Semi-analytical Algorithm Research Articles

Numerical solution of a fractional differential equation arising in optics

The aim of the article is to apply an efficient semi-analytical algorithm for the approximate solution of a fractional differential equation that governs ray tracing through the crystalline lens. A Laplace decomposition method (LDM) is implemented for the solution of the nonlinear ray equation of fractional order that is complemented with initial conditions. This method is a combination of Laplace transform and a decomposition method. The nonlinear term is decomposed and a recursive algorithm is constructed for the determination of the infinite series solution. Numerical tests are conducted, and the results confirm that this strategy converges rapidly with high accuracy using only few iterates of the recursive scheme.

Novel Semi-Analytical Solutions for the Transient Behaviors of Functionally Graded Material Plates in the Thermal Environment.

The primary objective of this article is to present a semi-analytical algorithm for the transient behaviors of Functionally Graded Materials plates (FGM plates) considering both the influence of in-plane displacements and the influence of temperature changes. Based on the classical plate theory considering the effect of in-plane displacements, the equilibrium equations of the motion system are derived by Hamilton’s principle. Here, we propose a novel, accurate, and efficient semi-analytical method that incorporates the Fourier series expansion, the Laplace transforms, and its numerical inversion and the Differential Quadrature Method (DQM) to simulate the transient behaviors. This paper validates the proposed method by comparisons with semi-analytical natural frequency results and those from the literature. Expressly, the results of dynamic response also agree well with those generated by the Navier’s method and Finite Element Method (FEM). A convergence study that utilizes the different numbers of sampling points shows that the process can converge quickly, and a few sampling points can achieve high accuracy. The effects of various boundary conditions at the ends, material graded index, and temperature change are further investigated. From the detailed parametric study, it is seen that the peak displacement increases as the edge degrees of freedom, the gradient index of the material, and temperature change increase.

Accelerating Streamline Tracking in Groundwater Flow Modeling on GPUs.

Streamline simulation in groundwater flow modeling is a time-consuming process when a large number of streamlines are analyzed. We develop a parallelization method on graphics processing units (GPUs) for the semi-analytical particle tracking algorithm developed by Pollock (1988). Compute Unified Device Architecture was used to implement the parallel method. Forward and backward tracking of a streamline is handled by an individual thread. A GPU includes a grid of blocks where a block handles 32 threads. We use multi-GPUs to accelerate streamline tracking in a flow model with millions of particles. The method was examined to simulate streamlines for identifying three-dimensional (3D) flow systems in a Tóthian basin. The speedup exceeds 1000 when 8 NVIDIA GPUs are used to simulate 5 million or more streamlines.

Dynamics and complexity analysis of the conformable fractional‐order two‐machine interconnected power system

In this paper, based on the Adomian decomposition method (ADM) semi analytical solution algorithm, dynamics and complexity of the conformable fractional‐order two‐machine interconnected power system are investigated numerically by the bifurcation diagram, Lyapunov exponents (LEs), chaos diagram, and modified multiscale sample entropy (MM‐SampEn) algorithm separately. The results show that the system has rich dynamics. The angular instability is found and its frequency of occurrence can be judged by the number of scrolls of the chaotic attractor. The coexisting attractors are observed by changing the initial value and the reason is discussed. The high‐complexity region is determined, and MM‐SampEn complexity can indicate different coexisting attractors of the system. The research results in this paper lay a theoretical basis for the application of the conformable fractional‐order two‐machine interconnected power system.

Empirical Formulas for Estimating Backscattering and Absorption Coefficients in Complex Waters from Remote-Sensing Reflectance Spectra and Examples of Their Application.

Many standard methods used for the remote sensing of ocean colour have been developed, though mainly for clean, open ocean waters. This means that they may not always be effective in complex waters potentially containing high concentrations of optically significant constituents. This paper presents new empirical formulas for estimating selected inherent optical properties of water from remote-sensing reflectance spectra Rrs(λ), derived, among other things, for waters with high concentrations of dissolved and suspended substances. These formulas include one for estimating the backscattering coefficient bb(620) directly from the magnitude of Rrs in the red part of the spectrum, and another for estimating the absorption coefficient a(440) from the hue angle α. The latter quantity represents the water’s colour as it might be perceived by the human eye (trichromatic colour vision); it is easily calculated from the shape of the Rrs spectrum. These new formulas are based on a combined dataset. Most of the data were obtained in the specific, optically complex environment of the Baltic Sea. Additional data, taken from the NASA bio-Optical Marine Algorithm Dataset (NOMAD) and representing various regions of the global oceans, were used to widen the potential applicability of the new formulas. We indicate the reasons why these simple empirical relationships can be derived and compare them with the results of straightforward modelling; possible applications are also described. We present, among other things, an example of a simple semi-analytical algorithm using both new empirical formulas. This algorithm is a modified version of the well-known quasi-analytical algorithm (QAA), and it can improve the results obtained in optically complex waters. This algorithm allows one to estimate the full spectra of the backscattering and absorption coefficients, without the need for any additional a priori assumptions regarding the spectral shape of absorption by dissolved and suspended seawater constituents.

Nature of optical products inverted semianalytically from remote sensing reflectance of stratified waters

AbstractAnalytical expressions between inherent optical properties (IOPs) retrieved using semi‐analytical algorithms (SAAs) and the vertical distributions of IOPs are derived based on an analytical model for subsurface remote sensing reflectance (rrs). These expressions provide a theoretical and comprehensive understanding of the inverted IOPs for stratified waters and lay the foundation to obtain equivalent products from profiling measurements for apple‐to‐apple comparison of the remotely sensed properties. It is found that the backscattering coefficient (bb) derived from an rrs spectrum via an SAA is governed by rrs in the longer wavelengths, consequently the inverted bb has less (or negligible) contributions from deeper depths, such as the subsurface chlorophyll maximum. In addition, the inverted bb spectrum does not have the spectral feature of the vertically weighted average bb. The SAA‐derived absorption coefficients, on the other hand, are found related to both the weighting profile at the wavelength of interest and the weighting profile in the longer wavelength(s). As a result, the absorption coefficients in the shorter wavelengths inverted from a perfect SAA are generally lower (can be 40% or more) than its vertically weighted average unless the surface layer dominates the diffuse attenuation of light. Furthermore, unlike the historical perception of remotely sensed chlorophyll concentration (Chl) for stratified waters, SAA‐derived Chl is also sensitive to the vertical profile of bb and tends to be lower than its vertically weighted average assuming perfect bio‐optical relationships, unless surface properties dominate the water column.

Wirelessly-Powered Full-Duplex AF MIMO Relay Systems Based on Antenna Switching

This letter proposes a novel wirelessly-powered full-duplex amplify-and-forward (AF) relaying scheme, where the multi-antenna relay can harvest energy from the source and recycle energy from itself by using antenna switching protocol. Unlike conventional fixed antennas relay that can only harvest and recycle energy at the second transmission phase, in the proposed scheme, the relay can also harvest energy at the first transmission phase with dynamic antenna combination. To maximize the spectral efficiency of the system, the transmit power of the source, the beamforming matrix at the relay and its antenna allocation are jointly optimized. Also, to tackle the discontinuity and nonconvexity of the optimization problem, a semi-analytical iterative algorithm is developed. In particular, for a fixed antenna combination at the relay, a closed-form solution to the source power allocation and the beamforming matrix at the relay are derived. Then, a greedy antenna switching algorithm is designed to obtain the best antenna combination. Simulation results demonstrate the near optimality of the proposed algorithm and the superiority of the proposed scheme.

Biogeochemical Response of Apalachicola Bay and the Shelf Waters to Hurricane Michael Using Ocean Color Semi-Analytic/Inversion and Hydrodynamic Models

Hurricanes are increasingly being recognized as important episodic drivers in ocean biogeochemical cycling; however, spatiotemporal response of their impacts on coastal and estuarine ecosystems are limited. Hurricane Michael, which made landfall just west of Apalachicola Bay (ApB) on October 10, 2018 as a Category 5 hurricane with sustained winds of 250 km h-1, caused widespread damage to the northwest Florida coast, and adverse effects on oyster reefs and water quality in ApB due to winds and coastal flooding associated with a strong storm surge. The impact of wind forcing and retreating storm surges on coastal and shelf biogeochemical properties remains however, largely unknown. In this study, we use a combination of pre-hurricane field observations, ocean-color satellite imagery and the outputs (salinity, currents, sea surface height and temperature) of a nested high-resolution 3-dimensional hydrodynamic model (NCOM) to examine the biogeochemical response of ApB and the surrounding shelf waters to Hurricane Michael. MODIS-derived optical proxies (e.g., absorption of colored dissolved organic matter or CDOM and particle backscattering coefficients) of dissolved and particulate organic carbon (DOC and POC) were derived for a series of clear-sky imagery (prior to and following the hurricane) using a combination of estuarine-tuned semi-analytic and empirical algorithms. Following the hurricane, spatiotemporal distribution of both DOC and POC in ApB and the nearshore coastal waters showed a strong response to storm surge, increasing river discharge, currents, and wind field. Average flux estimates of organic carbon exported from ApB between October 5-21, 2018 to the coastal ocean were much greater for DOC (0.86106 kg C d-1) than POC (0.21106 kg C d-1) and increased with increasing river discharge and the wind field. A bio-optical inversion algorithm applied to Sentinel-3A OLCI imagery of 13 October, 2018 immediately following the hurricane’s passage, showed a strong, week-long biological response with spatially distinct phytoplankton blooms of Karenia brevis and Emiliania Huxleyi, as detected by satellite imagery of pigments, an approach that could revolutionize our understanding of environmental impacts on phytoplankton. This study revealed spatiotemporal changes in estuarine and coastal ocean biogeochemistry reflective of a systematic regional ecosystem response to Hurricane Michael.

Design and modeling of third-harmonic plasmonic metasurfaces coupled to multi-quantum well structures

In this paper, we analyze plasmonic nonlinear metasurfaces based on multi-quantum well (MQW) substrates designed for the third-harmonic generation (THG) nonlinear process. MQW plasmonic metasurfaces have become one of the most noticeable structures in the implementation of nonlinear processes due to the high conversion efficiency for low pump intensities. We propose a semi-analytical algorithm to model TH metasurfaces using the Lorentz reciprocity theorem. Also, in order to predict the electromagnetic behavior of these structures accurately, the THG saturation effects are taken into account. In the next step, two MQW heterostructures and an efficient plasmonic resonator are designed for THG at 30 THz fundamental frequency. In our best design, power conversion efficiency up to 2.1×10−3% is achieved by applying pump intensity below the optical damage threshold (about 200 kW/cm2). The proposed TH metasurfaces can find application used in spectroscopy and as mid-infrared optical sources.

Removal of Chlorophyll-a Spectral Interference for Improved Phycocyanin Estimation from Remote Sensing Reflectance

Monitoring cyanobacteria is an essential step for the development of environmental and public health policies. While traditional monitoring methods rely on collection and analysis of water samples, remote sensing techniques have been used to capture their spatial and temporal dynamics. Remote detection of cyanobacteria is commonly based on the absorption of phycocyanin (PC), a unique pigment of freshwater cyanobacteria, at 620 nm. However, other photosynthetic pigments can contribute to absorption at 620 nm, interfering with the remote estimation of PC. To surpass this issue, we present a remote sensing algorithm in which the contribution of chlorophyll-a (chl-a) absorption at 620 nm is removed. To do this, we determine the PC contribution to the absorption at 665 nm and chl-a contribution to the absorption at 620 nm based on empirical relationships established using chl-a and PC standards. The proposed algorithm was compared with semi-empirical and semi-analytical remote sensing algorithms for proximal and simulated satellite sensor datasets from three central Indiana reservoirs (total of 544 sampling points). The proposed algorithm outperformed semi-empirical algorithms with root mean square error (RMSE) lower than 25 µg/L for the three analyzed reservoirs and showed similar performance to a semi-analytical algorithm. However, the proposed remote sensing algorithm has a simple mathematical structure, it can be applied at ease and make it possible to improve spectral estimation of phycocyanin from space. Additionally, the proposed showed little influence from the package effect of cyanobacteria cells.

Optimal Cyanobacterial Pigment Retrieval from Ocean Colour Sensors in a Highly Turbid, Optically Complex Lake

To date, several algorithms for the retrieval of cyanobacterial phycocyanin (PC) from ocean colour sensors have been presented for inland waters, all of which claim to be robust models. To address this, we conducted a comprehensive comparison to identify the optimal algorithm for retrieval of PC concentrations in the highly optically complex waters of Lake Balaton (Hungary). MEdium Resolution Imaging Spectrometer (MERIS) top-of-atmosphere radiances were first atmospherically corrected using the Self-Contained Atmospheric Parameters Estimation for MERIS data v.B2 (SCAPE-M_B2). Overall, the Simis05 semi-analytical algorithm outperformed more complex inversion algorithms, providing accurate estimates of PC up to ±7 days from the time of satellite overpass during summer cyanobacteria blooms (RMSElog < 0.33). Same-day retrieval of PC also showed good agreement with cyanobacteria biomass (R2 > 0.66, p < 0.001). In-depth analysis of the Simis05 algorithm using in situ measurements of inherent optical properties (IOPs) revealed that the Simis05 model overestimated the phytoplankton absorption coefficient [aph(λ)] by a factor of ~2. However, these errors were compensated for by underestimation of the mass-specific chlorophyll absorption coefficient [a*chla(λ)]. This study reinforces the need for further validation of algorithms over a range of optical water types in the context of the recently launched Ocean Land Colour Instrument (OLCI) onboard Sentinel-3.

A Particle Tracking Algorithm for Arbitrary Unstructured Grids.

A new semi-analytical algorithm for generating the local groundwater velocity field within irregular three-dimensional grid cells (such as a Voronoi cell) is presented. The method supports particle tracking in the MODFLOW-USG software package for a variety of supported unstructured grid formats. It is shown to replicate the results from the fully analytical Pollock's method when used on a rectilinear grid, and is demonstrated for difficult test cases where there is high spatial variability in the flow field within individual cells. The method is now available as an option in the mod-PATH3DU software package and may be used with most MODFLOW-USG packages without additional intervention.

Semi-analytical algorithms of ocean color remote sensing under high solar zenith angles.

With the increasing interest in ocean color remote sensing in polar oceans and geostationary ocean color satellite with diurnal observations, it is unavoidable to encounter ocean color retrievals under high solar zenith angles. Under these scenarios, the capability of current remote sensing algorithms is poorly known. In this study, the performance of the two widely used semi-analytical algorithms for the water inherent optical properties (QAA and GSM01) under high solar zenith angle conditions were firstly evaluated based on global in situ data set (SeaBASS-NOMAD). The results showed that the performances of both QAA and GSM01 degraded significantly with the increasing in solar zenith angle (SZA), and the biases increased about 1.3-fold when SZA varied from 30° to 80°. The high uncertainties at high SZA was mainly induced by the systematic overestimation of the key parameter u (ratio of backscattering coefficient to the sum of absorption and backscattering coefficients) at high solar zenith angles. Based on the Hydrolight-simulated data set, a new model (NN-algorithm) for retrieving u from remote sensing reflectance was developed for high solar zenith angle conditions using the neural network method. The validation results revealed that the NN-algorithm could improve the estimation of parameter u and further ocean color products. In addition, our results indicate that a more accurate atmosphere correction is needed to deal with ocean color remote sensing data acquired under large solar zenith angle conditions.

Monitoring cyanobacteria occurrence in freshwater reservoirs using semi-analytical algorithms and orbital remote sensing

Cyanobacterial blooms pose a serious threat to the multiple uses of inland waters because of their adverse effects on the environment and human health. Monitoring cyanobacteria concentrations using traditional methods can be expensive and impractical. Recently, alternative efforts using remote sensing techniques have been successful. In particular, semi-analytical modelling approaches have been used to successfully predict chlorophyll (Chl)-a concentrations from remote sensing reflectance. The aims of this study were to test the performance of different semi-analytical algorithms in the estimation of Chl-a concentrations and the applicability of Sentinel-2 multispectral instrument (MSI) imagery, and its atmospheric correction algorithms, in the estimation of Chl-a concentrations. For our dataset, phycocyanin concentration was strongly correlated with Chl-a concentration and the inversion model of inland waters (IIMIW) semi-analytical algorithm was the best performing model, achieving a root mean square error of 4.6mgm–3 in the prediction of Chl-a. When applying the IIMIW model to MSI data, the use of top-of-atmosphere reflectance performed better than the atmospheric correction algorithm tested. Overall, the results were satisfactory, demonstrating that even without an adequate atmospheric correction pipeline, the monitoring of cyanobacteria can be successfully achieved by applying a semi-analytical bio-optical model to MSI data.

Effects of forward models on the semi-analytical retrieval of inherent optical properties from remote sensing reflectance.

Inherent optical properties (IOPs) play a key role in modulating an aquatic light field; they are the core link for remotely sensing water constituents based on ocean color remote sensing. Many semi-analytical algorithms (SAAs) have been developed to obtain IOPs from remote sensing reflectance (Rrs) data; these algorithms require a forward model (FM) to link the IOPs to Rrs. Most currently available SAAs use the FM presented by Gordon etal.[J. Geophys. Res.93, 10909 (1988)JGREA20148-022710.1029/JD093iD09p10909] (G88 hereafter) without knowledge of how other models would impact the retrieval of IOPs from Rrs. This study evaluates the effects of two popular SAAs, namely, the quasi-analytical algorithm (QAA) and the generalized IOP algorithm (GIOP), combined with six different FMs on the retrieval of IOPs from a synthetic data set generated with Hydrolight software. The results indicated that different FMs can have quite different effects on the computed Rrs(λ), and the effects were not uniform across the Rrs spectrum. Of the six FMs tested, G88 and P05 [Appl. Opt.44, 1236 (2005)APOPAI0003-693510.1364/AO.44.001236] produced the best estimates of Rrs(λ) at 350, 440, and 550nm in both oceanic and coastal sub-datasets; they also were less impacted by changes in the particle phase function. M02 also produced a good estimation of Rrs but only at 440nm, and L04 performed well only in the oceanic condition. When the two SAAs were combined with the six FMs, in the oceanic condition, QAA and GIOP combined with M02 (QM02 and GM02) provided better quality for the absorption coefficient [a(λ)] at 350, 440, and 550nm when compared with the SAAs combined with the other models. However, for the retrieval of the particle backscattering coefficient [bbp(λ)] in the oceanic condition, QAA and GIOP combined with L04 (QL04 and GL04) performed better than the others, and GL04 always provided a better estimation of bbp(λ) than QL04. In the coastal condition, QAA and GIOP combined with G88 or P05 produced slightly better quality of IOPs compared with the other four FMs. Compared with GIOP in the coastal condition, QAA combined with G88 or P05 always showed better quality of retrieval of a(λ) but weaker quality of retrieval of bbp(λ).

Sensitivity of inherent optical properties from ocean reflectance inversion models to satellite instrument wavelength suites.

The Earth science community seeks to develop climate data records (CDRs) from satellite measurements of ocean color, a continuous data record which now exceeds twenty years. Space agencies will launch additional instruments in the coming decade that will continue this data record, including the NASA PACE spectrometer. Inherent optical properties (IOPs) quantitatively describe the absorbing and scattering constituents of seawater and can be estimated from satellite-observed spectroradiometric data using semi-analytical algorithms (SAAs). SAAs exploit the contrasting optical signatures of constituent matter at spectral bands observed by satellite sensors. SAA performance, therefore, depends on the spectral resolution of the satellite spectroradiometer. A CDR spanning SeaWiFS, MODIS, OLCI, and PACE, for example, would include IOPs derived using varied wavelength suites if all available wavelengths were considered. Here, we explored differences in derived IOPs that stem simply from the use of (eight) different wavelength suites of input radiometric measurements. Using synthesized data and SeaWiFS Level-3 mission-long composites, we demonstrated equivalent SAA performance for all wavelength suites, but that IOP retrievals vary by several percent across wavelength suites and as a function of water type. The differences equate to roughly ≤ 6, 12, and 7% for adg (443), aph (443), and bbp (443), respectively, for waters with Ca ≤ 1 mg m-3. These values shrink for sensors with similar wavelength suites (e.g., SeaWiFS, MODIS, and MERIS) and rise to substantially larger values for higher Ca waters. Our results also indicate that including 400 nm (in the case of OLCI) influences the derived IOPs, using longer wavelengths (>600 nm) influences the derived IOPs when there is a red signal, and, including additional spectral information shows potential for improved IOP estimation, but not without revisiting SAA parameterizations and execution. While modest in scope, we believe this study contributes to the knowledge base for CDR development. The implication of ignoring such an analysis as CDRs continue to be developed is a prolonged inability to distinguish between algorithmic and environmental contributions to trends and anomalies in the IOP time-series.

Cochlear mechanics with fluid viscosity and compressibility.

We extend the one-dimensional cochlear model to include the effect of viscosity and compressibility of the cochlear fluid. The resulting boundary-value problem is solved exactly using numerical techniques and semianalytically using the WKB approximation. Our results indicate the general trend of basilar membrane response increasing with an increase in compressibility or a decrease in viscosity. However, in the physiologically relevant range of these parameters, the change in the response is insignificant, justifying the assumption made in one-dimensional cochlear models that these effects are unimportant. Using the semianalytical WKB algorithm, we also demonstrate the simultaneous existence of forward- and backward-traveling waves on the basilar membrane.

Deriving inherent optical properties from decomposition of hyperspectral non-water absorption

Semi-analytical algorithms (SAAs) developed for multispectral ocean color sensors have benefited from a variety of approaches for retrieving the magnitude and spectral shape of inherent optical properties (IOPs). SAAs generally follow two approaches: 1) simultaneous retrieval of all IOPs, resulting in pre-defined bio-optical models and spectral dependence between IOPs and 2) retrieval of bulk IOPs (absorption and backscattering) first followed by decomposition into separate components, allowing for independent retrievals of some components. Current algorithms used to decompose hyperspectral remotely-sensed reflectance into IOPs follow the first strategy. Here, a spectral deconvolution algorithm for incorporation into the second strategy is presented that decomposes at-w(λ) from in situ measurements and estimates absorption due to phytoplankton (aph(λ)) and colored detrital material (adg(λ)) free of explicit assumptions. The algorithm described here, Derivative Analysis and Iterative Spectral Evaluation of Absorption (DAISEA), provides estimates of aph(λ) and adg(λ) over a spectral range from 350 to 700 nm. Estimated aph(λ) and adg(λ) showed an average normalized root mean square difference of <30% and <20%, respectively, from 350 to 650 nm for the majority of optically distinct environments considered. Estimated Sdg median difference was <20% for all environments considered, while distribution of Sdg uncertainty suggests that biogeochemical variability represented by Sdg can be estimated free of bias. DAISEA results suggest that hyperspectral satellite ocean color data will improve our ability to track biogeochemical processes affiliated with variability in adg(λ) and Sdg free of explicit assumptions.

An Overall Evaluation of Water Transparency in Lake Malawi from MERIS Data

Lake Malawi is an important water resource in Africa. However, there is no routine monitoring of water quality in the lake due to financial and institutional constraints in the surrounding countries. A combination of satellite data and a semi-analytical algorithm can provide an alternative for routine monitoring of water quality, especially in developing countries. In this study, we first compared the performance of two semi-analytical algorithms, Doron11 and Lee15, which can estimate Secchi disk depth (SD) from satellite data in Lake Malawi. Our results showed that even though the SD estimations from the two algorithms were very highly correlated, the Lee15 outperformed the Doron11 in Lake Malawi with high estimation accuracy (RMSE = 1.17 m, MAPE = 18.7%, R = 0.66, p &lt; 0.05). We then evaluated water transparency in Lake Malawi using the SD values estimated from nine years of Medium Resolution Imaging Spectrometer (MERIS) data (2003–2011) with the Lee15 algorithm. Results showed that Lake Malawi maintained four water transparency levels throughout the study period (i.e., level 1: SD &gt; 12 m; level 2: SD between 6–12 m; level 3: SD between 3–6 m; level 4: SD between 1.5–3 m). The level 1 and 2 water areas tended to shift or trade places depending on year or season. In contrast, level 3 and 4 water areas were relatively stable and constantly distributed along the southwestern and southern lakeshores. In general, Lake Malawi is dominated by waters with SD values larger than 6 m (&gt;95%). This study represents the first overall and comprehensive analysis of water transparency status and spatiotemporal variation in Lake Malawi.

New semi-analytical method for solving two point nth order fuzzy boundary value problem

In this paper, a semi analytical algorithm, namely homotopy analysis method (HAM) is presented for the first time to obtain approximate analytical solutions of nth order two point fuzzy boundary value problems (TPFBVP) involving ordinary differential equations. This method allows for the solution of the TPFBVP to be calculated in the form of an infinite series with the components that can be easily calculated. The HAM utilises a convergence control parameter to control the convergence region of the infinite series solution. Numerical experiment is tested to highlight the important features of the HAM algorithm. Comparison of the HAM results and available exact solution reveal that this algorithm is effective and simple and for solving TPFBVP.