",isbn:"978-1-83881-119-8",printIsbn:"978-1-83881-118-1",pdfIsbn:"978-1-83881-120-4",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"8bd4f03c89e63ef15984ee1b7f1485c4",bookSignature:"Prof. Andrew James Manning",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10407.jpg",keywords:"Hydrodynamics, Suspension/Saltation/Bedload, Numerical Modeling / CFD, Deposition, Flocculation, Sediment Types, Regional/Temporal Variability, Turbidity Currents, Dust Storms, Socio-Economic Effects, Contaminants, Storm / Severe Weather Effects",numberOfDownloads:205,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"May 27th 2020",dateEndSecondStepPublish:"September 11th 2020",dateEndThirdStepPublish:"November 10th 2020",dateEndFourthStepPublish:"January 29th 2021",dateEndFifthStepPublish:"March 30th 2021",remainingDaysToSecondStep:"6 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Dr. Manning is a highly published and world-renowned scientist in the field of depositional sedimentary flocculation processes.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"23008",title:"Prof.",name:"Andrew James",middleName:null,surname:"Manning",slug:"andrew-james-manning",fullName:"Andrew James Manning",profilePictureURL:"https://mts.intechopen.com/storage/users/23008/images/system/23008.jpeg",biography:"Professor Andrew J. Manning is a Principal Scientist (Rank Grade 9) in the Coasts & Oceans Group at HR Wallingford (UK) and has over 23 years of scientific research experience (in both industry and academia) examining natural turbulent flow dynamics, fine-grained sediment transport processes, and assessing how these interact, (including both field studies and controlled laboratory flume simulations). Andrew also lectures in Coastal & Shelf Physical Oceanography at the University of Plymouth (UK). Internationally, Andrew has been appointed Visiting / Guest / Adjunct Professor at five Universities (Hull, UK; Delaware, USA; Florida, USA; Stanford, USA; TU Delft, Netherlands), and is a highly published and world-renowned scientist in the field of depositional sedimentary flocculation processes. 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\n
1. Introduction
\n
Breastmilk is the physiologic norm for infant nutrition, offering multiple health benefits and protection for babies and mothers [1, 2, 3, 4]. WHO recommends that breastfeeding be initiated within 1 hour of birth, that it continue with no other foods or liquids for the first 6 months of life, and that it be continued with complementary feeding (breastfeeding with other age-appropriate foods) until at least 24 months of age. However, global breastfeeding rates remain far below international targets. In most high-income countries, the prevalence of breastfeeding at 12 months is lower than 20%, and it is highest in sub-Saharan Africa, south Asia, and parts of Latin America [5]. In Mexico, rates of breastfeeding are particularly low, 38.3% of Mexican women initiate breastfeeding soon after giving birth, only 14.4% of these women report exclusively breastfeeding at 6 months postpartum, 35.5% report any breastfeeding at 12 months postpartum, and 14.1% report any breastfeeding at 2 years postpartum. Importantly, breastfeeding rates vary by demographic area and are highest in rural over urban area [6].
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Breastfeeding has protective effects on maternal health, including a reduced risk of breast cancer and ovarian cancer, obesity, hypertension, stroke, hyperlipidemia, metabolic syndrome, and type 2 diabetes mellitus (T2DM) [7, 8, 9, 10, 11].
\n
One of the strongest risk factors for T2DM is a history of gestational diabetes mellitus (GDM). Among women with a history of GDM, the cumulative risk of developing T2DM at 10 years postpartum ranges from 20 to 50% [12]. Infant feeding method is a modifiable risk factor for the development of diabetes; breastfeeding confers short- and long-term benefits on metabolism reducing the risk of developing T2DM.
\n
Despite the important benefits of breastfeeding, there is evidence to suggest that lower rates of breastfeeding occur in women with GDM and that the duration of breastfeeding is shorter compared with that of healthy mothers [13, 14]. The explanations to account for these rates are higher rates of cesarean sections and neonatal intensive care unit admissions, which include increased recovery time for the mother, prolonged separation of mother and baby, and decreased or delayed bonding [15]. Other factors influencing the lower rates of breastfeeding in GDM are insulin therapy during pregnancy and obesity; women with insulin-treated diabetes have less intention to breastfeed and women with high BMI may have different hormonal patterns, delaying onset of milk production [16, 17].
\n
Insulin treatment is related to severity of GDM, and this marked gestational disturbance in insulin and glucose metabolism may interfere with the hormonal pathways for initiation of lactogenesis. Results from a study of gene expression profiles at different stages of lactation suggested that decreased insulin sensitivity may delay milk production as a result of protein tyrosine phosphatase, receptor type F overexpression in the mammary gland [18].
\n
On the other hand, the GDM treatment with oral hypoglycemic agents has not been related to milk production by the mothers [16]. Glyburide and metformin are the two oral hypoglycemic agents most commonly used during pregnancy, and both are safe with breastfeeding [19].
\n
\n
\n
2. Short-term benefits of breastfeeding
\n
Lactation confers favorable metabolic changes, including lower fasting and postprandial blood glucose, as well as insulin, and triglycerides, and greater insulin sensitivity, and plasma HDL-C [20]. Glucose is diverted for milk production via noninsulin-mediated pathways of uptake by the mammary gland and, thus, lactating women exhibit lower blood glucose [21].
\n
On the other hand, lactation promotes postpartum weight loss [22]; lactogenesis increases maternal total energy expenditure by 15–25% [23]. Prospective studies have reported more rapid weight loss within 6 months postpartum, and lower weight retention at 1 year postpartum [24].
\n
Studies in women with recent GDM report more favorable glucose tolerance and lipid metabolism during 4 months postpartum for lactating compared with non-lactating women [25]. At 6–9 weeks postpartum, the SWIFT cohort in a racially and ethnically diverse group found a dose-response relationship between increasing intensity of lactation and decreasing fasting plasma glucose and both fasting and 2-h insulin, as well as improved insulin sensitivity [26].
\n
In a retrospective cohort among Latinas with recent GDM, Kjos et al. reported 5 mg/dL lower fasting blood glucose for any intensity of lactation versus no lactation, and an improved glucose tolerance determined by the glucose area under the curve from the oral glucose tolerance test (OGTT) [27].
\n
In a recent study of women with previous GDM diagnosed according to the new “International Association of Diabetes and Pregnancy Study Groups” (IADPSG) criteria, that included women with a milder metabolic impairment, breastfeeding for almost 3 months improved the metabolic outcomes, such as fasting and 2 h glycemia at OGTT, an index of insulin resistance (HOMA-IR) and triglycerides [28].
\n
It has been demonstrated that the favorable effects of lactation on glucose metabolism persist after weaning. Chouinard-Castonguay et al., in a cross-sectional study, showed that lactation duration was an independent predictor of fasting insulin concentrations and insulin sensitivity indices up to a mean of 4 years after delivery [29].
\n
\n
\n
3. Long-term benefits of breastfeeding
\n
It has been suggested that a longer duration of breastfeeding is associated with a lower risk of T2DM. However, results in some studies have been inconsistent. Chouinard-Castonguay found that women who reported lactating for >10 months had impaired glucose tolerance less frequently compared with women who lactated for <10 months at 4 years postpartum [29].
\n
In a longitudinal analysis, Buchanan found no difference in the prevalence of diabetes at 11–26 months postpartum [30]. Similarly, Kjos, in a retrospective study, reported that breastfeeding was not associated with the progression to T2DM within a follow-up of 7.5 years after delivery [27], and in a retrospective cohort study, the Nurses’ Health Study, breastfeeding did not affect the risk of diabetes at 14 years [31].
\n
However, of interest, one prospective study that assessed the development of T2DM in women with GDM for up to 20 years after delivery found that breastfeeding reduced the risk of diabetes by 46%; median time to postpartum diabetes was 12.3 years for women who breastfed versus 2.3 years for women who did not breastfeed, independently of maternal BMI and insulin use during pregnancy [32]. Moreover, women who breastfed for >3 months had lower risk of diabetes than women who breastfed for ≤3 months (P = 0.029).
\n
Also, the positive metabolic impact of breastfeeding has been reported in women with mild forms of GDM. A recent study showed that women with glucose intolerance in early postpartum breastfed less often than women with a normal OGTT (69.5 vs. 84.2%, p = 0.041) [33].
\n
The discrepancy among the different studies could be a result of the differences in the design of the study, the severity of GDM, diagnosis of T2DM, breastfeeding assessment, follow-up time postpartum sample size, lifestyle behaviors, ethnic characteristics, and, finally, the use of oral contraceptives. Birth control with progestin-only oral contraceptive pills has been associated with an increased risk of T2DM during the first 2 years of use. Kjos reported a threefold increase in the risk of T2DM at 7.5 years postpartum in breastfeeding women with recent GDM. By contrast, use of low dose progestin/estrogen combination oral contraceptive pills during breastfeeding does not increase the risk of T2DM [27].
\n
\n
\n
4. Mechanisms in the protective effects of breastfeeding
\n
The potential mechanisms involved in the protective effects of breastfeeding on glucose metabolism include breastfeeding-related hormones such as prolactin and estrogen [34]. Prolactin levels are elevated and estradiol is lower in breastfeeding women than in non-lactating women. In vitro experiments of rat pancreatic islets cultured with prolactin have shown enhanced stimulated insulin secretion through stimulation of b-cell proliferation by downregulating the expression of menin [35, 36]. Also, prolactin modulates the transcription factors STAT5 and PPARg, and the expression of lipoprotein lipase, which are co-expressed in breast, adipose tissue, and skeletal muscle [37].
\n
On the other hand, it has been suggested that lactation improves insulin sensitivity by mobilizing lipids derived from liver and muscle for lactogenesis rather than by redirecting lipids into adipocytes [34].
\n
Another mechanism is the influence of lactation on regional fat tissue metabolism; some studies have reported enhanced fat mass mobilization from the trunk and thighs for lactating women [38, 39]. In keeping with this, another study reported that lactation history is associated with a smaller visceral fat area in women who reported they lactated for at least 3 months [40].
\n
There is also evidence that leptin, which is an adipokine positively associated with body adiposity and insulin resistance, is modified in breastfeeding women with previous GDM [41]. Leptin directly affects whole-body insulin sensitivity by regulating the efficiency of insulin-mediated glucose metabolism by skeletal muscle and by hepatic regulation of gluconeogenesis through its action on gene expression of phosphoenolpyruvate carboxykinase [42, 43]. Moreover, it exerts an acute inhibitory effect on insulin secretion, and upregulates inflammatory mediators like TNFα and interleukin-6, which contribute to excessive insulin resistance both at the level of the whole body and in specific organs, including in the liver, muscle, and brain [44].
\n
Leptin is predominantly produced by adipocytes, but is also produced by non-adipose tissues such as stomach, intestine, ovaries, and in particular, the placenta [45]. Maternal leptin levels increase two- to threefold in pregnancy, with a peak occurring around 28 weeks of gestation and decreasing to pre-pregnancy concentrations after delivery [46]. It has been suggested that the rise in maternal leptin concentration during pregnancy may result from an upregulation of adipocyte leptin synthesis in the presence of increasing insulin resistance and hyperinsulinemia in the second half of pregnancy [47]. However, there is strong evidence that the placenta, rather than maternal adipose tissue, contributes to the increase of maternal leptin concentrations during pregnancy [48]. Leptin induces human chorionic gonadotropin production, regulates placental growth, angiogenesis, trophoblast invasion, and nutrient transfer [49]. Leptin enhances the mobilization of maternal fat stores to increase availability and to support transplacental transfer of lipid substrates [50]. Moreover, leptin upregulates placental System A amino acid transport, to increase fetal nutrient availability [51]. Leptin serves as a mitogen for a growing number of cell types, including endothelial cells, hemopoietic cells, lung epithelial cells, and pancreatic b-cells in vitro [52]. Leptin could therefore be stimulating growth of tissues in the developing fetus.
\n
Leptin contributes to the pathophysiologic relationship between GDM and subsequent T2DM. In our previous study, we found that women with previous GDM persisted with insulin resistance in the postpartum period, in association with higher leptin levels compared with the control group [53]. It is possible that postpartum insulin resistance may be contributing to these elevated levels. A positive association between leptinemia and insulinemia has been reported in numerous studies of obese and non-obese humans [54, 55]. Experimentally, increased insulin levels may stimulate leptin production in adipocytes, and vice versa, an increase in leptin levels may lead to insulin resistance and alter b-cell secretory capacity [56, 57]. Interestingly, we recently reported that breastfeeding was associated with better metabolic profile in the early postpartum period in women with previous GDM, showing that women with longer duration of lactation had greater weight loss at postpartum and lower leptin levels compared with women who lactated for a short period. This difference remained statistically significant after adjustment for weight [58]. Similarly, a previous study with a large cohort of women with recent GDM that utilized a quantitative measure of lactation intensity found that mean leptin concentrations were inversely associated with lactation intensity (lower by 15–21%) independent of maternal pre-pregnancy obesity, race, weight loss, sociodemographics, and postpartum insulin resistance [41]. It has been suggested that during lactation, prolactin suppresses leptin secretion [59].
\n
On the other hand, leptin is a long-term regulator of appetite that serves as an anorexigenic signal when adipose stores are high [60]. It has been detected in breast milk at concentrations of 0.35–4.6 μg/L [61], and the concentration of leptin in breast milk is correlated to indices of maternal adiposity, including body mass index (r = 0.65, P < 0.02), and fat mass (r = 0.65, P < 0.02) [62]. This evidence provides an attractive explanation for the ability of breast milk to regulate infant body weight. Bouret has recently suggested that the presence of appetite hormones may permanently affect the appetite-regulating system of the infant by affecting the development of the hypothalamus. These differences in appetite hormone exposure may create permanent changes in the way the brain reacts to appetite hormones and satiety cues [63].
\n
Another adipokine related to abnormal glucose metabolism during pregnancy is adiponectin. It has been suggested that low levels of adiponectin may induce severe insulin resistance prior to the onset of GDM [64]. Gunderson evaluated the relationship between lactation intensity and plasma adiponectin among postpartum women with previous GDM and found that higher lactation intensity was associated with 6% lower adiponectin. This inverse association remained after adjustment for insulin resistance [41]. This observation is consistent with the action of prolactin in suppressing the production and secretion of adiponectin from human adipocytes [13].
\n
\n
\n
5. Implications of breastfeeding in the offspring of the GDM mother
\n
Infants of mothers with GDM are at increased risk of prematurity, macrosomia, hypoglycemia, respiratory distress, hypocalcemia, polycythemia, and hyperbilirubinemia. Breastfeeding has many established benefits for child health; it prevents child morbidity due to diarrhea, respiratory infections, and otitis media [5]. In particular, in the offspring of the GDM mother, lactation has been associated with lower episodes of hypoglycemia [65].
\n
On the other hand, exposure to maternal gestational diabetes has been shown to increase the risk of obesity, childhood-onset type 2 diabetes in offspring, as well as the risk of adult-onset type 2 diabetes and gestational diabetes in those offspring [66]. Breastfeeding confers protection against these medical complications; exclusive breastfeeding decreases the risk of the development of childhood-onset type 2 diabetes and obesity [67].
\n
Benefits of breastfeeding on children’s health are likely due to the unique composition of breast milk. Human milk is a source of immunoglobulins, hormones and growth factors including leptin, adiponectin, ghrelin, peptide YY, glucagon-like peptide-1, resistin, and obestatin, which are involved in food intake regulation and energy balance, and may have a role in the regulation of growth and development in the neonatal period and infancy, as well as long-term effects on metabolic programming [68].
\n
\n
\n
6. Suggestions for mothers with GDM regarding breastfeeding
\n
Women whose pregnancy is affected by GDM should be educated early as to the benefits of breastfeeding their offspring. An increase in breastfeeding duration among women with GDM has been demonstrated with prenatal education. Breastfeeding support in the hospital immediately after delivery and during the postpartum period as well as community support that encourages breastfeeding are also essential. Electronic alerts via text message or email, automated letters, and nurse phone contact may increase uptake. This targeted breastfeeding support for women with GDM is feasible and efficacious, and could be integrated into GDM management [69].
\n
Likewise, insulin treatment during pregnancy should be considered a targeting indicator for providing extra skilled breastfeeding support to GDM women who decide to breastfeed [16].
\n
\n
\n
7. Conclusions
\n
Breastfeeding is recommended and encouraged for mothers, as it has multiple benefits for both women and children. Mothers who breastfeed have been shown to have reduced risk of developing subsequent breast cancer and ovarian cancer, obesity, hypertension, stroke, hyperlipidemia, metabolic syndrome, and T2DM. In women with GDM, several studies suggest that breastfeeding is associated with reduced risk of T2DM. Despite this important benefit, there is evidence to suggest that lower rates of breastfeeding occur in women with GDM. Evidence has shown that healthcare provider support of breastfeeding along with patient education has a significant impact on breastfeeding rates. The medical and behavioral communities should be better able to design, implement, and administer public health programs that may promote healthy lifestyle behaviors including breastfeeding among GDM women, and mitigating T2DM risk.
\n
\n
Acknowledgments
\n
RS holds a fellowship from the National System of Investigators. We thank the Hospital of Gynecology and Obstetrics, Medical Center La Raza, Instituto Mexicano del Seguro Social, for providing patient care services.
\n
This work was supported by scientific grants from IMSS.
\n
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this chapter.
\n
Thanks
\n
This chapter is dedicated to the memory of Dr. Arturo Zárate (1936–2018), pioneer in the field of Gynecological Endocrinology in Mexico.
\n
\n',keywords:"gestational diabetes, breastfeeding, leptin, insulin resistance, type 2 diabetes mellitus",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/64264.pdf",chapterXML:"https://mts.intechopen.com/source/xml/64264.xml",downloadPdfUrl:"/chapter/pdf-download/64264",previewPdfUrl:"/chapter/pdf-preview/64264",totalDownloads:478,totalViews:0,totalCrossrefCites:2,totalDimensionsCites:1,hasAltmetrics:0,dateSubmitted:"July 10th 2018",dateReviewed:"October 13th 2018",datePrePublished:"November 12th 2018",datePublished:"February 5th 2020",dateFinished:null,readingETA:"0",abstract:"Breastfeeding is recommended as the preferred method of feeding for infants for at least 1 year, because of its multiple immediate and long-term benefits for both the mother and child. Among women with a history of gestational diabetes mellitus (GDM), breastfeeding is associated with increased insulin sensitivity, improved insulin secretion, improved glucose tolerance, and a reduced incidence of type 2 diabetes mellitus (T2DM). Lactation has also been associated with postpartum weight loss, reduced long-term obesity risk, a lower prevalence of the metabolic syndrome, hypertension, and cardiovascular disease. The mechanisms underlying the benefits of breastfeeding for the mother are unclear. However, a role of adipose tissue-produced cytokines (adipokines) has been suggested. Lactation appears to mobilize adipose tissue accrued during pregnancy, and some changes in adipokine levels have been reported. Higher lactation intensity has been associated with lower plasma leptin, a peptide mainly associated with appetite regulation and insulin resistance.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/64264",risUrl:"/chapter/ris/64264",book:{slug:"gestational-diabetes-mellitus-an-overview-with-some-recent-advances"},signatures:"Renata Saucedo, Jorge Valencia, María Isabel Peña-Cano, Enrique Morales-Avila and Arturo Zárate",authors:[{id:"58977",title:"Prof.",name:"Arturo",middleName:null,surname:"Zarate",fullName:"Arturo Zarate",slug:"arturo-zarate",email:"zaratre@att.net.mx",position:null,institution:null},{id:"89876",title:"Dr.",name:"Enrique",middleName:null,surname:"Morales-Avila",fullName:"Enrique Morales-Avila",slug:"enrique-morales-avila",email:"enrimorafm@yahoo.com.mx",position:null,institution:{name:"Universidad Privada del Estado de México",institutionURL:null,country:{name:"Mexico"}}},{id:"267142",title:"Ph.D.",name:"Renata",middleName:null,surname:"Saucedo",fullName:"Renata Saucedo",slug:"renata-saucedo",email:"sgrenata@yahoo.com",position:null,institution:null},{id:"278768",title:"Mr.",name:"Jorge",middleName:null,surname:"Valencia",fullName:"Jorge Valencia",slug:"jorge-valencia",email:"j.valencia.o@hotmail.com",position:null,institution:null},{id:"278771",title:"MSc.",name:"Maria Isabel",middleName:null,surname:"Peña-Cano",fullName:"Maria Isabel Peña-Cano",slug:"maria-isabel-pena-cano",email:"isabelpenacano@hotmail.com",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Short-term benefits of breastfeeding",level:"1"},{id:"sec_3",title:"3. Long-term benefits of breastfeeding",level:"1"},{id:"sec_4",title:"4. Mechanisms in the protective effects of breastfeeding",level:"1"},{id:"sec_5",title:"5. Implications of breastfeeding in the offspring of the GDM mother",level:"1"},{id:"sec_6",title:"6. Suggestions for mothers with GDM regarding breastfeeding",level:"1"},{id:"sec_7",title:"7. Conclusions",level:"1"},{id:"sec_8",title:"Acknowledgments",level:"1"},{id:"sec_11",title:"Conflict of interest",level:"1"},{id:"sec_8",title:"Thanks",level:"1"}],chapterReferences:[{id:"B1",body:'Bartick MC, Stuebe AM, Schwarz EB, Luongo C, Reinhold AG, Foster EM. Cost analysis of maternal disease associated with suboptimal breastfeeding. Obstetrics and Gynecology. 2013;122:111-119. DOI: 10.1097/AOG.0b013e318297a047\n'},{id:"B2",body:'Gunderson EP, Lewis CE, Wei GS, Whitmer RA, Quesenberry CP, Sidney S. 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Endocrine Research Unit, National Medical Center, Instituto Mexicano del Seguro Social, Mexico
Endocrine Research Unit, National Medical Center, Instituto Mexicano del Seguro Social, Mexico
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1. Introduction
Partial differential equations play a dominant role in applied mathematics. The classical heat conduction equation is second order linear partial differential equation. The solutions of which are obtained by using various analytical and numerical methods [1, 2, 3]. This equation describes the heat distribution in each domain over some time. Jean-Joseph Fourier was the first to formulate and describe the heat conduction process [1, 4]. Perturbation methods depending upon small/large parameters have been encountered from past few years. Perturbation methods are analytical approximation method to understand physical phenomena which depends on perturbation quantity. But these methods do not provide an easy way to find out the rapid convergence of approximate series. Therefore, this method is simple, suitable and appropriate method to provide the rapid convergence of series [5, 6, 7]. The perturbation method along with the homotopy method has been employed to develop a hybrid method known as homotopy perturbation method (HPM) [1, 2, 3, 4]. Ji-Huan was the first to introduce HPM. Homotopy perturbation method provides analytical approximation to linear/nonlinear problems without linearization or discretization. It helps in formulating simpler equations by breaking down the complex problems, which can be solved easily. Since HPM does not depend on small parameters, therefore drawbacks of the existing perturbation methods can be abolished [8, 9, 10, 11]. The solution obtained by HPM converges to exact solution, which are in the form of an infinite function series. Various problems are modeled by linear and non-linear partial differential equations problems in the fields of physics, engineering etc. To solve such kind of partial differential equations (PDE), many methods are used to find the numerical or exact solutions. Homotopy perturbation method (HPM) is one of the methods used in recent years to solve various linear and non-linear PDE [12, 13, 14, 15]. Initial and boundary value problems can be solved using HPM extensively. Many researchers and scientists show great interest in homotopy perturbation method. Huan was the first who described homotopy perturbation method. He showed that this method is a one of the powerful tools used to investigate various problems which are arising nowadays. HPM is used for solving linear and non-linear ordinary and partial differential equations [16].
In HPM, complex linear or non-linear problem can be continuously distorted into simpler ones. Perturbation theory and homotopy theory in topology is combined to develop homotopy perturbation method [1]. HPM is applicable to linear and non-linear boundary value problems. The solution obtained by HPM gives the solution approximately near to the universally accepted method of separation of variable [17, 18, 19].
Recently, Biazar and Eslami proposed the new homotopy perturbation method (NHPM). Construction of an appropriate homotopy equation and selection of appropriate initial approximation guess are two important steps of NHPM [19, 20]. The study reveals that with less computational work, we can construct proper homotopy by decomposition of source function in a correct way. New homotopy perturbation method is the most powerful tool which can be used to obtain analytical solution of various kinds of linear and nonlinear PDE’s. This method is widely used by researchers to obtain solution of various functional Equations [20, 21, 22].
To develop this new technique, HPM is combined with the decomposition of source function. The decomposition of a source function is the basis of homotopy used in this method because convergence of a solution is affected by the decomposition of source functions [23]. Different kind of homotopy can be formed using various decomposition of a source functions. This study is aimed at constructing suitable homotopy by decomposition of a source function which requires less computational efforts and made calculations in simpler form unlike other perturbation methods. The obtained results directly imply the fact that NHPM is very influential as compared to HPM or any other perturbation technique. To establish exact solution of linear and non-linear problem with boundary and initial condition, new homotopy method is most appropriate method to apply [23].
The two most important steps in application of new homotopy perturbation method to construct a suitable homotopy equation and choose a suitable initial guess, we aim in this work to effectively employ the (NHPM) to establish exact solution for two-dimensional Laplace equation with Dirichlet and Neumann boundary condition, the difference between (NHPM) and standard (HPM) is starts from the form of initial approximation of the solution.
In this chapter, the semi analytic solution of one-dimensional heat conduction equation is obtained by means of homotopy perturbation method and new homotopy perturbation method. These methods are effectively applied to obtain the exact solution for the problem in hand which reveals the effectiveness and simplicity of the method. Numerical results have also been analyzed graphically to show the rapid convergence of infinite series expansion. The obtained analytic solution for one dimensional heat conduction equation with boundary and initial conditions using NHPM is same as the universally accepted exact solution. This tells us about the capability and reliability of this method. The solution obtained using NHPM is considered in the form of an infinite series. The convergence of solution to the exact solution is very rapid.
2. Heat conduction equation
The one-dimensional heat equation
∂U∂θ=β∂2U∂z2E1
with boundary conditions
U0θ=0,U1θ=0,E2
and initial condition
Uz0=hz,0≤z≤1.E3
3. Basic idea of Homotopy perturbation method
First, we outline the general procedure of the homotopy perturbation method developed and advanced by He. We consider the differential Eq. [2]
Au−fr=0,r∈ΩE4
Bu∂u∂x=0,r∈ΓE5
where A is a general differential operator, linear or nonlinear, fr is a known analytic function, B is a boundary operator and Γ is the boundary of the domain Ω. The operator A can be generally divided into two operators, L and N, where Lis linear and N is a nonlinear operator. Eq. (4) can be written as
Lu+Nu−fr=0E6
Using the homotopy technique, we can construct a homotopy [1,2]
vrp:Ω×01→Rwhich satisfies the relation
Hvp=1−pLv−Lu0+pAv−fr=0,r∈ΩE7
Here p∈01 is called the homotopy parameter and u0 is an initial approximation for the solution of Eq. (4), which satisfies the boundary conditions. Clearly, from Eq. (7), we have
Hv0=Lv−Lu0E8
Hv1=Av−frE9
We assume that the solution of Eq. (7) can be expressed as a series in p as follows:
v=v0+pv0+p2v2+p3v3+⋯E10
On setting p=1, we obtain the approximate solution of Eq. (10) as
u=limp→1v=v0+v0+v2+v3+⋯E11
4. Basic idea of new Homotopy perturbation method
First, following homotopy is constructed for solving heat conduction equation using NHPM
1−p∂T∂θ−U0+p∂T∂θ−β∂2T∂z2=0E12
Taking L−1=∫θ0θ.dθ i.e. inverse operator on Eq. (12), then
Suppose solution given by Eq. (14) is the solution of Eq. (13). On comparing the coefficients of powers of p and equating to zero and using Eq. (14) in Eq. (13), following are obtained:
So Uzθ=T0zθ=∑n=0∞cnzPnθ is obtained solution which is found to be exactly same as the exact solution obtained through method of separation of variable.
If U0zθ is analytic at θ=θ0,
U0zθ=∑n=0∞cnzθ−θ0n is the taylor series expansion which can be used in Eq. (9).
5. Applications of Homotopy perturbation method and new Homotopy perturbation method
For understanding the application of HPM and NHPM, we will solve the one-dimensional heat equation given by
∂U∂θ=β∂2U∂z2E18
with boundary conditions
U0θ=0,U1θ=0,E19
and initial condition
Uz0=sin2πzL,0≤z≤L.E20
The homotopy for the diffusion equation given by (18) is obtained as follows [2].
∂v∂θ−∂u0∂θ+ƥ∂u0∂θ−β∂2v∂z2=0E21
Let u0=sin2πzLcosπ2θ be the initial approximation, which satisfies boundary conditions given by (19).
Let solution of (18) has the following form
v=v0+ƥv1+ƥ2v2+ƥ3v3+ƥ4v4+…E22
On substituting the value of v in Eq. (21) and comparing the coefficients of like powers of ƥ we obtain
ƥ0:∂v0∂θ=∂u0∂θ
ƥ1:∂v1∂θ=β∂2v0∂z2,v10θ=0=v1Lθ
ƥ2:∂v2∂θ=β∂2v1∂z2,v20θ=0=v2Lθ
ƥ3:∂v3∂θ=β∂2v2∂z2,v30θ=0=v3Lθ
ƥn:∂vn∂θ=β∂2vn−1∂z2,vn0θ=0=vnLθE23
On solving the system of Eq. (23) using Mathematica 5.2
The approximate solution of (1) by setting ƥ=1 in (23) is given by
u=limp→1v=v0+v1+v2+v3+v3+…E25
On substituting values of vi′s in Eq. (25), solution is obtained in terms of a summation of infinite series which gives results near to the exact solution.
Now we will solve the Eq. (18) using NHPM. First of all, following homotopy is constructed for solving heat conduction equation using NHPM
1−p∂T∂θ−U0+p∂T∂θ−β∂2T∂z2=0E26
Taking L−1=∫θ0θ.dθ i.e. inverse operator on Eq. (26), then
Tzθ=∫0θU0zθdθ−p∫0θU0−β∂2T∂z2dθ+Tz0.E27
Let the solution of the (27) is
T=T0+pT1+p2T2+p3T3+…,E28
where, T0,T1,T2,… are to be determined.
Suppose Eq. (25) is the solution of Eq. (24). Comparing the coefficients of powers of p and equating to zero and using Eq. (25) in Eq. (24), following are obtained:
which is same as the universally accepted exact solution for the problem which is shown in Figure 1.
Figure 1.
Solution using NHPM.
The solution of one-dimensional heat conduction equation is solved using HPM and NHPM and then compared with the universally accepted exact solution obtained from method of separation of variable. Figure 2 represents the comparison of solution of heat equation using HPM, NHPM and method of separation of variable. It is found that the solution obtained using HPM gives result near to the exact solution whereas solution using NHPM gives same results as the exact solution.
Figure 2.
Comparison of HPM, NHPM and the exact solution.
6. Conclusion
The analytical approximate solutions of one-dimensional heat conduction equation are obtained by applying new homotopy perturbation method and new homotopy perturbation method. It is found that new homotopy perturbation method (NHPM) converges very rapidly as compared to homotopy perturbation method (HPM) and other traditional methods. The exact solutions are obtained up to more accuracy using NHPM. An infinite convergent series solution for particular initial conditions are obtained using these methods which shows the effectiveness and efficiency of NHPM and HPM. The convergence rate of NHPM is much faster than traditional methods which directly indicates that this method is better than other methods. The solution of heat equation obtained by homotopy perturbation method and new homotopy perturbation method are exactly same and very close to the solution obtained by universally accepted and tested analytical method of separation of variables. If the initial guess in homotopy perturbation method is effective and properly chosen which satisfy boundary and initial condition, homotopy perturbation method provides solution with rapid convergence. It is illustrated that NHPM is very prominent, when accuracy has a vital role to play. The numerical results also reflect the remarkable applicability of NHPM to linear and non-linear initial and boundary value problems. NHPM provides the rapid convergence of the series solution for linear as well as non-linear problems with less computational work.
\n',keywords:"heat conduction equation, homotopy perturbation method, new homotopy perturbation method, specific heat, diffusivity",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/74774.pdf",chapterXML:"https://mts.intechopen.com/source/xml/74774.xml",downloadPdfUrl:"/chapter/pdf-download/74774",previewPdfUrl:"/chapter/pdf-preview/74774",totalDownloads:115,totalViews:0,totalCrossrefCites:0,dateSubmitted:"August 7th 2020",dateReviewed:"December 21st 2020",datePrePublished:"January 12th 2021",datePublished:null,dateFinished:"January 12th 2021",readingETA:"0",abstract:"Many physical and engineering problems can be modeled using partial differential equations such as heat transfer through conduction process in steady and unsteady state. Perturbation methods are analytical approximation method to understand physical phenomena which depends on perturbation quantity. Homotopy perturbation method (HPM) was proposed by Ji Huan He. HPM is considered as effective method in solving partial differential equations. The solution obtained by HPM converges to exact solution, which are in the form of an infinite function series. Biazar and Eslami proposed new homotopy perturbation method (NHPM) in which construction of an appropriate homotopy equation and selection of appropriate initial approximation guess are two important steps. In present work, heat flow analysis has been done on a rod of length L and diffusivity α using HPM and NHPM. The solution obtained using different perturbation methods are compared with the solution obtained from most common analytical method separation of variables.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/74774",risUrl:"/chapter/ris/74774",signatures:"Neelam Gupta and Neel Kanth",book:{id:"10413",title:"Chaotic Systems",subtitle:null,fullTitle:"Chaotic Systems",slug:null,publishedDate:null,bookSignature:"Prof. Paul Bracken and Dr. Dimo I. Uzunov",coverURL:"https://cdn.intechopen.com/books/images_new/10413.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:[{id:"92883",title:"Prof.",name:"Paul",middleName:null,surname:"Bracken",slug:"paul-bracken",fullName:"Paul Bracken"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Heat conduction equation",level:"1"},{id:"sec_3",title:"3. Basic idea of Homotopy perturbation method",level:"1"},{id:"sec_4",title:"4. Basic idea of new Homotopy perturbation method",level:"1"},{id:"sec_5",title:"5. Applications of Homotopy perturbation method and new Homotopy perturbation method",level:"1"},{id:"sec_6",title:"6. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'N. Gupta and N. Kanth (2019). Study of heat flow in a rod using homotopy analysis method and homotopy perturbation method. 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Jeffrey, Comparison of Homotopy Analysis Method and Homotopy Perturbation Method through an Evolution Equation, Commun. Nonlinear Sci. Numer. Simul., vol. 14, pp. 4057–4064, Dec. 2009'},{id:"B8",body:'A. Demir, S. Erman, B. Ozgur, E. Korkmaz, Analysis of the new homotopy perturbation method for linear and nonlinear problems. Boundary Value Problems, 2013(1), 1–11, 2013'},{id:"B9",body:'J-H. He, Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation, 135(1) 73–79, 2003'},{id:"B10",body:'J. Biazar, M. Eslami, A new homotopy perturbation method for solving systems of partial differential equations. Computers and Mathematics with Applications, 62(1), 225–234, 2011'},{id:"B11",body:'J-H. He, Homotopy perturbation method for solving boundary value problems. Physics Letters A. 350(1), 87–88, 2006'},{id:"B12",body:'M. Mirzazadeh, Z. Ayati, New homotopy perturbation method for system of Burgers equations. Alexandria Engineering Journal. 55(3), 1619–1624, 2016'},{id:"B13",body:'M. Elbadri, A New Homotopy Perturbation Method for Solving Laplace Equation. Advances in Theoratical and Applied Mathematics. 8(3), 237–242, 2013'},{id:"B14",body:'M. Elbadri, T.M. Elzaki, New Modification of Homotopy Perturbation Method and the Fourth - Order Parabolic Equations with Variable Coefficients. Pure Appl. Math. J. 4(6) 242–247, 2015'},{id:"B15",body:'J.H. He, A coupling method of homotopy technique and a perturbation technique for non linear problems, Int. J. Non linear Mech., 35, pp. 37–43, 2000'},{id:"B16",body:'J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals, 26, pp. 295–700,2005'},{id:"B17",body:'A. Yıldırım, Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method, Comput. Math. Appl., 56 (12), 3175–3180, 2008'},{id:"B18",body:'S. 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