An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition.

In this paper, a linear singularly perturbed Fredholm integro-differential initial value problem with integral condition is being considered. On a Shishkin-type mesh, a fitted finite difference approach is applied using a composite trapezoidal rule in both; in the integral part of equation and in the initial condition. The proposed technique acquires a uniform second-order convergence in respect to perturbation parameter. Further provided the numerical results to support the theoretical estimates.

Introduction

Singularly perturbed differential equations are described by a small parameter multiplying all or some of the differential equation’s highest order terms, as boundary layers are generally present in their solutions. These equations are crucial for sophisticated scientific computations in the twenty-first century. Singularly perturbed problems (SPPs) are used to express a variety of mathematical models, ranging from chemical reactions to problems in mathematical engineering, fluid dynamics, electrical networks, control theory, aerodynamics, biology and neuroscience. Further information on SPPs may be found in the works [18, 26, 27, 29] and their references. Numerical analysis of SPPs has always been difficult because of the solution’s boundary layer behavior. Within some thin layers at the inside or boundary of the problem domain, such a problem exhibits fast changes [26, 29]. Standard numerical techniques for resolving such problems are widely recognized for being unstable and failing to produce exact results when the perturbation parameter is small. On account of this, it is critical to design numerical methods for solving problems whose accuracy is independent on parameter value. The references [18, 22, 26, 33, 35, 40] cover a variety of techniques for numerically solving this type differential equations.

Differential equations with integral boundary conditions have also been utilized to describe a variety of processes in the applied sciences, such as subsurface water flow, chemical engineering and heat conduction [11, 21, 28]. Therefore, many authors have studied boundary value problems with integral boundary conditions. Researchers have considered the singularly perturbed cases of these problems. The authors in [9, 10, 25, 36] investigated first-order convergent finite difference schemes on non-uniform meshes for various problems with integral boundary conditions.

Integro-differential equations have emerged in most engineering applications and several fields of sciences. Plasma physics, financial mathematics, epidemic models, population dynamics, biology, artificial neural networks, fluid mechanics, electromagnetic theory, financial mathematics, oceanography and physical processes are among these (see, e.g., [8, 39]). For instance, in [23], the integro-differential equation used to modelling infectious diseases in optimal control strategies for policy decisions and applications in COVID-19 has been expressed as follows:whereThat’s why, many researchers have been pondering the Fredholm integro-differential equations (FIDEs) for a long time. An overview of existence and uniqueness results for the solution of FIDEs can be found in some references such as [1, 19] (see also references therein). Furthermore, researchers employed fitted analytical approaches because of the difficulty of obtaining accurate solutions to these types of problems. Some of these methods are reproducing kernel Hilbert space method [7], Nyström method [38], Touchard polynomials method [2], Tau method [20, 32], Collocation and Kantorovich methods [37], Galerkin method [12, 41, 43], Boole collocation method [14], parameterization method [17], Legendre collocation matrix method[44], variational iteration technique [19]. The increasing interest in recent years is not limited to only FIDEs, but also the numerical solutions of linear and nonlinear Volterra or Volterra-Fredholm integro-differential equations are increasing in popularity. Recently, Turkyilmazoglu presented an effective technique for solving the linear FIDEs and nonlinear Volterra-Fredholm-Hammerstein integro-differential equations based on the Galerkin method [41, 42] (see also references therein).

is the set of features characterizing dissimilar styles of populations (e.g. sex, age),

the aggregate number of people aforethought,

represent a parametrization of different courses of diseases and the probability of a person with property suffering from disease .

the basic breeding number, i.e. the number of people infected by a single infectious individual in a completely responsive population.

, with ,   the probability of an infection event between a person with property infected at time infecting a person with property p at time t.

and is the initial datum. Further, the Incubation Period has been defined by , and the infectious (COntagious) period by .

We consider a singularly perturbed Fredholm integro-differential equation (SPFIDE) with integral boundary condition as follows:where . is a perturbation parameter. , A and are given constants. We assume that , , f(x) and K(x, s) are the sufficiently smooth functions satisfying certain regularity conditions to be specified. Under these conditions, the solution u(x) of the problem (1)-(2) has in general initial layer at for small values of . This means that the derivatives of the solution become unbounded for small values of perturbation parameter near .

The above-mentioned papers, related to FIDEs, were dealt mainly with the regular cases (i.e., when the boundary layers are absent). Scientists have also given numerical approaches to singular perturbation situations of FIDEs in recent years. Amiraliyev et al. [3, 5] proposed an exponentially fitted difference method on a uniform mesh for solving first and second-order linear SPFIDEs, demonstrating that the approach is first-order convergent uniformly in . Difference schemes of the fitted homogeneous type with an accuracy of on a piecewise uniform mesh for this type of problems are given in [4, 15]. It should also be noted that in [30, 31], for the numerical solution of singularly perturbed Volterra integro-differential equations, first-order difference schemes on a piecewise uniform mesh are given, followed by Richardson extrapolation to obtain the second order of accuracy.

The aim of this work is to present a homogeneous (non-hybrid) type difference scheme for the numerical solution of SPFIDE with an integral condition. A special technique is necessary to establish the appropriate difference scheme and investigate the error analysis for the numerical solution of such problems. The scheme is built using the integral identity method and suitable quadrature rules, with the remainder terms in integral form. The goal is to develop an -uniformly second-order homogeneous finite difference method that produces uniform convergent numerical approximations in order to solve problem (1)-(2).

The content is arranged as follows: Some properties of the solution of (1)-(2) are given in Sect. 2. A finite difference scheme and a special piecewise uniform mesh are presented in Sect. 3. The stability and convergence analysis of this scheme are shown in Sect. 4. The numerical results of two examples to verify the theoretical estimates are presented in Sect. 5. Finally, the work ends with a summary of the conclusions in Sect. 6.

Properties of the exact solution

We now present some properties of the solution of (1)-(2), which are needed in later sections for the analysis of the appropriate numerical solution. Here, we will use the following notations:

Lemma 1

Assume that and , MoreoverThen the solution u(x) of the problem (1)-(2) satisfies the bounds

Proof

From (1) we have the following relation for :By using the boundary condition (2) we getSince and , the denominator is bounded below by one.

Also, we can write the numerator of (5) asConsidering (5) and (6) together, we obtainLater on, according to the maximum principle for from (1), we haveNow, considering the estimate of (7) instead of in the above inequality by virtue of (3), we acquirewhich implies the validity of (4) for . The proof of (4) for can be proved in a similar way as in [3, 4].

Designing of the numerical method

Let be any non-uniform mesh on [0, l] : andPrior to describing our numerical technique, we present certain notations for the mesh functions. To any mesh function v(x) described on , we utilizeWe construct the numerical method using the identitywith the basis functionsandWe note that the function is the solution of the problemUsing the method of exact difference schemes [6, 13, 24, 45] (see also [34], pp. 207-214), for the differential part from (9), we obtainwithBy Newton interpolation formula with respect to mesh point we haveTherefore we getAlso usingin the first term at the right side of (12), we havewhereSimple calculation giveswithIt is easy to see that So, the identity (10) degrades towhereand is given by (14). Analogously we derivewhereIt remains to obtain an approximation for integral term from (1). Using the Taylor expansionwe getwhereNext, if the first term at the right side of (20) is operated by applying the composite trapezoidal integration rule with the remainder term in the integral form [4], we getwhereandTo approximate the boundary condition (2), using again the composite trapezoidal integration rule, we havewhereAfter taking into consideration (15), (17), (20) and (23) in (9) we obtain the following discrete identity for u(x):with remainder termwhere and are defined by (13), (19), (22), (24) and (26) respectively.

Based on (27) we propose the following difference scheme for approximating (1)-(2):where and are given by (11), (16), (18) and (21) respectively.

To discretize the interval [0, l],  we will use the piecewise-uniform Shishkin type mesh. As the problem (1)-(2) has an exponential initial layer in the neighborhood at , we divide [0, l] into two subinterval and For an even N,  a uniform mesh with N/2 intervals is placed on each subinterval, where the transition point which separates the fine and coarse portions of , that is defined asHence, if we denote by and the stepsizes in and respectively, our piecewise-uniform mesh can be expressed as

The convergence

We proceed to estimate the error of the approximate solution , From (27) and (29) we havewhere the truncation error functions and is given by (26) and (28).

It should be noted that since and then exist a number such that for sufficiently large values of N will be ( is defined by (14)).

Lemma 2

Assume that and Then the truncation error functions and satisfy the estimates

First, we estimate the remainder term . From the explicit expression (26), under the condition of Lemma 1, we obtainNow we find a convergence error estimate for the first term in the right-side of (35) in our special piecewise-uniform meshNote that the above estimate is valid for values both and .

For the second two term in the right-side of (35), we find the estimate for the case Then it has the form and . Thus we getFor two term in the right-side of (35), we find the estimate for the case . From this inequality, we can writeFor the first term in the right-side of (38), we haveFor the second term in the right-side of (38), we obtainTherefore, the estimates (36), (37), (39) and (40) along with (35) yield (34).

Further, to confirm (33), we will estimate the remainder terms and separately. For , taking into account the boundedness of , from (24) similar to above, we getNext, we will estimate Since , and by using Lemma 1, it follows thatWe find the estimate for the case Then and Hence we haveWe now consider the case in (42) on The inequalitiesimply thatTherefore, from (43) and (44), we deduce thatThird, we will estimate Since , and by using Lemma 1, it follows thatNote that the above estimate is valid for values both and . Fourth, we will estimate . By taking into account the boundedness of , from (22) it follows thatNote that the above estimate is valid for values both and . The inequalities (41), (45), (46) and (47) finish the proof of (33).

Theorem 1

Let a, c and K satisfy the assumptions from Lemma 2. MoreoverThen for the solution z of the difference problem (31)-(32) holds the estimate

Equation (31) may be rewritten aswhereFrom (49) we getThe solution to the above first-order difference equation will be as follows:whereThen, from (32) and (50), we obtainSince, the denominator is bounded below by one and the equality (51) reduces toConsidering (51) and (52) together, we haveNow, applying discrete maximum principle for (49), we getFinally, instead of in the above inequality, considering the estimate of (53), we getThereforeThis inequality together with (33) and (34) produces the desired result.

Numerical results

Here, we have considered two specific problems to demonstrate the feasibility of the proposed approach. The following iterative technique will be used.where are the given initial iterations.

Example 1

We consider the test problem

The exact solution of test problem is given byWe define the exact errors as follows:The results of the problem obtained by using different and N values for both the present method and solving exact of SPFIDE are given in the following tables 1-6. In addition, in tables, exact errors are shown according to the exact solutions and approximate solutions.

Table 1

The numerical results of Example 1 for and

x_i	u_i	y_i	ei64
0.0081	1.8704	1.8701	0.0003
0.0162	1.7557	1.6543	0.1014
0.0243	1.6541	1.4174	0.2367
0.0486	1.4132	0.9774	0.4358
0.0729	1.2435	0.8890	0.3545
0.1377	0.9894	0.8380	0.1514
0.2187	0.8508	0.7877	0.0631
0.4447	0.6930	0.6778	0.0152
0.6757	0.5968	0.5989	0.0021
0.8605	0.5375	0.5509	0.0134

Table 6

The numerical results of Example 1 for and

x_i	u_i	y_i	ei512
0.0002	1.9522	1.9292	0.0230
0.0010	1.7638	1.1965	0.5673
0.0014	1.6924	1.0489	0.6435
0.0061	1.2042	0.9904	0.2138
0.0100	1.0675	0.9867	0.0808
0.1004	0.9088	0.9089	0.0001
0.3018	0.7682	0.7760	0.0078
0.5032	0.6652	0.6805	0.0153
0.7008	0.5880	0.6104	0.0224
0.9022	0.5257	0.5555	0.0298

Figs. 1 and 2 represent the solution plots for different values of and N in Example 1, according to the table values. The figures clearly show that the exact solution and the approximated solution for Example 1 overlap, thereby showing the aptness of the proposed techniques.

Fig. 1

Numerical results of Example 1 for and

Fig. 2

Numerical results of Example 1 for and

Example 2

Consider the other problem:

The exact solution to this problem is unknown. For this reason, we estimate errors and calculate solutions using the double-mesh method, which compares the obtained solution to a solution computed on a mesh that is twice as fine. We introduce the maximum point-wise errors and the computed aswhere is the approximate solution of the respective method on the meshwithWe also describe the rates of convergence and computed -uniform rate of convergence of the form

The values of and N for which we resolve the Example 2 are and . From Table 7, we observe that the -uniform rate of convergence is monotonically increasing towards two, therefore in agreement with the theoretical rate given by Theorem 1.

Table 7

Maximum point-wise errors and the rates of convergence for different vales of and N

ε	N=64	N=128	N=256	N=512	N=1024
2^0	0.05368	0.01607	0.00452	0.00117	0.00029
	1.74	1.83	1.95	2.01
2^-4	0.05558	0.01687	0.00481	0.00127	0.00032
	1.72	1.81	1.92	1.99
2^-8	0.05610	0.01703	0.00496	0.00132	0.00034
	1.72	1.78	1.91	1.96
2^-12	0.05544	0.01683	0.00497	0.00135	0.00035
	1.72	1.76	1.88	1.95
2^-16	0.05680	0.01736	0.00516	0.00142	0.00037
	1.71	1.75	1.86	1.94
e^N	0.05680	0.01736	0.00516	0.00142	0.00037
p^N	1.66	1.74	1.86	1.93

Conclusion

This article comprises a numerical method employed to solve a linear SPFIDE of the form (1)-(2). On a special piecewise uniform mesh, the differential equation is discretized by using a fitted finite difference operator. The composite trapezoidal integration rule with the remainder term in integral form has been used for the integral part in (1) and initial condition (2), yielding uniform second-order convergence. Specific test problems have been performed to assess and test the performance of the numerical scheme. The obtained results can be presented to more complicated FIDEs.

Table 2

The numerical results of Example 1 for and

x_i	u_i	y_i	ei128
0.0047	1.9229	1.9223	0.0006
0.0094	1.8511	1.7829	0.0682
0.0235	1.6636	1.2688	0.3948
0.0376	1.5117	0.9836	0.5281
0.0799	1.2045	0.8777	0.3268
0.1363	0.9930	0.8375	0.1555
0.2162	0.8537	0.7879	0.0658
0.4450	0.6929	0.6776	0.0153
0.6630	0.6013	0.6028	0.0015
0.8592	0.5379	0.5515	0.0136

Table 3

The numerical results of Example 1 for and

x_i	u_i	y_i	ei256
0.0027	1.9550	1.9549	0.0001
0.0108	1.8306	1.6244	0.2062
0.0162	1.7557	1.3597	0.3960
0.0297	1.5929	0.9796	0.6133
0.0864	1.1714	0.8720	0.2994
0.1620	0.9355	0.8208	0.1147
0.2565	0.8124	0.7659	0.0465
0.4486	0.6911	0.6757	0.0154
0.6730	0.5977	0.5995	0.0018
0.8566	0.5386	0.5518	0.0132

Table 4

The numerical results of Example 1 for and

x_i	u_i	y_i	ei128
0.0006	1.8587	1.7957	0.0630
0.0012	1.7372	1.4661	0.2711
0.0024	1.5429	1.0599	0.4830
0.0056	1.2312	0.9909	0.2403
0.0649	0.9391	0.9379	0.0012
0.1108	0.9003	0.9009	0.0006
0.3097	0.7635	0.7718	0.0083
0.5086	0.6629	0.6785	0.0156
0.7075	0.5857	0.6086	0.0229
0.9064	0.5245	0.5547	0.0302

Table 5

The numerical results of Example 1 for and

x_i	u_i	y_i	ei256
0.0003	1.9167	1.8776	0.0391
0.0010	1.7701	1.3999	0.3702
0.0019	1.6191	1.0523	0.5668
0.0054	1.2446	0.9911	0.2535
0.0100	1.0677	0.9867	0.0810
0.0977	0.9110	0.9109	0.0001
0.3029	0.7675	0.7749	0.0074
0.5005	0.6664	0.6811	0.0147
0.7057	0.5863	0.6083	0.0220
0.9033	0.5254	0.5546	0.0292