MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2016/9418730 9418730 Research Article Sherman-Morrison-Woodbury Formula for Linear Integrodifferential Equations http://orcid.org/0000-0003-4665-0124 Wu Feng 1 Sadarangani Kishin State Key Laboratory of Structural Analysis of Industrial Equipment Faculty of Vehicle Engineering and Mechanics Dalian University of Technology Dalian 116023 China dlut.edu.cn 2016 25102016 2016 22 09 2016 04 10 2016 2016 Copyright © 2016 Feng Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The well-known Sherman-Morrison-Woodbury formula is a powerful device for calculating the inverse of a square matrix. The paper finds that the Sherman-Morrison-Woodbury formula can be extended to the linear integrodifferential equation, which results in an unified scheme to decompose the linear integrodifferential equation into sets of differential equations and one integral equation. Two examples are presented to illustrate the Sherman-Morrison-Woodbury formula for the linear integrodifferential equation.

National Natural Science Foundation of China 11472076 51609034 China Postdoctoral Science Foundation 2016M590219
1. Introduction

This paper is devoted to the study of the linear integrodifferential equation (IDE):(1)Lvθ,x+kθ,xΘθQθ,x,s,Dvs,xds=fθ,x,xΩ,θΘ,ΘθΘwith the homogeneous boundary conditions:(2)Bvθ,x=0,xΩ,Cvθ,x=0,θΘ,where(3)θ=θ1θ2θmTΘ,x=x1x2xnTΩ,Lvθ,x, Bvθ,x, Cvθ,x, and Dvθ,x are all linear differential operators. Qθ,s,Dvs,x is a linear function with respect to Dvθ,x. ΘθΘ is dependent on the variable  θ. vθ,x is the unknown function of two types of variables, that is, θ and x. However, the integration in (1) is with respect to θ, without involving x.

The IDE can be found to describe various kinds of phenomena such as Bose-Einstein condensates , wave-power hydraulics , heterogeneous heat transfer , slow erosion of granular flow , and wind ripple in the desert . Solving this equation has been one of the interesting tasks for researchers. To date, there are many numerical methods such as finite element method  and finite difference method  that can be used to obtain an approximate solution for this problem. There are also some analytical methods such as the classical polynomial expansion method , the operational approach , the homotopy perturbation method , the separation of variables technique , and the resolvent method . When we employ these analytical methods to solve the IDE, one common feature is that the IDE is converted to a series of pure differential equations and pure integral equations. In this article we propose a new technique for (1), that is, the Sherman-Morrison-Woodbury (SMW) formula . The SMW formula is a powerful device for calculating the inverse matrix of a square matrix, which will be described briefly in Section 2. In Section 3, we extend the SMW formula to solve the IDE. We find that the SMW formula provides a unified procedure to decompose the linear IDE into several differential equations and one pure integral equation. In Section 4 two simple examples are used to explain this scheme.

2. Sherman-Morrison-Woodbury Formula

The SMW formula was developed for calculating the inverse matrix of a square matrix. Suppose that we have already obtained, by herculean effort, the inverse matrix A-1 of a square matrix A. Then a small change, for example, ΔA=abT, in A happens, and we need to calculate the inverse matrix A+ΔA-1. SMW formula is a powerful device for this problem.

Lemma 1 (Sherman-Morrison-Woodbury formula).

If ΔA=abT, the inverse matrix of A+ΔA is(4)A+ΔA-1=A-1-A-1aI+bTA-1a-1bTA-1.

Proof.

Refer to .

It can be observed from (4) that calculating A+ΔA-1 is converted to calculating I+bTA-1a-1. In the next section we will develop the IDE version of the SMW formula.

3. SMW Formula for the IDE

In this section we extend the SMW formula to the IDE.

Definition 2.

The function ws,c;θ,x is the Green function of the following differential equation (5), if it satisfies(5)Lws,c;θ,x=δx-cδθ-s,xΩ,θΘ,Bws,c;θ,x=0,xΩ,Cws,c;θ,x=0,θΘ,where(6)s=s1s2smTΘ,c=c1c2cnTΩ,δx-c=i=1nδxi-ci,δθ-s=j=1mδθi-si.δx is the Dirac function.

Lemma 3.

For the following differential equation:(7)Luθ,x=fθ,x,xΩ,θΘ,Buθ,x=0,xΩ,Cuθ,x=0,θΘ,its solution is(8)uθ,x=ΘΩws,c;θ,xfs,cdsdc.

Proof.

Substituting (8) into (7), we have (9)Luθ,x=LΘΩws,c;θ,xfs,cdsdc=ΘΩδx-cδθ-sfs,cdsdc=fθ,x,Buθ,x=BΘΩws,c;θ,xfs,cdsdc=ΘΩBws,c;θ,xfs,cdsdc=0,Cuθ,x=CΘΩws,c;θ,xfs,cdsdc=ΘΩCws,c;θ,xfs,cdsdc=0.So the proof of the lemma is completed.

Note that in the proof we assume that the linear differential operators L, B, and C and the integral operator  ΘΩ· are commutative. The assumption holds throughout the paper.

Theorem 4.

The solution of (5) can be written as(10)vθ,x=uθ,x-pθ,x,where uθ,x satisfies(11)Luθ,x=fθ,x,xΩ,θΘ,Buθ,x=0,xΩ,Cuθ,x=0,θΘ,pθ,x satisfies(12)Lpθ,x=kθ,xgθ,x,xΩ,θΘ,Bpθ,x=0,xΩ,Cpθ,x=0,θΘ,and gθ,x satisfies(13)gθ,x+ΘΩqs,c;θ,xks,cgs,cdsdc=ΘθQθ,x,s,Dus,xdsin which(14)qs,c;θ,x=ΘθQθ,x,α,Dws,c;α,xdα.

Proof.

It can be observed easily in terms of (11) and (12) that (10) satisfies the boundary conditions defined by (2). Substituting uθ,x-pθ,x into the left side of (1) and using (11) and (12), we have (15)Lvθ,x+kθ,xΘθQθ,x,s,Dvs,xds=Luθ,x-Lpθ,x+kθ,xΘθQθ,x,s,Dus,xds-kθ,xΘθQθ,x,s,Dps,xds=fθ,x-kθ,xgθ,x+kθ,xΘθQθ,x,s,Dus,xds-kθ,xΘθQθ,x,s,Dps,xds.In terms of Lemma 3, the solution of (12) can be written as(16)ps,x=ΘΩwα,c;s,xkα,cgα,cdαdc.Using (14) and (16) to the last term in the right side of (15) yields(17)ΘθQθ,x,s,Dps,xds=ΘθQθ,x,s,DΘΩwα,c;s,xkα,cgα,cdαdcds=ΘΩΘθQθ,x,s,Dwα,c;s,xdskα,cgα,cdαdc=ΘΩqα,c;θ,xkα,cgα,cdαdc=ΘΩqs,c;θ,xkα,cgs,cdsdc.Substitute (17) into (15) and then use (29) to the result(18)Lvθ,x+kθ,xΘθQθ,x,s,Dvs,xds=Luθ,x-Lpθ,x+kθ,xΘθQθ,x,s,Dus,xds-kθ,xΘθQθ,x,s,Dps,xds=fθ,x-kθ,xgθ,x+ΘΩqs,c;θ,xks,cgs,cdsdc-ΘθQθ,x,s,Dus,xds=fθ,x.So the proof of the theorem is completed.

Note that in the proof we assume that the linear differential operators Q, D  and the integral operator  ΘΩ·  are commutative. The assumption holds throughout the paper.

Equation (10) can be seen as the IDE version of the Sherman-Morrison-Woodbury formula. In order to explain this point more clearly, we introduce some notations.

Definition 5.

The following linear operators are defined by(19)Lvθ,x=Lvθ,x,Svθ,x=ΘθQθ,x,s,Dvs,xds.

Definition 6.

The inverse linear operator of L is defined by(20)L-1fx,θ=ux,θ=ΘΩws,c;θ,xfs,cdsdc,where ws,c;θ,x is the Green function defined by (5).

Using (19) and (20) to (1) yields  (21)Lvθ,x+kθ,x·Svθ,x=fθ,x,xΩ,θΘ.

Lemma 7.

Equation (29) can be rewritten as  (22)gθ,x+SL-1kθ,x·gθ,x=SL-1fθ,x.

Proof.

In terms of (19), we have (23)ΩΘqs,c;θ,xfs,cdsdc=ΩΘΘθQθ,x,α,Dws,c;α,xdαfs,cdsdc=ΘθQθ,x,α,DΩΘws,c;α,xfs,cdsdcdα=ΘθQθ,x,α,DL-1fx,θdα=SL-1fx,θ,Substituting (23) into (5) yields (22).

We denote the solution  gθ,x  of (22) by  (24)gθ,x=1+SL-1kθ,x-1SL-1fθ,x.

Theorem 8 (Sherman-Morrison-Woodbury formula).

The solution of (1) can be written as (25)vθ,x=L-1fθ,x-L-1kθ,x1+SL-1kθ,x-1SL-1fθ,x.

Proof.

In terms of (20), we have (26)uθ,x=L-1fθ,x,pθ,x=L-1kθ,x·gθ,x.Combining (26) with (24) gives (27)vθ,x=uθ,x-pθ,x=L-1fθ,x-L-1kθ,xgθ,x=L-1fθ,x-L-1kθ,x1+SL-1kθ,x-1SL-1fθ,x.So the proof of the theorem is completed.

It can be observed clearly that (25) is the SMW formula for the IDE. Theorems 4 and 8 show that solving the IDE (1) could be converted to solving the integral equation (13) or (29).

According to Lemma 3, the solution of (11) can be written as(28)us,x=ΘΩwα,c;s,xfα,cdαdc.Using (28) to the right-hand side of (13) yields(29)gθ,x+ΘΩqs,c;θ,xks,cgs,cdsdc=ΩΘqs,c;θ,xfs,cdsdc.Multiply both sides of (29) by  kθ,x  and rearrange the equation:  (30)kθ,xgθ,x+ΘΩqs,c;θ,xks,cgs,c-fs,cdsdc=0.Let(31)hθ,x=fθ,x-kθ,xgθ,xthen (30) can be rewritten as(32)hθ,x+ΘΩqs,c;θ,xhs,cdsdc=fs,c.Noting (10)–(12) and (31), we have the following corollary.

Corollary 9.

The solution of (1) satisfies (33)Lvθ,x=hθ,x,xΩ,θΘ,Bvθ,x=0,xΩ,Cvθ,x=0,θΘ,where hθ,x is defined by (32).

In terms of Theorem 4, we can obtain the following corollary.

Corollary 10.

If kθ,x=kx is independent of the variable θ and Qθ,x,s,Dvs,x=Qx,s,Dvs,x is also independent of the variable θ, then the solution is(34)vθ,x=L-1fθ,x-L-1kx·gx,where gx satisfies(35)gx+ΘΩqs,c;xkcgcdsdc=ΩΘqs,c;xfs,cdsdcin which (36)qs,c;x=ΘQx,α,Dws,c;α,xdα.

Corollary 10 shows that if kθ,x and Qx,s,Dvs,x are independent of the variable θ, solving the IDE is converted to solving the integral equation (35) which only involves the variable x. Hence, in this case, the problem is reduced by using Corollary 10.

4. Examples

In this section we present two simple examples. These examples are considered to illustrate the SMW formula for the IDE.

Example 1.

We consider the following linear IDE firstly:(37)xxvθ,x+01e-θxxvθ,xdθ=π2sinπxeθ+1,vθ,0=vθ,1=0.According to Theorem 4, the solution of (37) can be written as (38)vθ,x=uθ,x-pθ,x,where uθ,x satisfies(39)xxuθ,x=π2sinπxeθ+1,uθ,0=uθ,1=0,pθ,x satisfies(40)xxpθ,x=gx,pθ,0=pθ,1=0,and gx satisfies  (41)gx+01qc,xgcdc=01e-θxxuθ,xdθin which(42)qc,x=01e-θxxwc,θ,xdθ=01e-θδx-cdθ=δx-c1-e-1.Solving (39) gives(43)uθ,x=-sinπxeθ+1.Substituting (42) into (41) yields  (44)gx+1-e-1gx=π2sinπx2-e-1.In terms of (44), we have  (45)gx=π2sinπx.Once gx is obtained, (40) can be solved:  (46)pθ,x=-sinπx.So the exact solution is (47)vθ,x=uθ,x-pθ,x=-sinπxeθ-sinπx+sinπx=-sinπxeθ.

Example 2.

Consider the following linear IDE:(48)v˙t+0πcost+svsds=1,v0=0.In terms of Corollary 9, (48) can be converted to the following equations:(49)v˙t=ht,v0=0,where ht satisfies(50)ht+0+qτ;thτdτ=1in which(51)qτ;t=0πcost+swτ;sds=sint+π-sint+τ,τπ,0,τ>π.Substituting (51) into (50) yields(52)ht-sint0πhτdτ-sint0πcosτhτdτ-cost0πsinτhτdτ=1.Solving (52) yields(53)ht=1+sint-8π4+π2+cost8-2π24+π2.Combining (53) with (49) yields  (54)vt=t+8π4+π2cost-1+8-2π24+π2sintwhich is the exact solution.

5. Conclusion

In this work, we extend the SMW formula for the linear IDE. By using the SMW formula, the linear IDE can be decomposed into several linear differential equations and one pure integral equation. The SMW formula could be a powerful tool to solve the linear IDE. Combining the SMW formula with some analytical methods developed for the integral equation may provide the analytical approach for the linear IDE. The SMW formula can also be combined with the numerical methods for solving the linear IDE numerically.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful for the financial support of the Natural Science Foundation of China (nos. 11472076 and 51609034) and China Postdoctoral Science Foundation (no. 2016M590219).

Lekala M. L. Rampho G. J. Sofianos S. A. Adam R. M. An integro-differential equation for Bose-Einstein condensates Few-Body Systems 2011 50 1–4 427 429 10.1007/s00601-010-0137-1 2-s2.0-79955158245 Elliott C. M. McKee S. On the numerical solution of an integro-differential equation arising from wave-power hydraulics BIT Numerical Mathematics 1981 21 3 317 325 10.1007/bf01941466 MR640931 2-s2.0-0019714685 Kostoglou M. Theoretical analysis of the warm-up of monolithic reactors under non-reacting conditions Chemical Engineering Science 1999 54 17 3943 3953 10.1016/s0009-2509(98)00525-9 2-s2.0-0033522778 Guerra G. Shen W. Existence and stability of traveling waves for an integro-differential equation for slow erosion Journal of Differential Equations 2014 256 1 253 282 10.1016/j.jde.2013.09.003 MR3115842 ZBL1320.35138 2-s2.0-84885379762 Bo T.-L. Xie L. Zheng X. J. Numerical approach to wind ripple in desert International Journal of Nonlinear Sciences and Numerical Simulation 2007 8 2 223 228 2-s2.0-34249895025 Chen C. M. Shih T. M. Finite Element Methods for Integrodifferential Equations 1998 Singaore World Scientific 10.1142/9789812798138 MR1638130 Huang C. M. Stability of linear multistep methods for delay integro-differential equations Computers & Mathematics with Applications 2008 55 12 2830 2838 10.1016/j.camwa.2007.09.005 MR2401434 2-s2.0-42749083946 Turkyilmazoglu M. An effective approach for numerical solutions of high-order Fredholm integro-differential equations Applied Mathematics and Computation 2014 227 384 398 10.1016/j.amc.2013.10.079 MR3146325 2-s2.0-84890101056 Borhanifar A. Sadri K. A new operational approach for numerical solution of generalized functional integro-differential equations Journal of Computational and Applied Mathematics 2015 279 80 96 10.1016/j.cam.2014.09.031 MR3293311 ZBL1325.65182 2-s2.0-84913525263 Aminikhah H. A new analytical method for solving systems of linear integro-differential equations Journal of King Saud University—Science 2011 23 4 349 353 10.1016/j.jksus.2010.07.016 2-s2.0-82955224994 Kostoglou M. On the analytical separation of variables solution for a class of partial integro-differential equations Applied Mathematics Letters 2005 18 6 707 712 10.1016/j.aml.2004.05.018 MR2131283 2-s2.0-15844373014 Hernández E. Dos Santos J. P. C. Asymptotically almost periodic and almost periodic solutions for a class of partial integrodifferential equations Electronic Journal of Differential Equations 2006 38 1 8 Press W. H. Teukolsky S. A. Vetterling W. T. Flannery B. P. Numerical Recipes in C: The Art of Scientific Computing 2002 Cambridge, UK Cambridge University Press