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ORIGINAL RESEARCH article

Front. Phys., 17 April 2023
Sec. Interdisciplinary Physics
Volume 11 - 2023 | https://doi.org/10.3389/fphy.2023.1167797

A modern analytic method to solve singular and non-singular linear and non-linear differential equations

  • 1Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt, Jordan
  • 2Department of Mathematics, Faculty of Science, Zarqa University, Zarqa, Jordan

This article circumvents the Laplace transform to provide an analytical solution in a power series form for singular, non-singular, linear, and non-linear ordinary differential equations. It introduces a new analytical approach, the Laplace residual power series, which provides a powerful tool for obtaining accurate analytical and numerical solutions to these equations. It demonstrates the new approach’s effectiveness, accuracy, and applicability in several ordinary differential equations problem. The proposed technique shows the possibility of finding exact solutions when a pattern to the series solution obtained exists; otherwise, only rough estimates can be given. To ensure the accuracy of the generated results, we use three types of errors: actual, relative, and residual error. We compare our results with exact solutions to the problems discussed. We conclude that the current method is simple, easy, and effective in solving non-linear differential equations, considering that the obtained approximate series solutions are in closed form for the actual results. Finally, we would like to point out that both symbolic and numerical quantities are calculated using Mathematica software.

1 Introduction

A differential equation or a system of differential equations, along with proper boundary and initial conditions (ICs), is one of the most common outputs when mathematical modeling describes physical, biological, or chemical phenomena. Finding ordinary or partial differential equations and analyzing their solutions are at the heart of applied mathematics [1].

Since ancient times, differential equations have attracted the interest of researchers and scientists from two sides. The first is how to use them to express phenomena and issues that interest them in their specializations and research. On the other hand, there is the question of how to solve these equations. There are a limited number of differential equations, especially linear ones, whose solutions can be determined using the well-known traditional methods based on finite and simple algebraic operations. In contrast, there are many kinds of differential equations that still require the search for simple and accurate solutions. For this reason, the interest of mathematicians in previous decades was and is still in the investigation for analytical and sometimes numerical methods to solve these forms of differential equations.

Many analytical, numerical, and numero-analytical techniques have been proposed previously and recently to provide solutions for differential equations with initial or boundary conditions, such as the Laplace and Fourier transforms method [2], the Adomian decomposition method [35], the variational iteration method [68], the homotopy perturbation method [911], the homotopy analysis method [12, 13], the differential transformation method [1416], the finite difference method [17], the predictor–corrector method [18, 19], the first integral method [20, 21], the Adams–Bashforth Molten method [22], the new iterative method [23, 24], the Crank–Nicolson method [25, 26], the reproducing kernel method [27, 28], the Laplace Adomian decomposition method [11, 29, 30], the He–Laplace method [3133], and others [3436].

Recently, Eriqat et al. [37] presented a new hybrid method in which they combined the Laplace transform (LT) method with the residual power series (PS) method to establish series solutions of the pantograph equation. This method is called the LRPS method, and it simulates the residual PS method but with a different construction and view. It uses the limit concept instead of the concept of the derivative as in the residual PS method. The LRPS method uses the LT to transfer the given differential equation to a new algebraic equation in a new space. The obtained algebraic equation is solved by assuming that it is a solution that has a Laurent series (LS) form. The values of the coefficients of the LS are determined by utilizing the limit at infinity. Then, the inverse LT is used to transfer the LS, which is the solution of the algebraic equation in the Laplace space, to the initial space. Thus, we have obtained the solution to the original problem in the form of PS.

Indeed, the LRPS method is similar to the He–Laplace method’s [3133] idea of searching for the solution of differential equations in Laplace space. The He–Laplace technique uses the variational iteration method or homotopy perturbation method to solve the just-transformed Laplace space. In contrast, the LRPS method uses the PS method to solve that equation using the Laurent series instead of the Taylor series. In addition, the LRPS method is just an easy and fast technique for finding the PS solution coefficients of the differential equations.

The LRPS method has won the admiration and interest of many researchers due to its ease, speed, and efficiency in arriving at exact or accurate approximate solutions to many equations. In addition, the LT was employed in dealing with non-linear problems because it is known that the LT deals only with some categories of linear equations. In 2021, El-Ajou [38] adapted the LRPS method to establish solitary solutions of non-linear time-fractional dispersive partial differential equations and to introduce a vector series solution of some types of hyperbolic system of Caputo time-fractional partial differential equations with variable coefficients [39]. Recently, the LRPS method was used for solving time-fractional Navier–Stokes equations [40], fuzzy quadratic Riccati DEs [41], Lane–Emden equations of fractional order [42], a system of fractional initial value problems (IVPs) [43], autonomous n-dimensional fractional non-linear systems [44], and others [4549].

Despite the extensive publication of research dealing with the new method, all works dealt with specific problems devoid of complexity and generality. Therefore, we aim in this manuscript, first, to employ the LRPS method to provide exact or accurate approximate analytic series solutions to linear ordinary differential equations (ODEs) in their general form, whether their coefficients are constants or analytical functions, which have the following formula:

dnydtn=Ltyt+gt,t0.

Subject to the ICs,

y0=y0,y0=y1,,yn10=yn1,

where Lt is a linear differential operator of order n1 with coefficients being analytic functions. This general equation is difficult to solve by the direct PS method. Herein lays the importance and novelty of the aim of this research.

Since in our world, most events are essentially non-linear and modeled by non-linear equations, the study of non-linear issues is critical in mathematics and physics, engineering, economics, and other disciplines. Solving non-linear problems is difficult, and getting an analytical approximation of a given problem is often more complicated than getting a numerical one. Therefore, the second aim of this paper is to establish analytic approximate solutions to the general form of non-linear ODEs, which have the following form using the proposed method (LRPS method):

dnydtn=ft,yt,dydt,d2ydt2,,dn1ydtn1,t0.

Subject to the ICs,

y0=y0,y0=y1,,yn10=yn1,

where f is an analytic function on 0,).

The third objective of this article is to provide a series solution to the singular ODEs, whether linear or non-linear. This type of equation is of great interest to researchers in providing analytical solutions to it, as it appears in the models of many natural phenomena, as well as the difficulty of providing solutions to it.

To determine the efficiency and applicability of the method, we test for three types of errors: exact error, relative error, and residual error. We present the numerical results of the resulting solutions through prepared and organized tables. In addition, we sketch the obtained approximate solution by the proposed method along with the exact solution if we can obtain it to make the comparison on the one hand and to determine the period of convergence to solve the series on the other hand.

2 Basic facts of the LT and PS

In this section, we overview essential facts about the LT and the PS, along with some properties that are needed in this article.

Definition 2.1. [50]). We assume that yt is a continuous function defined for t0 and let sIR. Then, the LT of yt is the function Ys, denoted and defined as follows:

Ys=Lyts=0estytdt,

where the improper integral is the convergence on an interval of s, which represents the domain of Ys.Also, the inverse LT of a function Ys,sI is the function yt,t0 that is denoted and defined as

yt=L1Yst=cic+iestYsds,c=Res>c0,

where c0 lies in the right-half plane of the absolute convergence of the Laplace integral.

Lemma 2.1. [50]). Suppose that yt and xt are both continuous functions defined on 0,), Ys=Lyt, Xs=Lxt, and η,λ are constants. Then, we have the following properties:

1) Leλtyt=Ysλ

2) Ltnyt=1ndndsnYs

3) Lyλt=1λYsλ,λ>0

4) L1ηYs+λXs=ηL1Ys+λL1Xs=ηyt+λxt5) limssYs=y0

6) Lynt=snLyk=0n1snk1yk0

Definition 2.2. [51]). A series that has the representation

n=cnss0n=n=1cnss0n+n=0cnss0n

is called the LS about s=s0, where s is the variable and cn's are the coefficients of the series. The series n=0cnss0n is known as the analytic or regular part of the LS, while n=1cnss0n is known as the singular or the principal part of the LS.

Theorem 2.1. [50]). Let yt be an analytic function defined on the domain D:ξ1<tt0<ξ2. Then, yt can be expanded as a PS as follows:

yt=n=0cntt0n,

which is valid for ξ1<tt0<ξ2.

Theorem 2.2. If Ys=Lyt has an LS representation about s=0,

Ys=c0s+n=1cnsn+1,s>0,

then cn=yn0,n=0,1,2,..Proof. Suppose that Ys can be represented by the LS expansion as in Eq. 2.5. So,

sYs=c0+n=1cnsn,s>0.

According to part (5) of Lemma 2.1, we have c0=y0.Multiplying Eq. 2.6 by s gives the following expansion:

s2Ysy0s=c1+n=2cnsn,s>0.

Using part (5) of Lemma 2.1, it is obvious that

c1=limsc1+n=2cnsn=limss2Yssy0=limsssYsy0=limssLyt=y0.

Similarly, multiplying Eq. 2.7 by s gives the following expansion:

ss2Ysy0sy0=c2+n=3cnsn,s>0.

Again, by parts (5) and (6) of Lemma 2.1, we have

c2=limsc2+n=3cnsn=limsss2Yssy0y0=limssLyt=y0.

Now, we can find out the general formula for the coefficient cn. However, we can get it by multiplying Eq. 2.6 by sn+1 and taking the limit of the resulting equation as s; then, we find that cn=yn0,n=0,1,2,. Thus, the proof is now complete.

Theorem 2.3. assume that Lyt=Ys can be represented as in Eq. 2.6. If sLyntK, on 0<sd, then the reminder Rns of the expansion of the LS appearing in Theorem 2.2 will satisfy the relation

RnsKsn+2,0<sd.

Proof.First, we assume that for r=0,1,2,,n+1, Lyrts is defined on 0<sd. Also, we assume the following:

sLyn+1tK,0<sd.

From the definition of the reminder Rns=Ysi=0nyi0si+1, one can acquire

sn+2Rns=sn+2Ysm=0nsn+1mym0=ssn+1Ysi=0nsnmym0=sLyn+1t.

Eq. 2.10 and Eq. 2.11 lead to the conclusion that sn+1α+1RnsK. Thus,

Ksn+2RnsK,0<sd.

The inequality RnsKsn+2 can be discovered by reformulating Eq. 2.12, and so, we got the result.

3 Constructing series solutions to ODEs

In this section, we first use the LRPS method to solve linear ODEs in preparation for solving non-linear ODEs. What is worth noting is the possibility of solving non-linear ODEs, which cannot be carried out using the traditional LT method. We will use the construction that we will get to solve non-linear ODEs when solving singular ODEs, whether linear or non-linear, and this is what we will see in Section 3.3.

3.1 LRPS method for solving linear ODEs

In this section, we demonstrate the steps of the LRPS method for solving linear ODEs. The basic idea of the proposed method is to apply the LT to the linear ODEs and then use the LRPS approach to construct a series solution, in LS form, to the transformed equation. Then, we transform the obtained solution into the required solution in the original space.

To illustrate the idea of the LRPS method in constructing series solutions to the linear ODEs, we consider problems (1.1) and (1.2), considering that Lt is a linear differential operator given by

Lt=an1tdn1dtn1++a1tddt+a0t,

where a0t,a1t,...,an1t and gt are arbitrary analytic functions that depend only on t, yt is the unknown function of the independent variables t, and I is an open interval.

To generate the LRPS solution of the IVP (1.1) and (1.2), first, we apply the LT to both sides of Eq. 1.1 to obtain

Lynt=LLtyt+Lgt,tI.

Using ICs (1.2) and some properties of the LT, Eq. 3.2 becomes

Ys=i=0n1yisi+1+1snLLtL1Ys+Gssn,s>0.

We assume that Ys in Eq. 3.3 has an expansion in the LS form as

Ys=i=0cis1+i,s>0.

Depending on Theorem 2.3 and the given conditions in Eq. 1.2, the first n-coefficients of the expansion (3.4) can be determined, so it can be rewritten as follows:

Ys=i=0n1yisi+1+i=ncis1+i,s>0.

The kth-truncated series of Ys is given by

Yks=i=0n1yisi+1+i=nkcis1+i,s>0.

Thus, one can conclude

Yns=i=0n1yisi+1+cns1+n,s>0.

To find the values of the unknown coefficients in series (3.7), we define the Laplace residual function (LRF) of Eq. 3.3 as

LRess=Ysi=0n1yisi+11snLLtL1YsGssn,s>0

and the kth LRF as

LResks=Yksi=0n1yisi+11snLLtL1YksGssn,s>0.

It is clear that LimkLResks=LRess, LRess=0, and thus, skLRess=0 for s>0 and k=0,1,2,3,. Therefore, LimsskLRess=0. Moreover,

Limssk+1LRess=Limssk+1LResks=0,k=1,2,3,

Substituting the first nth-truncated series in Eq. 3.7 into the nth LRF to obtain

LResns=cns1+n1snLLtL1i=0n1yisi+1+cns1+nGssn,s>0.

Running the inverse LT in Eq. 3.11, we get

LResns=cns1+n1snLLti=0n1yii!ti+cnn!tnGssn,s>0.

Since the coefficients of the linear operator in Eq. 3.1 are analytic functions, they can be expressed as

art=j=0λrjtj,r=0,1,,n1,

where

λrj=arj0j!,r=0,1,,n1,j=0,1,2,.

So, the linear operator Lt in Eq. 3.1 can be expressed as

Lt=r=0n1j=0λrjtjdrdtr.

Running the operator Lt on Eq. 3.12 according to its new form in Eq. 3.15, we get

LResns=cns1+n1snLr=0n1j=0cnλrjnr!tn+jr+i=rn1λrjyiir!ti+jrGssn.

Finally, we run the LT in Eq. 3.16 to obtain the required form of the nth LRF:

LResns=cns1+nGssn1snr=0n1j=0cnλrjnr!n+jr!s1+n+jr+i=rn1λrjyiir!i+jr!s1+j+ir.

Now, multiplying Eq. 3.17 by sn+1, we get the following function:

sn+1LResns=cnsGsr=0n1j=0cnλrjnr!n+jr!sn+jrr=0n1j=0i=rn1λrjyiir!i+jr!sj+ir.

Taking the limit at infinity to Eq. 3.18, according to Eq. 3.10, we get

cn=g0+r=0n1λr0yr.

Thus, the first approximation of the solution of Eq. 3.3 is

Yns=y0s+y1s2+y2s3++yn1sn+1sn+1g0+r=0n1λr0yr.

Following that, one can find the value of the coefficient cn+1; to do that, we substitute the n+1th-truncated series, Yn+1s=y0s+y1s2+y2s3++yn1sn+cnsn+1+cn+1sn+2, into the n+1th LRF to get the following:

LResn+1s=cnsn+1+cn+1sn+21snLLtL1i=0n1yisi+1+cns1+n+cn+1sn+2Gssn.

Performing the previous steps, we obtain the final form of the n+1th LRF:

LResn+1s=cnsn+1+cn+1sn+2Gssn1snr=0n1j=0cn+1λrjn+1r!n+1+jr!s2+n+jr+cnλrjnr!n+jr!s1+n+jr+i=rn1λrjyiir!i+jr!s1+j+ir.

Again, we multiply Eq. 3.22 by sn+2 to obtain

sn+2LResn+1s=cn+1s2Gs+sg0+sr=0n1λr0yrr=0n1j=0cn+1λrjn+1r!n+1+jr!sn+jr+cnλrjnr!n+jr!sn+jr1+i=rn1λrjyiir!i+jr!si+jr1.

Computing the limit at infinity to both sides of the last equation and using Eq. 3.10, we get

cn+1=g0+cnλn10+1!r=0n1λr1yr+r=0n2λr0yr+1.

So, the second approximation of the solution of Eq. 3.3 is

Yn+1s=y0s+y1s2+y2s3++yn1sn+1sn+1g0+r=0n1λr0yr+1sn+2g0+g0+r=0n1λr0yrλn10+r=0n1λr1yr+r=0n2λr0yr+1.

Like the previous steps, we have

cn+2=g0+cn+1λn10+cnλn20+2!1!cnλn11+2!r=0n1λr2yr+r=0n2λr1yr+1+r=0n3λr0yr+2.

Repeating the steps, one can obtain

cn+3=g0+3!i=02r=0icn+2iλn1i+rr3r!+3!i=03r=0n1iλr3iyr+ii!.

Considering a pattern of the obtained coefficients, we easily deduce the coefficient cn+k as follows:

cn+k=gk0+k!i=0k1r=0icn+k1iλn1i+rrkr!+k!i=0kr=0n1iλrkiyr+ii!,k=0,1,.

According to Eq. 3.14, the recurrence relation (3.28) becomes as follows:

cn+k=gk0+i=0k1r=0ikrcn+k1ian1i+rr0+i=0kr=0n1ikiyr+iarki0.

Thus, we can express the k+1)th-approximate solution of Eq. 3.3 by the following formula:

Yn+ks=i=0n1yisi+1+i=0kcn+is1+i+n,s>0,k=0,1,..

Therefore, the exact solution of Eq. 3.3 can be expressed as

Ys=i=0n1yisi+1+i=0cn+is1+i+n.

Substituting the result in Eq. 3.29 into Eq. 3.31 and running the inverse LT gives the solutions of IVP (1.1) and (1.2) as follows:

yt=i=0n1yii!ti+i=0ti+ni+n!gi0+j=0i1r=0jircn+i1jan1j+rr0+j=0ir=0n1jijyr+jarij0.

3.2 The LRPS method for solving non-linear ODEs

This section introduces the steps of the LRPS approach in solving non-linear ODEs. To explain the methodology of the proposed method in constructing series solutions to this class, we consider IVP (1.3) and (1.4).

To generate the LRPS solution of the IVP (1.3) and (1.4), we consider the first step; that is, operating the LT to Eq. 1.3 and utilizing conditions (1.4), we obtain

Ys=i=0n1yisi+1+1snΨs,Ys,dYds,d2Yds2,,dmYdsm,s>0,

where Ψ is a multivariable function of s,Ys,dYds,d2Yds2, and dmYdsm, mN.

We assume that Ys given in Eq. 3.33 can be expanded as in Eq. 3.4. According to the conditions given in Eq. 1.4 and Theorem 2.3, series (3.4) also has the form in Eq. 3.5, and the kth-truncated series of Ys will be like Eq. 3.6.

To set the values of the unknown coefficients in Eq. 3.6, according to Eq. 3.33, we define the LRF of Eq. 3.33 as

LRess=Ysi=0n1yisi+11snΨs,Ys,dYds,d2Yds2,,dmYdsm,s>0

and the kth LRF as

LResks=Yksi=0n1yisi+11snΨs,Yks,dYkds,d2Ykds2,,dmYkdsm,s>0.

According to the form of Yks as in Eq. 3.6, it is clear that Ψs,Yks,dYkds,d2Ykds2,,dmYkdsm has a finite LS as follows:

Ψs,Yks,dYkds,d2Ykds2,,dmYkdsm=i=0kϕcn,cn+1,,cn+is1+i,s>0,

where ϕ is a multivariable function of cn,cn+1,,cn+i, for i=0,1,,k.

Substituting the expansions (3.6) and (3.36) in (3.35) gives the following expansion form of the kth LRF:

LResks=i=nkcis1+ii=0kϕcn,cn+1,,cn+is1+n+i,s>0.

Thus, the nth LRF is

LResns=cns1+ni=0kϕcn,cn+1,,cn+is1+n+i,s>0.

Now, we multiply Eq. 3.38 by sn+1 to get

sn+1LResns=cni=0kϕcn,cn+1,,cn+isi,s>0.

Now, applying the limit as s → ∞ to both sides of Eq. 3.39 and using the fact in Eq. 3.10, we can easily determine the value of cn by solving the following equation for cn:

cn=ϕcn.

In the same manner, we find the value of the coefficient cn+1 by substituting the n+1th-truncated series, Yns=i=0n1yis1+i+cns1+n, into the n+1th LRF to get the following:

LResn+1s=cn+1s2+ni=0kϕcn,cn+1,,cn+is2+n+i,s>0.

Multiplying sn+2 by both sides of Eq. 3.41, we get the following function:

sn+2LResn+1s=cn+1i=1kϕcn,cn+1,,cn+isi1,s>0.

Applying the limit at infinity to Eq. 3.42, we obtain the algebraic equation:

cn+1=ϕcn+1.

Solving Eq. 3.43 implicitly for cn+1 determines the second unknown coefficient in Eq. 3.6.

Similarly, we compute the third coefficient cn+2 by substituting the n+2th-truncated series, Yn+2s=y0s+y1s2+y2s3++yn1sn+cnsn+1+cn+1sn+2+cn+2sn+3, into the n+2th LRF to get the following function:

LResn+2s=cn+2sn+3i=0kϕcn,cn+1,,cn+is1+n+i,s>0.

Multiplying Eq. 3.44 by sn+3 gives

sn+3LResn+2s=cn+2ϕcnϕcn+1ϕcn+2i=2kϕcn,cn+1,,cn+isi2,s>0.

According to fact (3.10), we obtain

cn+2=ϕcn+ϕcn+1+ϕcn+2.

Solving Eq. 3.46 for cn+2 sets another coefficient in Eq. 3.6.

The value of the third unknown coefficient cn+3 can be obtained by similar arguments and by solving the following equation:

cn+3=ϕcn+ϕcn+1+ϕcn+2+ϕcn+3.

Considering the pattern of the obtained coefficients, we easily conclude the coefficient cn+k from the following implicit formula of cn+k:

cn+k=ϕcn+ϕcn+1+ϕcn+2+ϕcn+3++ϕcn+k.

Thus, we can express the k+1)th-approximate solution of Eq. 3.33 in the following shape:

Yn+ks=i=0n1yisi+1+i=0kϕcn,cn+1,,cn+is1+n+i,s>0,k=0,1,.

Therefore, the exact analytic solution of Eq. 3.33 is written in a series form:

Ys=i=0n1yisi+1+i=0ϕcn,cn+1,,cn+is1+n+i.

Running the inverse LT to Eq. 3.50 gives the solution of Eq. 1.3 and Eq. 1.4 in a series expansion as

yt=i=0n1yii!ti+i=0ϕcn,cn+1,,cn+ii+n!ti+n.

3.3 The LRPS method for solving singular-value problems

This section presents the LRPS method’s procedure for handling singular-value problems. To do this, let us consider the following singular-value problem:

1tkdnydtn=ft,yt,dydt,d2ydt2,,dn1ydtn1,tI,m,kN.

Subject to the ICs

y0=y0,y0=y1,,yn10=yn1.

To solve the initial-singular value problems (3.52) and (3.53), we first multiply Eq. 3.52 by tk to get

dnydtn=tkft,yt,dydt,d2ydt2,,dn1ydtn1.

Now applying LT to Eq. 3.54 and using ICs (3.53), we get

Ys=i=0n1yisi+1+1snLL1k!sk+1L1Ψs,Ys,dYds,d2Yds2,,dmYdsm,s>0.

Now, suppose that the function Y(s) can be expressed in the form of the expansion of (3.4), and so on. We can complete the steps described in the previous Section 3.2 to obtain the required solution.

4 Applications to linear and non-linear problems

This section presents seven interesting problems with wide applications in physics and other sciences that are discussed and solved by the LRPS method.

Problem 4.1. consider the following composite oscillation equation:

d2ydt2adydtbyt=8,t0,

with respect to the initial condition

y0=0,y0=0.

Comparing Eq. 4.1 with Eq. 1.1 concludes that a1t=a,a0t=b, and gx=8. Using the results obtained in Section 3.1, we can deduce λ10=a,λ11=λ12=λ13==0 and λ00=b,λ01=λ02=λ03==0. According to the recurrence relation in Eq. 3.29, we can see that c2=8, c3=8a, c4=8b+a2,c5=8a3+2ab, c6=8a4+3a2b+b2, c7=8a5+4a3b+3ab2, c8=8a6+5a4b+6a2b2+b3, c9=8a7+6a5b+10a3b2+4ab3, and c10=8a8+7a6b+15a4b2+10a2b3+b4. Therefore, the 10th approximation of the solution of the IVP (4.1) and (4.2) will be as follows:

y10t=82!t2+8a3!t3+8b+a24!t4+8a3+2ab5!t5+8a4+3a2b+b26!t6+8a5+4a3b+3ab27!t7+8a6+5a4b+6a2b2+b38!t8+8a7+6a5b+10a3b2+4ab3t99!+8a8+7a6b+15a4b2+10a2b3+b4t1010!.

It is easy to check if the exact solution of Eq. 4.1 and Eq. 4.2 is as follows:

yt=4bcc+ae12act+cae12a+ct8b,c=a2+4b.

To analyze the accuracy of the approximate solution in Eq. 4.3 and determine the interval of convergence, we introduce and compute two types of error, actual and relative errors that are defined, respectively, as follows:

Act.Err.t=yty10t

and

Rel.Err.t=yty10tyt.

For analysis and comparison of the exact and approximate solutions of IVP (4.1) and (4.2), Table 1 shows the numerical results of this problem. It displays the exact and approximate results in addition to the actual and relative errors at different values of t within the interval 0,1. The results indicate that the errors increase when the value of t increases. It is known that by increasing the number of terms in the series solution, the error decreases and the convergence period of the truncated series increases. It should be noted that we can extend the convergence period using the multi-stage technique.

TABLE 1
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TABLE 1. The exact and the 10th approximate solutions of the IVP (4.1) and (4.2) and the actual and relative errors at a=3 and b=2.

Problem 4.2. consider the following Bessel’s equation:

1t2yt2tyt+2yt=0.

Subject to the ICs,

y0=1,y0=1.

According to the existence and uniqueness theorem, it is clear that the IVPs (4.7) and (4.8) have a unique solution in the interval 1,1, so we seek to get this solution via the LRPS method. To reach our goal and be able to rely on the construction obtained in Section 3.1, it is necessary to rewrite Eq. 4.7 as follows:

yt=2t1t2yt21t2yt,0t<1.

Comparing Eq. 4.7 with Eq. 1.1, we find that

Lt=2t1t2ddt21t2,a1t=2t1t2,a0t=21t2,

where a0t and a1t are analytic functions on 0,1).Since the coefficients of the linear operator in Eq. 4.10 are analytic functions, they can be expressed as McLaurin expansions as follows:

a0t=j=02t2j,a1t=j=02t2j+1.

So, according to Eq. 4.11 and Eq. 3.14, we have

λ02j=a02j02j!=2,λ02j+1=a02j+102j+1!=0,j=0,1,2,λ12j=a12j02j!=0,λ12j+1=a12j+102j+1!=2,j=0,1,2,.

Comparing with the general formula (3.29), we can find the values of the coefficients as follows:

c2=λ00y0+λ10y1=2,c3=c2λ10+λ01y0+λ11y1+λ00y1=0,c4=c3λ10+c2λ00+2!c2λ11+2λ02y0+λ12y1+λ01y1=8,c5=c4λ10+c3λ00+3c3λ11+3c2λ01+6c2λ12+6λ03y0+λ13y1+λ02y1=0c6=c5λ10+c4λ00+4c4λ11+4c3λ01+12c3λ12+12c2λ02+24c2λ13+24λ04y0+λ14y1+λ03y1=144c7=c6λ10+c5λ00+5c5λ11+5c4λ01+20c4λ12+20c3λ02+60c3λ13+60c2λ03+120c2λ14+120λ04y1+λ05y0+λ15y1=0c8=c7λ10+c6λ00+6c6λ11+6c5λ01+30c5λ12+30c4λ02+120c4λ13+120c3λ03+360c3λ14+360c2λ04+720c2λ15+720λ05y1+λ06y0+λ16y1=5760c9=c7λ00+7c6λ01+42c5λ02+210c4λ03+840c3λ04+2520c2λ05+c8λ10+7c7λ11+42c6λ12+210c5λ13+840c4λ14+2520c3λ15+5040c2λ16+5040λ07y0+λ17y1+λ06y1=0c10=c8λ00+8c7λ01+56c6λ02+336c5λ03+1680c4λ04+6720c3λ05+20160c2λ06+c9λ10+8c8λ11+56c7λ12+336c6λ13+1680c5λ14+6720c4λ15+20160c3λ16+40320c2λ17+40320λ08y0+λ18y1+λ07y1=403200.

Therefore, the LRPS solution to Problem 4.2 can be expressed in the following series form:

yt=1ttt+t33+t55+t77+t99+.

The expansion in (4.13) is the same expansion as that of the function tan1t. Therefore, the exact solution of the IVP (4.7) and (4.8) has the following closed form:

yt=1tttan1t,0t<1.

Table 2 shows the numerical results of Problem 4.2. It shows the exact and approximate results in addition to the actual and relative errors at different values of t0,09. The displayed data are acceptable and can be improved by increasing the order of the approximation.

TABLE 2
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TABLE 2. The exact and the 10th approximate solutions of the IVP (4.7) and (4.8) and the actual and relative errors.

Problem 4.3. consider the following non-linear nonhomogeneous ODE:

y3t+y2t+costyt=1cost.

Subject to the ICs,

y0=0,y0=1,y0=0.

Similar to the previous problems, we operate the LT on both sides of Eq. 4.15 and employ the ICs (4.16). Then, we obtain the following equation in the Laplace space:

Ys=1s21s3LL1Ys21s3LL1s1+s2L1sYs+1s41s21+s2,s>0.

We assume that the solution of Eq. 4.17 has the same LS expansion as in Eq. 3.4. According to ICs (4.16), the kth-truncated series of Ys becomes

Yks=1s2+i=3kcis1+i,s>0.

To set the value of the unknown coefficients in series (4.18), we utilize the kth LRF of Eq. 3.17, which is defined as

LResks=Yks1s2+1s3LL1Yks2+1s3LL1s1+s2L1sYks1s4+1s21+s2,s>0.

To determine the coefficient c3, we substitute Y3s=1s2+c3s4 into LRes3s and run the operators in Eq. 4.19 to get the following rational function:

LRes3s=c3s41s4+2s21+s2+2s6+8c3s8+c31+s233c3s21+s23+20c32s10.

Employing fact (3.10), the solution of the equation limss4LRes3s=0 for c3 introduces c3=1.Similarly, to find out the value of the second unknown coefficient c4, we substitute Y4s=1s2+1s4+c4s5 into the 4th-LRF to get the following:

LRes4s=c3s5+2s68s8+20s1011+s23+3s21+s232s4+s670c3s11+10c3s9+c3s31+s246c3s1+s24+sc31+s24+70c32s12,s>0.

Utilizing fact (3.10) via Eq. 4.21 gives c4=0. Using the same procedure as mentioned above, we can find more coefficients for series (4.18). Some of them are c5=1,c6=0,c7=1,c8=0,c9=1,c10=0,c11=1. So, the series solution to Eq. 4.31 has the following LS:

Ys=1s21s4+1s61s8+1s101s12+.

Therefore, the LRPS solution to Eq. 4.15 and Eq. 4.16 can be expressed in the following series form:

yt=tt33!+t55!t77!+t99!t1111!+,

which is the expansion of the exact solution yt=sint.

Problem 4.4. consider the following non-linear pantograph equation:

d2ydt2=2y2t2,t0.

Subject to the ICs,

y0=1,y0=0.

Following the same procedure as in the previous problems, we can express the LRPS solution to IVP (4.24) and (4.25) in the form

yt=1t2+t4127t61440+127t8129024010879t107431782400+.

Since we cannot predict the pattern in the coefficients of the series solution in Eq. 4.26, we cannot reach the exact solution. Therefore, we test the results using the residual and relative errors, which are defined as follows, respectively:

Res.Err.t=L1LResks=d2ykdt2+2yk2t2,
Rel.Err.t=yktyk/2tykt.

Table 3 shows the numerical results of Problem 4.4. It displays the 10th approximate solution in addition to the residual and relative errors at different values of t within the interval 0,1. The results indicate that the LRPS solution is acceptable mathematically in the period 0,1.

TABLE 3
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TABLE 3. The 10th approximate LRPS solution of the IVP (4.24) and (4.25) and the residual and relative errors.

Problem 4.5. consider the following homogenous linear singular ODE:

sinty2costysinty=0,0<t<π,

with respect to the ICs:

y0=2,y0=0.

We apply the LT on both sides of Eq. 4.29 and use the ICs in Eq. 4.30 to obtain the following symbolic algebraic equation in the Laplace space:

LL111+s2L1s2Ys2s2LL111+s2L1sYs2LL111+s2L1Ys=0,s>0.

Suppose that the solution of Eq. 4.31 has a LS expansion as in Eq. 3.4. According to ICs (4.30), the kth-truncated series (3.6) can be expressed as

Yks=2s+i=2kcis1+i,s>0.

To set the unknown coefficient in series (4.32), we define the kth LRF of Eq. 4.31 as follows:

LResks=LL111+s2L1s2Yks2s2LL111+s2L1sYks2LL111+s2L1Yks,s>0.

We substitute Y2s=2s+c2s3 into LRes2s and run the operators in Eq. 4.34 to get the following function:

LRes2s=21+s23+4c21+s234s21+s23c2s21+s232s41+s23c2s41+s23.

Solving the equation limss2LRes2s=0 gives c2=2. Thus, the first approximation of the solution of Eq. 4.31 is Y2s=2s2s3.Again, we substitute 3rd-truncated series, Y3s=2s2s3+c3s4, into the 3rd LRF to get the following:

LRes3s=101+s24+12c3s1+s2412s21+s24+4c3s31+s242s41+s24.

Consequently, the equation limss3LRes3s=0 gives c3=0.Likewise, we substitute the 4th-truncated series, Y4s=2s2s3+c4s5, into the 4th LRF to get the following:

LRes4s=101+s254c41+s2522s21+s25+21c4s21+s2514s41+s25+10c4s41+s252s61+s25+c4s61+s25.

Solving the equation limss4LRes4s=0 gives c4=2. Applying the same procedure for k=5,6,7,8 leads to c5=0, c6=2, c7=0, and c8=2. Thus, we conclude that the solution of Eq. 4.31 has the following expansion:

Ys=2s2s3+2s52s7+2s9.

Applying the inverse LT to Eq. 4.38 gives the LRPS solution to the IVP (4.29) and (4.30) in the following PS form:

yt=21t22!+t44!t66!+t88!.

It is clear that the closed form of the exact solution of IVP (4.50) and (4.51) is yt=2cost.

Problem 4.6. consider the following non-homogeneous nonlinear Lane–Emden singular ODE:

yt+2tyttsinyt=e2t,t0,2,

considering the ICs:

y0=π,y0=0.

Using similar arguments to the previous problem, one can obtain the series solution of the LT of the IVP (4.40) and (4.41) as follows:

Ys=πs+13s3+1s4+125s5+143s6+607s7+15s8+28s9+258845s10+.

Applying the inverse LT on Eq. 4.42 gives the LRPS solution of the IVP (4.40) (4.41) in the following series form:

yt=π+t26+t36+t410+7t5180+t684+t7336+t81440+647t94082400+.

There is no pattern between the series terms in Eq. 4.43. So, it is difficult to predict the exact solution formula. Thus, we suffice with the approximate solution we got for Problem 4.6. It is worth noting that the more terms we calculate for the LRPS solution, the longer the series convergence interval and the higher the accuracy of the solution. Therefore, we test the 8th approximate LRPS solution for Problem 4.6 using the residual and relative errors, which are defined as follows, respectively:

Res.Err.t=L1LRes8s=t2tt+2t2+2t3+4t43+2t53+4t615+4t745+8t8315+t9151213t1043201003t11255150113221t1232659200,
Rel.Err.t=y8ty4ty8t=t684+t7336+t81440+647t94082400π+t26+t36+t410+7t5180+t684+t7336+t81440+647t94082400.

Table 4 shows the numerical results of Problem 4.6. It illustrates the 8th approximate solution in addition to the residual and relative errors at different values of t within the interval 0,1. Similar to the results in the previous tables, the data are good for the period 0,1].

TABLE 4
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TABLE 4. The 8th approximate LRPS solution of the IVP (4.40) and (4.41) and the residual and relative errors.

Problem 4.7. We consider the following micro-electromechanical system (MEMS) [52]:

y+y+θy1=0,θ>0,

with the ICs:

y0=y0=0.

This dynamic differential equation is used to describe the wire’s movement as a point mass, where y is the dimensionless distance and θ is a voltage-related parameter.

Simulating the previous examples, the LT of the IVP (4.46) and (4.47) is given by the following algebraic equation:

LL1s2YsL1Yss2Ys+LL1Ys2Ys+1s2=0,s>0.

Applying the arguments and processes of the LRPS method, one can obtain the LRPS solution to the algebraic Eq. 4.48 as follows:

Ys=θs3+θθ1s5+θ12θ+7θ2s7θ13θ+39θ2127θ3s9+θ14θ+168θ21678θ3+4369θ4s11+.

Applying the inverse LT to Eq. 4.49 gives the LRPS solution of the IVP (4.46) (4.47) as follows:

yt=θt22+θθ1t44!+θ12θ+7θ2t66!θ13θ+39θ2127θ3t88!+θ14θ+168θ21678θ3+4369θ4t1010!+.

To test the accuracy of the obtained solution given in (4.50), we compute the residual and relative errors to the 10th approximation of the solution. Table 5 shows the 10th approximate solution in addition to the residual and relative errors at different values of t within the interval 0,1. The results indicate that the obtained solution is acceptable mathematically.

TABLE 5
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TABLE 5. The 10th approximate LRPS solution of the IVP (4.46) and (4.47) and the residual and relative errors.

On the other hand, what specialists in MEMS system implementations are most interested in is the pull-in phenomenon analysis. The MEMS system in Eq. 4.46 and Eq. 4.47 conducts either periodically or unsteadily. This behavior depends on the value of the voltage-related parameter, θ. At small values of θ, the solution of the system is stable and periodic, whereas at large values of θ, it becomes unstable, called pull-in instability. Figure 1 shows that the system is stable and periodic at θ values less than or equal to the critical value (θ=0.203632188) [52], and it becomes unstable at θ values greater than the critical one.

FIGURE 1
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FIGURE 1. Phase trajectories at different values of θ. (A) Solid line: θ=0.2; dotted line: θ=0.203; and dashed line: θ=0.203632188. (B) Solid line: θ=0.2037; dotted line: θ=0.25; and dashed line: θ=0.3.

5 Conclusion

This study aims to test the efficiency of the LRPS method in finding series solutions for ODEs, which are difficult to solve in the analytical methods. We have succeeded in providing a solution to the general form of linear ODEs whose coefficients are analytical functions as an exact solution in a PS form. We also dealt with non-linear ODEs in the proposed technique and found approximate solutions with high accuracy. The biggest surprise is the success of the LRPS method in providing series solutions for the equations about the singular points that coincide with the exact results in some examples. Using the LRPS method, there is no longer an obstacle to obtaining a PS solution for a broad class of ODEs. In addition, the idea of the method circumvented the use of the LT to solve non-linear equations to which the LT is difficult to apply. In addition to the method’s efficiency in arriving at exact solutions, LRPS is easy and fast in finding the coefficients of a series solution. There is no doubt that we can use the new method to solve other sets of equations that we did not have to deal with in previous studies, such as using it to solve partial differential equations, integral equations, integrodifferential equations, and linear or non-linear, as well as algebraic equations. We should not forget that the method has not been applied to solve differential equations with boundary conditions. All these and other topics will be under research by our research team in the next stage.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: ordinary differential equations, Laplace transforms, power series, approximate solutions, laurent series

Citation: El-Ajou A, Al-ghananeem H, Saadeh R, Qazza A and Oqielat MN (2023) A modern analytic method to solve singular and non-singular linear and non-linear differential equations. Front. Phys. 11:1167797. doi: 10.3389/fphy.2023.1167797

Received: 16 February 2023; Accepted: 23 March 2023;
Published: 17 April 2023.

Edited by:

Ji-Huan He, Soochow University, China

Reviewed by:

Muhammad Nadeem, Qujing Normal University, China
Naveed Anjum, Government College University Faisalabad, Pakistan
Guangqing Feng, Henan Polytechnic University, China

Copyright © 2023 El-Ajou, Al-ghananeem, Saadeh, Qazza and Oqielat. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Rania Saadeh, rsaadeh@zu.edu.jo, raniaraed2011@gmail.com

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