaces. In this manuscript, a new approach based on the generalized Taylor series and residual function is proposed to predict and analyze Buckmaster and KdV type models. This algorithm estimates convergent series with an easy-to-use way of finding solution components through symbolic computation. The proposed algorithm is tested against the Buckmaster and KdV equations, and the results are compared with available solutions in the literature. At first, proposed algorithm is applied to Buckmaster-type linear and nonlinear equations, and attained the closed-form solutions. In the next phase, the proposed algorithm is applied to highly nonlinear KdV equations (namely, classical, modified, and generalized KdV) and approximate solutions are obtained. Simulations of the test problems clearly reassert the dominance and capability of the proposed methodology in terms of accuracy. Analysis reveals that the projected scheme is reliable, and hence, can be utilized for more complex problems in engineering and the sciences.


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## Body

## 1. Introduction

Many real-world systems are translated into mathematical models as differential equations [1, 2]. For more accurate analysis and predictions, it is recommended to use partial differential equations instead of ordinary differential equations for modeling various physical phenomena. These equations are widely used for describing different complex situations, such as fluid flow [3, 4], signal processing, control and information theory [5, 6], entropy generations [7], and waves on shallow water surfaces [8–10]. Examples include radioactive decay, spring-mass systems, population growth, and predator-prey models. As nonlinear model, KdV equations has enormous effect on many aspects in theoretical and mathematical physics [11], quantum and string theory [12, 13]. KdV equations are famous for capturing nonlinear dispersive waves. For the solution of such complex equations, scientist needs different methods and tools. In literature, various analytical and numerical techniques are available, such as the Darboux transformation [14], the tanh method [15], the separation of variable [16], and Sine–Cosine method [17], and Lie symmetry [18, 19] for the solution of DEs, but these methods are limited to linear problems only. Due to nonlinearity in most of the physical phenomena, researchers have switched their attention towards approximate solutions of DEs. These solutions can be obtained through different numerical schemes like radial basis function (RBF) methods [20], finite element methods (FEM) [21], and so on and seminumerical schemes include the homotopy perturbation method (HPM) [22, 23] and its different modifications, the variational iteration method (VIM) [24], and the Adomian decomposition method (ADM) [25].Every numerical scheme has its own restriction, such as linearization, discretization, or perturbation. To overcome these difficulties, we propose RPSM, which can start working with initial conditions and lead us to the convergent series solutions of initial and boundary value problems (IBVPs). In this method, truncated series and residual function concepts are essential for the solution process. Scientists have used RPSM to solve various types of problems. At first, RPSM was proposed by Arqub et al. for higher-order IVPs [26, 27]. Aqub also extended RPSM to fractional and fuzzy type DEs in [28, 29]. Al-Smadi also applies RPSM to different classes of IVPs in [30]. Later on, Kamoshynska utilize solved coupled Burger equations through RPSM [31]. Zhang et al. modify RPSM by mixing it with the least square method for time fractional PDEs in [32]. El-Ajou et al. extend this method to time fractional Burger-type equations in [33]. Alquran investigated the fractional drainage equation through RPSM [34]. Qayyum and Fatema utilized RPSM for the solution of stiff systems in [35]. Qayyum et al. extended this technique to higher order BVPs [36]. Other researchers also used RPSM to various problems arise in science and engineering [37–39].The objective of this paper is the application of RPSM to Buckmaster and KdV type nonlinear PDEs for improved results with less computational cost. In the rest of the paper, the basic idea of RPSM for PDEs is given in Section2. Convergence analysis of RPSM is in Section 3 while application of RPSM to the Buckmaster and KdV families are in Sections 4 and 5, respectively. Section 6 contains a discussion of results while Section 7 presents a conclusion.


## 2. Basic Idea of Residual Power Series Algorithm for Partial Differential Equations

To explain the proposed scheme, let us take the following PDE(1)Φtr,t+LΦr,t=fr,t,r∈Ω,t∈0,T,where f and L are source term and differential operator, respectively. The initial and boundary conditions are(2)Φr,0=Φ0r,r∈Ω,Φr,t=hr,t,r∈∂Ω,t∈0,T.Let the following power series as a solution of the problem(3)Φr,t=∑i=0kCirti,k=0,1,2,3,…,where Ci are unknowns to be computed. By using initial condition(4)C0r=Φ0r.Consider thekth truncated series as(5)Φr,t=Φinitial+∑i=1kCirti,where(6)Φinitial=C0r=Φ0r.Next, we use (3) in (1) to obtain the following kth residual function(7)Reskt=∂∂t∑i=1kCirti+L∑i=0kCirti−fr,t.In next step of implementation, we use the following fundamental concept with residual function to get coefficients of series solution.(8)dk−ndrk−nResk0=0,where n represents the number of initial conditions. This iterative process is repeated for higher order solution. By computing more coefficients improved accuracy can be achieved.


## 3. Convergence Analysis of Residual Power Series Algorithm

In this section, we introduce necessary definition and theorem of residual power series.Definition 1.
[26] A power series (PS) aboutr=r0is defined as(9)∑i=0∞βir−r0i=β0+β1x+β2r2+β3r3+…;r0≤r≤r+1,where constants betai, i=0,1,2,⋯ are coefficients of power series.Theorem 1 (see [30]).
letfhave a power series representation atr=r0of the form(10)fx=∑i=0∞βir−R0i;R0≤R≤R0+δ.If∂i/∂rifr, i=0,1,2,⋯ are continuous on r0,r0+δ, then βi=∂i/∂rifr/i! and δ is the radius of convergence.Theorem 2.
For residual power series∑i=0∞βir−r0i, there are following three possibilities:(1)
If convergence radius is zero, then the series will converge only forr=r0.(2)
If convergence radius is equal to∞, then series will converge for all r≥r0.(3)
The series converges forr∈r0,r0+δ, for some positive real number δ and diverges for r>r0+δ. Where δ is the radius of convergence in this case.Proof.
Let we assume that Case (1) and (2) are not true. Then, there exist nonzero numbers δ1 and δ2 such that series ∑βir−r0i converges for r=δ1 and diverges for r=δ2. Therefore, the convergence set S=x|∑βir−r0i is not empty. Thus using completeness axiom, S has at least upper bound r0+δ. If r>r0+δ, then r does not belong to S, and hence, ∑βir−r0i series diverges. If 0≤r<r0+δ, then, r is not an upper bound for S and so there exists δ1∈S such that δ1>r. Since δ1∈S and ∑βir−r0i series converges, so ∑βir−r0i converges, hence, proof of the theorem is complete.


## 4. Application and Simulations in Buckmaster Family of Equations

### 4.1. Test Problem 1

Consider the following nonlinear and nonhomogeneous Buckmaster equation(11)Φtr,t=Φr,r4r,t+Φr3r,t−12r2e4t−3r2e3t+ret.With initial and boundary conditions(12)Φr,0=r,r∈Ω=0,1,(13)Φ0,t=0,Φ1,t=et,r∈∂Ω,t∈0,T.Exact solution of the problem is(14)Φr,t=ret.To start solution process, let we assume the followingkth truncated series as(15)Φr,t=∑i=0kCirti,k=0,1,2,⋯.Using (12) with (15) gives(16)Φinitial=Φr,0=C0r=r.Next, re-writing (11) as(17)Φtr,t−Φ4r,rr,t−Φ3rr,t+12r2e4t+3r2e3t−ret=0.And plugkth truncated series in (17) to get the following residual function(18)Reskt=∂∂tr+∑i=1kCirti+∂2∂r2r+∑i=1kCirti4+∂∂rr+∑i=1kCirti3+12r2e4t+3r2e3t−ret.Now, puttingk=1 in (18) provide(19)Res1t=∂∂tr+∑i=11Cirti+∂2∂r2r+∑i=11Cirti4+∂∂rr+∑i=11Cirti3+12r2e4t+3r2e3t−ret.Pluging (19) in (8) gives(20)C1r=r.Repeat the process by puttingk=2 in (8) and (18), we get(21)∂∂rRes2t=∂∂r∂∂tr+∑i=12Cirti+∂3∂r3r+∑i=12Cirti4+∂2∂r2r+∑i=12Cirti3+∂∂r12r2e4t+3r2e3t−ret,C2r=r2.Continuing the recursive process, the series solution is(22)Φr,t=r+rt+12rt2+16rt3+124rt4+1120rt5+⋯Φr,t=r1+t+12t2+16t3+124t4+1120t5+⋯Φr,t=ret.Which is closed form (exact) solution of Test Problem 1.


### 4.2. Test Problem 2

Consider the nonlinear and nonhomogeneous Buckmaster equation(23)Φtr,t=Φr,r4r,t+Φr3r,t−12r2cos4t−3r2cos3t+rsint.With IBCs(24)Φr,0=r,r∈Ω=0,1,Φ0,t=0,Φ1,t=et,r∈∂Ω,t∈0,T.Exact solution is(25)Φr,t=rcost.Using basic theory of RPSM recursively, following are the unknown coefficients of thekth truncated series(26)C0r=r,C1r=0,C2r=−12r,C3r=0,C4r=124r,C5r=0,C6r=−1720r,⋯.Hence, series solution is(27)Φr,t=r−12rt2+124rt4−1720rt6+⋯Φr,t=r1−12t2+124t4−1720t6+⋯Φr,t=rcost.Which is closed form solution of Test Problem 2.


## 4.1. Test Problem 1

Consider the following nonlinear and nonhomogeneous Buckmaster equation(11)Φtr,t=Φr,r4r,t+Φr3r,t−12r2e4t−3r2e3t+ret.With initial and boundary conditions(12)Φr,0=r,r∈Ω=0,1,(13)Φ0,t=0,Φ1,t=et,r∈∂Ω,t∈0,T.Exact solution of the problem is(14)Φr,t=ret.To start solution process, let we assume the followingkth truncated series as(15)Φr,t=∑i=0kCirti,k=0,1,2,⋯.Using (12) with (15) gives(16)Φinitial=Φr,0=C0r=r.Next, re-writing (11) as(17)Φtr,t−Φ4r,rr,t−Φ3rr,t+12r2e4t+3r2e3t−ret=0.And plugkth truncated series in (17) to get the following residual function(18)Reskt=∂∂tr+∑i=1kCirti+∂2∂r2r+∑i=1kCirti4+∂∂rr+∑i=1kCirti3+12r2e4t+3r2e3t−ret.Now, puttingk=1 in (18) provide(19)Res1t=∂∂tr+∑i=11Cirti+∂2∂r2r+∑i=11Cirti4+∂∂rr+∑i=11Cirti3+12r2e4t+3r2e3t−ret.Pluging (19) in (8) gives(20)C1r=r.Repeat the process by puttingk=2 in (8) and (18), we get(21)∂∂rRes2t=∂∂r∂∂tr+∑i=12Cirti+∂3∂r3r+∑i=12Cirti4+∂2∂r2r+∑i=12Cirti3+∂∂r12r2e4t+3r2e3t−ret,C2r=r2.Continuing the recursive process, the series solution is(22)Φr,t=r+rt+12rt2+16rt3+124rt4+1120rt5+⋯Φr,t=r1+t+12t2+16t3+124t4+1120t5+⋯Φr,t=ret.Which is closed form (exact) solution of Test Problem 1.


## 4.2. Test Problem 2

Consider the nonlinear and nonhomogeneous Buckmaster equation(23)Φtr,t=Φr,r4r,t+Φr3r,t−12r2cos4t−3r2cos3t+rsint.With IBCs(24)Φr,0=r,r∈Ω=0,1,Φ0,t=0,Φ1,t=et,r∈∂Ω,t∈0,T.Exact solution is(25)Φr,t=rcost.Using basic theory of RPSM recursively, following are the unknown coefficients of thekth truncated series(26)C0r=r,C1r=0,C2r=−12r,C3r=0,C4r=124r,C5r=0,C6r=−1720r,⋯.Hence, series solution is(27)Φr,t=r−12rt2+124rt4−1720rt6+⋯Φr,t=r1−12t2+124t4−1720t6+⋯Φr,t=rcost.Which is closed form solution of Test Problem 2.


## 5. Application and Simulations in KdV Family of Equations

### 5.1. Test Problem 3 (Classical KdV)

Consider the following nonlinear and nonhomogeneous classical KdV equation (25):(28)Φtr,t+ΦΦrr,t+Φr,r,rr,t=−er−t2e2r.With initial condition(29)Φr,0=1.Exact solution is(30)Φr,t=1−ter.Using basic idea of RPSM recursively, unknown coefficients of the required series are(31)C0r=1,C1r=−er,C2r=0,C3r=0,C4r=0,C5r=0,C6r=0,⋯.Hence, series solution is(32)Φr,t=1−ter+0t2+0t3+0t4+⋯.Which leads to a closed form solution of Test Problem 3.


### 5.2. Test Problem 4 (Classical KdV)

Consider the following nonlinear classical KdV equation [40]:(33)Φtr,t+6Φr,tΦrr,t+Φr,r,rr,t=0.With initial condition(34)Φr,0=6r2.Exact solution of this problem is(35)Φr,t=6rr3−24tr3+122.After applying basic theory of RPSM recursively, third order solution is obtained. Numerical results of Test Problem 4 are shown in Table1 and Figure 1.Table 1
Comparison of exact and RPS solution at varioust in Test Problem 4.
trExact solutionRPS solutionAbsolute error0.0001100.060.062×10−7150.030.033×10−8200.0150.0159×10−9250.010.012×10−9300.0070.0071×10−9350.0050.0055×10−10400.0040.0042×10−10450.0030.0031×10−10500.00240.00249×10−110.001100.060.062×10−6150.030.033×10−7200.0150.0158×10−8250.0150.012×10−8300.0070.0071×10−8350.0050.0055×10−9400.0040.0042×10−9450.0030.0031×10−9500.00240.00249×10−100.1100.060.062×10−4150.030.033×10−5200.0150.0158×10−6250.010.012×10−6300.0070.0071×10−6350.0050.0055×10−7400.0040.0042×10−7450.0030.0031×10−7500.00240.00249×10−8Figure 1
Graphical comparison of exact and RPS solutions along with error in Test Problem 4 where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


### 5.3. Test Problem 5 (Classical KdV)

Consider the following nonlinear classical KdV equation [41]:(36)Φtr,t+6Φr,tΦrr,t+Φr,r,rr,t=0.With initial condition(37)Φr,0=12sech212r.Exact solution is(38)Φr,t=12sech212r−t.Sixth order RPS solution is obtained. Numerical results of Test Problem 5 are shown in Table2 and Figure 2.Table 2
Comparison of exact and RPS solutions at varioust in Test Problem 5.
trExact solutionRPS solutionAbsolute error0.0010.10.50.52×10−180.150.50.51×10−160.20.50.52×10−160.250.50.51×10−170.30.50.51×10−160.350.50.56×10−170.40.50.52×10−170.450.50.56×10−170.50.50.56×10−180.010.10.50.54×10−170.150.50.51×10−160.20.50.55×10−180.250.50.55×10−170.30.50.54×10−180.350.50.51×10−160.40.50.55×10−180.450.50.55×10−170.50.50.51×10−160.10.10.50.52×10−110.150.50.54×10−110.20.50.55×10−110.250.50.56×10−110.30.50.57×10−110.350.50.58×10−110.40.50.58×10−110.450.50.59×10−110.50.50.59×10−11Figure 2
Graphical comparison of exact and RPS solutions along with error in Test Problem 5 where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


### 5.4. Test Problem 6 (Modified KdV)

Consider the following nonlinear modified KdV equation [40]:(39)Φtr,t+6Φ2r,tΦrr,t+Φr,r,rr,t=0,t>0,−∞<r<+∞.With initial condition(40)Φr,0=2sech2r.−∞<r<+∞.Exact solution of the problem is(41)Φr,t=2sech2r−8t.Fourth order RPS solution is obtained whose results are shown in Table3 and Figure 3.Table 3
Comparison of RPSM and HAM errors att=0.8 in Test Problem 6.
trExact solutionRPS solutionAbsolute errorHAM [40]0.861×10−31×10−21×10−21×10−4121×10−89×10−88×10−81×10−8186×10−145×10−134×10−136×10−10244×10−193×10−183×10−186×10−10Figure 3
Graphical comparison of exact and RPS solutions along with error in Test Problem 6 where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


### 5.5. Test Problem 7 (Modified KdV)

Consider the following nonlinear modified KdV equation [40]:(42)Φtr,t+6Φ2r,tΦrr,t+Φr,r,rr,t=0,t>0,−∞<r<+∞.With initial condition(43)Φr,0=−4cosh2rcos2r−sin2rsinh2rcosh2rcosh2r+sin2r−∞<r<+∞.The exact solution of the problem is(44)Φr,t=−4cosh2r−16tcos2r+16t−sinh2r−16tsin2rcosh2r+16tcosh2r+16t+sin2r−16t−∞<r<+∞.Fourth order RPS solution is obtained and results are shown in Table4 and Figure 4.Table 4
Comparison of RPSM and HAM errors att=0.04 in Test Problem 7.
trExact solutionRPS solutionAbsolute errorHAM [40]0.04154×10−131×10−132×10−132×10−13202×10−172×10−172×10−183×10−18258×10−221×10−213×10−222×10−22301×10−264×10−263×10−262×10−26Figure 4
Graphical comparison of exact and RPS solutions along with error in Test Problem 7 where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


### 5.6. Test Problem 8 (Generalized KdV)

Consider third-order nonlinear generalized KdV equation [42]:(45)Φtr,t+Φpr,tΦrr,t+Φr,r,rr,t=0.With initial condition(46)Φr,t=Asech2kr−z1/p,where 2≤p, K,m, and z are constants and A=2p+1p+2/m2K2.Exact solution of the problem is(47)Φr,t=Asech2kr−ct−z1/p.To obtain solution we fixedp=4 in generalized equation. Other values of p can also be used in this problem.Third order RPS solution is obtained and results are shown in Table5 and Figure 5.Table 5
Comparison of exact and RPS solutions at varioust in Test Problem 8.
trExact solutionRPS solutionAbsolute error0.010.10.7620280.7620281×10−170.150.7618150.7618152×10−180.20.7615150.7615154×10−170.250.7611310.7611311×10−170.30.7606620.7606624×10−170.350.7601080.7601089×10−170.40.759470.759474×10−170.450.7587480.7587485×10−170.50.7579430.7579434×10−170.10.10.7620350.7620351×10−140.150.7618250.7618251×10−140.20.7615290.7615291×10−140.250.7611480.7611481×10−140.30.7606820.7606821×10−140.350.7601320.7601321×10−140.40.7594970.7594971×10−140.4500.7587790.7587791×10−140.50.7579780.7579781×10−140.20.10.7620430.7620431×10−130.150.7618360.7618361×10−130.20.7615440.7615441×10−130.250.7611670.7611671×10−130.30.7607050.7607051×10−130.350.7601580.7601581×10−130.40.7595280.7595281×10−130.450.7588130.7588131×10−130.50.7580160.7580161×10−13Figure 5
Graphical comparison of exact and RPS solutions along with error when,A=0.3375,c=0.00675,K=0.3,p=4,z=0 in Test Problem 8, where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


### 5.7. Test Problem 9 (Generalized KdV)

Consider fourth-order nonlinear generalized KdV equation [42]:(48)Φtr,t+p+1Φpr,tΦrr,t+Φr,r,r,rr,t=0,where p>2, A and k are constants.With initial condition(49)Φr,0=Asech2kr1/p.The exact solution of the problem is(50)Φr,t=Asech2kr−ct1/p,where p>2,A,k and c are constants.Third order RPS solution is obtained by applying basic theory of RPSM and results are shown in Table6 and Figure 6.Table 6
Comparison of exact and RPS solutions at varioust in Test Problem 9.
trExact solutionRPS solutionAbsolute error0.010.10.1999950.1999958×10−70.150.1999890.1999881×10−60.20.1999810.199981×10−60.250.199970.1999681×10−60.30.1999560.1999551×10−60.350.1999410.1999382×10−60.40.1999220.199922×10−60.450.1999010.1998982×10−60.50.1998780.1998752×10−60.10.10.1999990.1999927×10−60.150.1999950.1999859×10−60.20.1999890.1999771×10−50.250.199980.1999651×10−50.30.1999690.1999521×10−50.350.1999550.1999361×10−50.40.1999390.1999172×10−50.450.199920.1998962×10−50.50.1998990.1998722×10−50.20.10.20.1999881×10−50.150.1999990.1999821×10−50.20.1999950.1999732×10−50.250.1999890.1999622×10−50.30.199980.1999493×10−50.350.1999690.1999323×10−50.40.1999550.1999144×10−50.450.1999390.1998934×10−50.50.199920.1998695×10−5Figure 6
Graphical comparison of exact and RPS solutions along with error when,A=0.2,c=0.05,k=0.1,p=4 in Test Problem 9, where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


## 5.1. Test Problem 3 (Classical KdV)

Consider the following nonlinear and nonhomogeneous classical KdV equation (25):(28)Φtr,t+ΦΦrr,t+Φr,r,rr,t=−er−t2e2r.With initial condition(29)Φr,0=1.Exact solution is(30)Φr,t=1−ter.Using basic idea of RPSM recursively, unknown coefficients of the required series are(31)C0r=1,C1r=−er,C2r=0,C3r=0,C4r=0,C5r=0,C6r=0,⋯.Hence, series solution is(32)Φr,t=1−ter+0t2+0t3+0t4+⋯.Which leads to a closed form solution of Test Problem 3.


## 5.2. Test Problem 4 (Classical KdV)

Consider the following nonlinear classical KdV equation [40]:(33)Φtr,t+6Φr,tΦrr,t+Φr,r,rr,t=0.With initial condition(34)Φr,0=6r2.Exact solution of this problem is(35)Φr,t=6rr3−24tr3+122.After applying basic theory of RPSM recursively, third order solution is obtained. Numerical results of Test Problem 4 are shown in Table1 and Figure 1.Table 1
Comparison of exact and RPS solution at varioust in Test Problem 4.
trExact solutionRPS solutionAbsolute error0.0001100.060.062×10−7150.030.033×10−8200.0150.0159×10−9250.010.012×10−9300.0070.0071×10−9350.0050.0055×10−10400.0040.0042×10−10450.0030.0031×10−10500.00240.00249×10−110.001100.060.062×10−6150.030.033×10−7200.0150.0158×10−8250.0150.012×10−8300.0070.0071×10−8350.0050.0055×10−9400.0040.0042×10−9450.0030.0031×10−9500.00240.00249×10−100.1100.060.062×10−4150.030.033×10−5200.0150.0158×10−6250.010.012×10−6300.0070.0071×10−6350.0050.0055×10−7400.0040.0042×10−7450.0030.0031×10−7500.00240.00249×10−8Figure 1
Graphical comparison of exact and RPS solutions along with error in Test Problem 4 where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


## 5.3. Test Problem 5 (Classical KdV)

Consider the following nonlinear classical KdV equation [41]:(36)Φtr,t+6Φr,tΦrr,t+Φr,r,rr,t=0.With initial condition(37)Φr,0=12sech212r.Exact solution is(38)Φr,t=12sech212r−t.Sixth order RPS solution is obtained. Numerical results of Test Problem 5 are shown in Table2 and Figure 2.Table 2
Comparison of exact and RPS solutions at varioust in Test Problem 5.
trExact solutionRPS solutionAbsolute error0.0010.10.50.52×10−180.150.50.51×10−160.20.50.52×10−160.250.50.51×10−170.30.50.51×10−160.350.50.56×10−170.40.50.52×10−170.450.50.56×10−170.50.50.56×10−180.010.10.50.54×10−170.150.50.51×10−160.20.50.55×10−180.250.50.55×10−170.30.50.54×10−180.350.50.51×10−160.40.50.55×10−180.450.50.55×10−170.50.50.51×10−160.10.10.50.52×10−110.150.50.54×10−110.20.50.55×10−110.250.50.56×10−110.30.50.57×10−110.350.50.58×10−110.40.50.58×10−110.450.50.59×10−110.50.50.59×10−11Figure 2
Graphical comparison of exact and RPS solutions along with error in Test Problem 5 where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


## 5.4. Test Problem 6 (Modified KdV)

Consider the following nonlinear modified KdV equation [40]:(39)Φtr,t+6Φ2r,tΦrr,t+Φr,r,rr,t=0,t>0,−∞<r<+∞.With initial condition(40)Φr,0=2sech2r.−∞<r<+∞.Exact solution of the problem is(41)Φr,t=2sech2r−8t.Fourth order RPS solution is obtained whose results are shown in Table3 and Figure 3.Table 3
Comparison of RPSM and HAM errors att=0.8 in Test Problem 6.
trExact solutionRPS solutionAbsolute errorHAM [40]0.861×10−31×10−21×10−21×10−4121×10−89×10−88×10−81×10−8186×10−145×10−134×10−136×10−10244×10−193×10−183×10−186×10−10Figure 3
Graphical comparison of exact and RPS solutions along with error in Test Problem 6 where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


## 5.5. Test Problem 7 (Modified KdV)

Consider the following nonlinear modified KdV equation [40]:(42)Φtr,t+6Φ2r,tΦrr,t+Φr,r,rr,t=0,t>0,−∞<r<+∞.With initial condition(43)Φr,0=−4cosh2rcos2r−sin2rsinh2rcosh2rcosh2r+sin2r−∞<r<+∞.The exact solution of the problem is(44)Φr,t=−4cosh2r−16tcos2r+16t−sinh2r−16tsin2rcosh2r+16tcosh2r+16t+sin2r−16t−∞<r<+∞.Fourth order RPS solution is obtained and results are shown in Table4 and Figure 4.Table 4
Comparison of RPSM and HAM errors att=0.04 in Test Problem 7.
trExact solutionRPS solutionAbsolute errorHAM [40]0.04154×10−131×10−132×10−132×10−13202×10−172×10−172×10−183×10−18258×10−221×10−213×10−222×10−22301×10−264×10−263×10−262×10−26Figure 4
Graphical comparison of exact and RPS solutions along with error in Test Problem 7 where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


## 5.6. Test Problem 8 (Generalized KdV)

Consider third-order nonlinear generalized KdV equation [42]:(45)Φtr,t+Φpr,tΦrr,t+Φr,r,rr,t=0.With initial condition(46)Φr,t=Asech2kr−z1/p,where 2≤p, K,m, and z are constants and A=2p+1p+2/m2K2.Exact solution of the problem is(47)Φr,t=Asech2kr−ct−z1/p.To obtain solution we fixedp=4 in generalized equation. Other values of p can also be used in this problem.Third order RPS solution is obtained and results are shown in Table5 and Figure 5.Table 5
Comparison of exact and RPS solutions at varioust in Test Problem 8.
trExact solutionRPS solutionAbsolute error0.010.10.7620280.7620281×10−170.150.7618150.7618152×10−180.20.7615150.7615154×10−170.250.7611310.7611311×10−170.30.7606620.7606624×10−170.350.7601080.7601089×10−170.40.759470.759474×10−170.450.7587480.7587485×10−170.50.7579430.7579434×10−170.10.10.7620350.7620351×10−140.150.7618250.7618251×10−140.20.7615290.7615291×10−140.250.7611480.7611481×10−140.30.7606820.7606821×10−140.350.7601320.7601321×10−140.40.7594970.7594971×10−140.4500.7587790.7587791×10−140.50.7579780.7579781×10−140.20.10.7620430.7620431×10−130.150.7618360.7618361×10−130.20.7615440.7615441×10−130.250.7611670.7611671×10−130.30.7607050.7607051×10−130.350.7601580.7601581×10−130.40.7595280.7595281×10−130.450.7588130.7588131×10−130.50.7580160.7580161×10−13Figure 5
Graphical comparison of exact and RPS solutions along with error when,A=0.3375,c=0.00675,K=0.3,p=4,z=0 in Test Problem 8, where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


## 5.7. Test Problem 9 (Generalized KdV)

Consider fourth-order nonlinear generalized KdV equation [42]:(48)Φtr,t+p+1Φpr,tΦrr,t+Φr,r,r,rr,t=0,where p>2, A and k are constants.With initial condition(49)Φr,0=Asech2kr1/p.The exact solution of the problem is(50)Φr,t=Asech2kr−ct1/p,where p>2,A,k and c are constants.Third order RPS solution is obtained by applying basic theory of RPSM and results are shown in Table6 and Figure 6.Table 6
Comparison of exact and RPS solutions at varioust in Test Problem 9.
trExact solutionRPS solutionAbsolute error0.010.10.1999950.1999958×10−70.150.1999890.1999881×10−60.20.1999810.199981×10−60.250.199970.1999681×10−60.30.1999560.1999551×10−60.350.1999410.1999382×10−60.40.1999220.199922×10−60.450.1999010.1998982×10−60.50.1998780.1998752×10−60.10.10.1999990.1999927×10−60.150.1999950.1999859×10−60.20.1999890.1999771×10−50.250.199980.1999651×10−50.30.1999690.1999521×10−50.350.1999550.1999361×10−50.40.1999390.1999172×10−50.450.199920.1998962×10−50.50.1998990.1998722×10−50.20.10.20.1999881×10−50.150.1999990.1999821×10−50.20.1999950.1999732×10−50.250.1999890.1999622×10−50.30.199980.1999493×10−50.350.1999690.1999323×10−50.40.1999550.1999144×10−50.450.1999390.1998934×10−50.50.199920.1998695×10−5Figure 6
Graphical comparison of exact and RPS solutions along with error when,A=0.2,c=0.05,k=0.1,p=4 in Test Problem 9, where (a) Exact solution, (b) RPS solution and (c) error.
(a)(b)(c)


## 6. Results and Discussion

In this article, a residual power series algorithm is proposed for predicting and analyzing Buckmaster and KdV type partial differential equations. The algorithm directly applies to PDEs without linearization, discretization or perturbation because it depends on the recursive differentiation of dispersal along with the use of given initial constraints to calculate coefficients of the assumed power series using nominal computations. This algorithm mainly based on residual functions which will be obtained after applying the generalized Taylor series. Initially, the proposed algorithm is tested against linear and nonlinear Buckmaster and KdV equations, and closed-form solutions are obtained. These exact solutions are depicted in Figures7–9 for reader connivance. In the second stage of testing, RPSM is applied to different nonlinear KdV type equations namely, classical, modified, and generalized KdVs, and approximate series solutions are obtained. Numerical results related to these problems are shown in Tables 1–6 and Figures 1–6. Tables 1 and 2 and Figures 1 and 2 depict the comparison of exact and RPS solutions along with corresponding error in Test Problem 4 and 5 (classical KdV equations). Analysis of tables and figures indicate that the obtained solution is accurate and consistent. Tables 3 and 4 and Figures 3 and 4 present the comparison of RPS and HAM solutions at fixed time in Test Problem 6 and 7 (modified KdV equations). An analysis of tables and figures shows that obtained RPS solutions are consistent and reliable. Tables 5 and 6 and Figures 5 and 6 present the comparison of exact and RPS in Test Problem 8 and 9 (generalized KdV equations). Observation shows that obtained RPS solutions are accurate and acceptable. Overall, analysis endorse that the proposed algorithm is easy to use, and hence, can also be used for other families of PDEs.Figure 7
Exact solution of Test Problem 1.Figure 8
Exact solution of Test Problem 2.Figure 9
Exact solution of Test Problem 3.


## 7. Conclusions

In this article, a residual power series algorithm is proposed for predicting and analyzing Buckmaster and KdV type partial differential equations. This algorithm mainly based on residual functions, which will be obtained after applying the generalized Taylor series. The algorithm is directly applies to PDEs without linearization, discretization, or perturbation because it depends on the recursive differentiation of dispersal along with the use of given initial constraints to calculate coefficients of the assumed power series using nominal computations. In the first phase of simulations, the proposed algorithm is tested against linear and nonlinear Buckmaster equations, and closed-form solutions are obtained. In the second stage of simulations, proposed methodology is applied to different nonlinear KdV type equations (classical, modified, and generalized), and approximate series solutions are obtained. For validity purpose, results are compared with exact and available solutions from literature. Analysis endorse that the proposed algorithm surpasses the other traditional methods in terms of naivety, speediness, and constraints. Hence, this algorithm can also be extended to other families of differential equations arise in different scientific phenomena.



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