Search results for: allometric equation

Authors: Abdullahi Jibrin, Aishetu Abdulkadir

Abstract:

The development of allometric models is crucial to accurate forest biomass/carbon stock assessment. The aim of this study was to develop a set of biomass prediction models that will enable the determination of total tree aboveground biomass for savannah woodland area in Niger State, Nigeria. Based on the data collected through biometric measurements of 1816 trees and destructive sampling of 36 trees, five species specific and one site specific models were developed. The sample size was distributed equally between the five most dominant species in the study site (Vitellaria paradoxa, Irvingia gabonensis, Parkia biglobosa, Anogeissus leiocarpus, Pterocarpus erinaceous). Firstly, the equations were developed for five individual species. Secondly these five species were mixed and were used to develop an allometric equation of mixed species. Overall, there was a strong positive relationship between total tree biomass and the stem diameter. The coefficient of determination (R2 values) ranging from 0.93 to 0.99 P < 0.001 were realised for the models; with considerable low standard error of the estimates (SEE) which confirms that the total tree above ground biomass has a significant relationship with the dbh. The F-test value for the biomass prediction models were also significant at p < 0.001 which indicates that the biomass prediction models are valid. This study recommends that for improved biomass estimates in the study site, the site specific biomass models should preferably be used instead of using generic models.

Keywords: allometriy, biomass, carbon stock , model, regression equation, woodland, inventory

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Authors: Aleksandr S. Ermolenko

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Allometry is an important factor of morphological integration, contributing to the organization of the phenotype and its variability. The allometric change in the shape of the hand is particularly important in primate evolution, as the hand has important taxonomic features. Some of these features are known to parts with the shape, especially the ratio of the lengths of the index and ring fingers (2d: 4d ratio). The hand is a fairly well-studied system in the context of the evolutionary development of complex morphological structures since it consists of various departments (basipodium, metapodium, acropodium) that form a single structure –autopodium. In the present study, we examined the allometric variability of acropodium. We tested the null hypothesis that there would be no difference in allometric variation between the two components. Geometric morphometry based on a procrustation of 16 two-dimensional (2D) landmarks was analyzed using multivariate shape-by-size regressions in samples from 100 people (50 men and 50 women). The results obtained show that men have significantly greater allometric variability for the ring finger (variability in the transverse axis prevails), while women have significantly greater allometric variability for the index finger (variability in the longitudinal axis prevails). The influence of the middle finger on the shape of the hand is typical for both men and women. The influence of the little finger on the shape of the hand, regardless of gender, was not revealed. The results of this study support the hypothesis that allometry contributes to the organization of variation in the human hand.

Keywords: human hand, size and shape, 2d:4d ratio, geometric morphometry

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Authors: S. O. Oyamakin, A. U. Chukwu

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Richard's growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richard's growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richard's growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov-Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richard's nonlinear growth models better than the classical Richard's growth model.

Keywords: height, Dbh, forest, Pinus caribaea, hyperbolic, Richard's, stochastic

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Authors: Sultan Haji Shube

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Woodland ecosystems of semiarid rift valley of Ethiopia play a significant role in climate change mitigation by sequestering and storing more carbon. This study was conducted in Gidabo river sub-basins southeastern rift-valley escarpment of Ethiopian. It aims to estimate biomass and carbon stocks of woodlands and its implications for climate change mitigation. A total of 44 sampling plots (900m²each) were systematically laid in the woodland for vegetation and environmental data collection. A composite soil sample was taken from five locations main plot. Both disturbed and undisturbed soil samples were taken at two depths using soil auger and core-ring sampler, respectively. Allometric equation was used to estimate aboveground biomass while root-to-shoot ratio method and Walkley-Black method were used for belowground biomass and SOC, respectively. Result revealed that the totals of the study site was 17.05t/ha, of which 14.21t/ha was belonging for AGB and 2.84t/ha was for BGB. Moreover, 2224.7t/ha total carbon stocks was accumulated with an equivalent carbon dioxide of 8164.65t/ha. This study also revealed that more carbon was accumulated in the soil than the biomass. Both aboveground and belowground carbon stocks were decreased with increase in altitude while SOC stocks were increased. The AGC and BGC stocks were higher in the lower slope classes. SOC stocks were higher in the higher slope classes than in the lower slopes. Higher carbon stock was obtained from woody plants that had a DBH measure of >16cm and situated at plots facing northwest. Overall, study results will add up information about carbon stock potential of the woodland that will serve as a base line scenario for further research, policy makers and land managers.

Keywords: allometric equation, climate change mitigation, soil organic carbon, woodland

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Authors: Samuel Oluwafemi Oyamakin, Angela Unna Chukwu

Abstract:

Richrads growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richards growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richards growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov-Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richards nonlinear growth models better than the classical Richards growth model.

Keywords: height, diameter at breast height, DBH, hyperbolic sine function, Pinus caribaea, Richards' growth model

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Authors: Titinan Pothong, Prasit Wangpakapattanawong, Stephen Elliott

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Shifting cultivation is an indigenous agricultural practice among upland people and has long been one of the major land-use systems in Southeast Asia. As a result, fallows and secondary forests have come to cover a large part of the region. However, they are increasingly being replaced by monocultures, such as corn cultivation. This is believed to be a main driver of deforestation and forest degradation, and one of the reasons behind the recurring winter smog crisis in Thailand and around Southeast Asia. Accurate biomass estimation of trees is important to quantify valuable carbon stocks and changes to these stocks in case of land use change. However, presently, Thailand lacks proper tools and optimal equations to quantify its carbon stocks, especially for secondary evergreen forests, including fallow areas after shifting cultivation and smaller trees with a diameter at breast height (DBH) of less than 5 cm. Developing new allometric equations to estimate biomass is urgently needed to accurately estimate and manage carbon storage in tropical secondary forests. This study established new equations using a destructive method at three study sites: approximately 50-year-old secondary forest, 4-year-old fallow, and 7-year-old fallow. Tree biomass was collected by harvesting 136 individual trees (including coppiced trees) from 23 species, with a DBH ranging from 1 to 31 cm. Oven-dried samples were sent for carbon analysis. Wood density was calculated from disk samples and samples collected with an increment borer from 79 species, including 35 species currently missing from the Global Wood Densities database. Several models were developed, showing that aboveground biomass (AGB) was strongly related to DBH, height (H), and wood density (WD). Including WD in the model was found to improve the accuracy of the AGB estimation. This study provides insights for reforestation management, and can be used to prepare baseline data for Thailand’s carbon stocks for the REDD+ and other carbon trading schemes. These may provide monetary incentives to stop illegal logging and deforestation for monoculture.

Keywords: aboveground biomass, allometric equation, carbon stock, secondary forest

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Authors: Riki Dutta, Sagardeep Talukdar, Gautam Kumar Saharia, Sudipta Nandy

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Fokas-Lenells equation (FLE) is one of the integrable nonlinear equations use to describe the propagation of ultrashort optical pulses in an optical medium. A 2x2 Lax pair has been introduced for the FLE and from that solving the Riccati equation yields infinitely many conserved quantities. Thereafter for a new field function (S) of the Landau-Lifshitz (LL) system, a gauge equivalence of the FLE with the generalised LL equation has been derived. We hope our findings are useful for the application purpose of FLE in optics and other branches of physics.

Keywords: conserved quantities, fokas-lenells equation, landau-lifshitz equation, lax pair

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Authors: Jian-Jun Shu

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It is common knowledge that many physical problems (such as non-linear shallow-water waves and wave motion in plasmas) can be described by the Korteweg-de Vries (KdV) equation, which possesses certain special solutions, known as solitary waves or solitons. As a marriage of the KdV equation and the classical Burgers (KdVB) equation, the Korteweg-de Vries-Burgers (KdVB) equation is a mathematical model of waves on shallow water surfaces in the presence of viscous dissipation. Asymptotic analysis is a method of describing limiting behavior and is a key tool for exploring the differential equations which arise in the mathematical modeling of real-world phenomena. By using variable transformations, the asymptotic expansion of the KdVB equation is presented in this paper. The asymptotic expansion may provide a good gauge on the validation of the corresponding numerical scheme.

Keywords: asymptotic expansion, differential equation, Korteweg-de Vries-Burgers (KdVB) equation, soliton

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Authors: Y. Pala, M. O. Ertas

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In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

Keywords: Riccati equation, analytical solution, proper solution, nonlinear

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Authors: Ahmed Bolaji Alarape, Oluwatobi Damilola Aba

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The growth pattern of Oreochromis niloticus and Sarotherodon galilaeus in Epe Lagoon Lagos State was investigated. One hundred (100) samples of each species were collected from fishermen at the landing site. They were transported to the Fisheries Laboratory of National Institute of Oceanography for identification, sexing morphometric measurement. The results showed that 58.0% and 56.0 % of the O.niloticus and S.galilaeus were female respectively while 42.0% and 44.0% were male respectively. The length-weight relationship of O.niloticus showed a strong regression coefficient (r = 0.944) (p<0.05) for the combined sex, (r =0.901) (p<0.05) for female and (r=0.985) (p<.05) for male with b-value of 2.5, 3.1 and 2.8 respectively. The S.galilaeus also showed a regression coefficient of r=0.970; p<0.05 for the combined sex, r=0.953; p<0.05 for the female and r= 0.979; p<0.05 for the male with b-value of 3.4, 3.1 and 3.6 respectively. O.niloticus showed an isometric growth pattern both in male and female. The condition factor in O.niloticus are 1.93 and 1.95 for male and female respectively while that of S.galilaeus is 1.95 for both sexes. Positive allometric was observed in both species except the male O.niloticus that showed negative allometric growth pattern. From the results of this study, the growth pattern of the two species indicated a good healthy environment.

Keywords: Epe Lagoon, length-weight relationship, Oreochromis niloticus, Sarotherodon galilaeus

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Authors: Sharon-Yasotha Veerayah-Mcgregor, Valipuram Manoranjan

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A backward or inverse problem is known to be an ill-posed problem due to its instability that easily emerges with any slight change within the conditions of the problem. Therefore, only a limited number of numerical approaches are available to solve a backward problem. This paper considers the Nagumo equation, an equation that describes impulse propagation in nerve axons, which also models population growth with the Allee effect. A creative operator splitting numerical scheme is constructed to solve the inverse Nagumo equation. Computational simulations are used to verify that this scheme is stable, accurate, and efficient.

Keywords: inverse/backward equation, operator-splitting, Nagumo equation, ill-posed, finite-difference

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Authors: Humphrey Agevi, Harrison Tsingalia, Richard Onwonga, Shem Kuyah

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Most of the world ecosystems have been affected by the effects of climate change. Efforts have been made to mitigate against climate change effects. While most studies have been done in forest ecosystems and pure plant plantations, trees on farms including agroforestry have only received attention recently. Agroforestry systems and tree cover on agricultural lands make an important contribution to climate change mitigation but are not systematically accounted for in the global carbon budgets. This study sought to: (i) determine tree diversity in different agroforestry practices; (ii) determine tree biomass in different agroforestry practices. Study area was determined according to the Land degradation surveillance framework (LSDF). Two study sites were established. At each of the site, a 5km x 10km block was established on a map using Google maps and satellite images. Way points were then uploaded in a GPS helped locate the blocks on the ground. In each of the blocks, Nine (8) sentinel clusters measuring 1km x 1km were randomized. Randomization was done in a common spreadsheet program and later be downloaded to a Global Positioning System (GPS) so that during surveys the researchers were able to navigate to the sampling points. In each of the sentinel cluster, two farm boundaries were randomly identified for convenience and to avoid bias. This led to 16 farms in Kakamega South and 16 farms in Kakamega North totalling to 32 farms in Kakamega Site. Species diversity was determined using Shannon wiener index. Tree biomass was determined using allometric equation. Two agroforestry practices were found; homegarden and hedgerow. Species diversity ranged from 0.25-2.7 with a mean of 1.8 ± 0.10. Species diversity in homegarden ranged from 1-2.7 with a mean of 1.98± 0.14. Hedgerow species diversity ranged from 0.25-2.52 with a mean of 1.74± 0.11. Total Aboveground Biomass (AGB) determined was 13.96±0.37 Mgha-1. Homegarden with the highest abundance of trees had higher above ground biomass (AGB) compared to hedgerow agroforestry. This study is timely as carbon budgets in the agroforestry can be incorporated in the global carbon budgets and improve the accuracy of national reporting of greenhouse gases.

Keywords: agroforestry, allometric equations, biomass, climate change

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Authors: Saeed Otarod

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In a different simple and straight forward analysis a closed-form integral solution is found for nonhomogeneous second order linear ordinary differential equations, in terms of a particular solution of their corresponding homogeneous part. To find the particular solution of the homogeneous part, the equation is transformed into a simple Riccati equation from which the general solution of non-homogeneouecond order differential equation, in the form of a closed integral equation is inferred. The method works well in manyimportant cases, such as Schrödinger equation for hydrogen-like atoms. A non-homogenous second order linear differential equation has been solved as an extra example

Keywords: explicit, linear, differential, closed form

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Authors: Yuan Yan Tang, Lina Yang, Hong Li

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Harmonic model is a very important approximation for the image transform. The harmanic model converts an image into arbitrary shape; however, this mode cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the image transform. In this paper, a novel Integral Equation-Wavelet based method is presented, which consists of three steps: (1) The partial differential equation is converted into boundary integral equation and representation by an indirect method. (2) The boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. (3) The plane integral equation and representation are then solved by a method we call wavelet collocation. Our approach has two main advantages, the shape of an image is arbitrary and the program code is independent of the boundary. The performance of our method is evaluated by numerical experiments.

Keywords: harmonic model, partial differential equation (PDE), integral equation, integral representation, boundary measure formula, wavelet collocation

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Authors: Tadas Telksnys, Zenonas Navickas, Minvydas Ragulskis

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Necessary and sufficient conditions for the existence of second order solitary solutions to the Hodgkin-Huxley equation are derived in this paper. The generalized multiplicative operator of differentiation helps not only to construct closed-form solitary solutions but also automatically generates conditions of their existence in the space of the equation's parameters and initial conditions. It is demonstrated that bright, kink-type solitons and solitary solutions with singularities can exist in the Hodgkin-Huxley equation.

Keywords: Hodgkin-Huxley equation, solitary solution, existence condition, operator method

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Authors: Nara Guimarães, Marcelo Aquino Martorano, Douglas Gouvêa

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An investigation into Cahn-Hilliard equation was carried out through numerical simulation to identify a possible phase separation for one and two dimensional domains. It was observed that this equation can reproduce important mass fluxes necessary for phase separation within the miscibility gap and for coalescence of particles.

Keywords: Cahn-Hilliard equation, miscibility gap, phase separation, dimensional domains

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Authors: Hesamoddin Abdollahpour, Roghayeh Abdollahpour, Elham Rahgozar

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This paper we deal with an analysis of the free vibrations of the governing partial differential equation that it is Danel equation in the shells. The problem considered represents the governing equation of the nonlinear, large amplitude free vibrations of the hinged shell. A new implementation of the new method is presented to obtain natural frequency and corresponding displacement on the shell. Our purpose is to enhance the ability to solve the mentioned complicated partial differential equation (PDE) with a simple and innovative approach. The results reveal that this new method to solve Danel equation is very effective and simple, and can be applied to other nonlinear partial differential equations. It is necessary to mention that there are some valuable advantages in this way of solving nonlinear differential equations and also most of the sets of partial differential equations can be answered in this manner which in the other methods they have not had acceptable solutions up to now. We can solve equation(s), and consequently, there is no need to utilize similarity solutions which make the solution procedure a time-consuming task.

Keywords: large amplitude, free vibrations, analytical solution, Danell Equation, diagram of phase plane

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Authors: S. Mousavian, F. Mousavian, V. Nikkhah Rashidabad

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Cubic equations of state like Redlich–Kwong (RK) EOS have been proved to be very reliable tools in the prediction of phase behavior. Despite their good performance in compositional calculations, they usually suffer from weaknesses in the predictions of saturated liquid density. In this research, RK equation was modified. The result of this study shows that modified equation has good agreement with experimental data.

Keywords: equation of state, modification, ammonia, genetic algorithm

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Authors: Muna Alghabshi, Edmana Krishnan

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A nonlinear Schrodinger equation has been considered for solving by mapping methods in terms of Jacobi elliptic functions (JEFs). The equation under consideration has a linear evolution term, linear and nonlinear dispersion terms, the Kerr law nonlinearity term and three terms representing the contribution of meta materials. This equation which has applications in optical fibers is found to have soliton solutions, shock wave solutions, and singular wave solutions when the modulus of the JEFs approach 1 which is the infinite period limit. The equation with special values of the parameters has also been solved using the tanh method.

Keywords: Jacobi elliptic function, mapping methods, nonlinear Schrodinger Equation, tanh method

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Authors: Benedict Barnes, Anthony Y. Aidoo

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A Divergence Regularization Method (DRM) is used to regularize the ill-posed Helmholtz equation where the boundary deflection is inhomogeneous in a Hilbert space H. The DRM incorporates a positive integer scaler which homogenizes the inhomogeneous boundary deflection in Cauchy problem of the Helmholtz equation. This ensures the existence, as well as, uniqueness of solution for the equation. The DRM restores all the three conditions of well-posedness in the sense of Hadamard.

Keywords: divergence regularization method, Helmholtz equation, ill-posed inhomogeneous Cauchy boundary conditions

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Authors: F. U. Rahman, R. Q. Zhang

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This work aims to develop a systematic numerical technique which can be easily extended to many-body problem. The Lippmann Schwinger equation (integral form of the Schrodinger wave equation) is solved for the nonrelativistic radial wave of hydrogen atom using iterative integration scheme. As the unknown wave function appears on both sides of the Lippmann Schwinger equation, therefore an approximate wave function is used in order to solve the equation. The Green’s function is obtained by the method of Laplace transform for the radial wave equation with excluded potential term. Using the Lippmann Schwinger equation, the product of approximate wave function, the Green’s function and the potential term is integrated iteratively. Finally, the wave function is normalized and plotted against the standard radial wave for comparison. The outcome wave function converges to the standard wave function with the increasing number of iteration. Results are verified for the first fifteen states of hydrogen atom. The method is efficient and consistent and can be applied to complex systems in future.

Keywords: Green’s function, hydrogen atom, Lippmann Schwinger equation, radial wave

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Authors: Ayaz Ahmad

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In this work, we present the effect of noise on the solution of a partial differential equation (PDE) in three different setting. We shall first consider random initial condition for two nonlinear dispersive PDE the non linear Schrodinger equation and the Kortteweg –de vries equation and analyse their effect on some special solution , the soliton solutions.The second case considered a linear partial differential equation , the wave equation with random initial conditions allow to substantially decrease the computational and data storage costs of an algorithm to solve the inverse problem based on the boundary measurements of the solution of this equation. Finally, the third example considered is that of the linear transport equation with a singular drift term, when we shall show that the addition of a multiplicative noise term forbids the blow up of solutions under a very weak hypothesis for which we have finite time blow up of a solution in the deterministic case. Here we consider the problem of wave propagation, which is modelled by a nonlinear dispersive equation with noisy initial condition .As observed noise can also be introduced directly in the equations.

Keywords: drift term, finite time blow up, inverse problem, soliton solution

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Authors: Praveen Kumar, R. Uma, R. P. Sharma

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This study investigates the generation of turbulence in a deep-fluid environment using a damped 1D-modified nonlinear Schrödinger equation model. The well-known damped modified nonlinear Schrödinger equation (d-MNLS) is solved using numerical methods. Artificial damping is added to the MNLS equation, and turbulence generation is investigated through a numerical simulation. The numerical simulation employs a finite difference method for temporal evolution and a pseudo-spectral approach to characterize spatial patterns. The results reveal a recurring periodic pattern in both space and time when the nonlinear Schrödinger equation is considered. Additionally, the study shows that the modified nonlinear Schrödinger equation disrupts the localization of structure and the recurrence of the Fermi-Pasta-Ulam (FPU) phenomenon. The energy spectrum exhibits a power-law behavior, closely following Kolmogorov's spectra steeper than k⁻⁵/³ in the inertial sub-range.

Keywords: water waves, modulation instability, hydrodynamics, nonlinear Schrödinger's equation

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Authors: Meruyert Zhassybayeva, Kuralay Yesmukhanova, Ratbay Myrzakulov

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Integrable nonlinear differential equations are an important class of nonlinear wave equations that admit exact soliton solutions. All these equations have an amazing property which is that their soliton waves collide elastically. One of such equations is the (1+1)-dimensional Fokas-Lenells equation. In this paper, we have constructed an integrable (2+1)-dimensional Fokas-Lenells equation. The integrability of this equation is ensured by the existence of a Lax representation for it. We obtained its bilinear form from the Hirota method. Using the Hirota method, exact one-soliton and two-soliton solutions of the (2 +1)-dimensional Fokas-Lenells equation were found.

Keywords: Fokas-Lenells equation, integrability, soliton, the Hirota bilinear method

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Authors: Ognjen Vukovic

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Chern-Simons equation represents the cornerstone of quantum physics. The question that is often asked is if the aforementioned equation can be successfully applied to the interaction in international financial markets. By analysing the time series in financial theory, it is proved that Chern-Simons equation can be successfully applied to financial time-series. The aforementioned statement is based on one important premise and that is that the financial time series follow the fractional Brownian motion. All variants of Chern-Simons equation and theory are applied and analysed. Financial theory time series movement is, firstly, topologically analysed. The main idea is that exchange rate represents two-dimensional projections of three-dimensional Brownian motion movement. Main principles of knot theory and topology are applied to financial time series and setting is created so the Chern-Simons equation can be applied. As Chern-Simons equation is based on small particles, it is multiplied by the magnifying factor to mimic the real world movement. Afterwards, the following equation is optimised using Solver. The equation is applied to n financial time series in order to see if it can capture the interaction between financial time series and consequently explain it. The aforementioned equation represents a novel approach to financial time series analysis and hopefully it will direct further research.

Keywords: Brownian motion, Chern-Simons theory, financial time series, econophysics

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Authors: Paschal A. Ochang, Emmanuel C. Oji

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The Duffing’s Equation is a second order system that is very important because they are fundamental to the behaviour of higher order systems and they have applications in almost all fields of science and engineering. In the biological area, it is useful in plant stem dependence and natural frequency and model of the Brain Crash Analysis (BCA). In Engineering, it is useful in the study of Damping indoor construction and Traffic lights and to the meteorologist it is used in the prediction of weather conditions. However, most Problems in real life that occur are non-linear in nature and may not have analytical solutions except approximations or simulations, so trying to find an exact explicit solution may in general be complicated and sometimes impossible. Therefore we aim to find out if it is possible to obtain one analytical fixed point to the non-linear ordinary equation using fixed point analytical method. We started by exposing the scope of the Duffing’s equation and other related works on it. With a major focus on the fixed point and fixed point iterative scheme, we tried different iterative schemes on the Duffing’s Equation. We were able to identify that one can only see the fixed points to a Damped Duffing’s Equation and not to the Undamped Duffing’s Equation. This is because the cubic nonlinearity term is the determining factor to the Duffing’s Equation. We finally came to the results where we identified the stability of an equation that is damped, forced and second order in nature. Generally, in this research, we approximate the solution of Duffing’s Equation by converting it to a system of First and Second Order Ordinary Differential Equation and using Fixed Point Iterative approach. This approach shows that for different versions of Duffing’s Equations (damped), we find fixed points, therefore the order of computations and running time of applied software in all fields using the Duffing’s equation will be reduced.

Keywords: damping, Duffing's equation, fixed point analysis, second order differential, stability analysis

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Authors: Ayda Nikkar, Roghayye Ahmadiasl

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In this letter, a new analytical method called homotopy perturbation method, which does not need small parameter in the equation is implemented for solving the nonlinear Whitham–Broer–Kaup (WBK) partial differential equation. In this method, a homotopy is introduced to be constructed for the equation. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of exact solution has led us to significant consequences. The results reveal that the HPM is very effective, convenient and quite accurate to systems of nonlinear equations. It is predicted that the HPM can be found widely applicable in engineering.

Keywords: homotopy perturbation method, Whitham–Broer–Kaup (WBK) equation, Modified Boussinesq, Approximate Long Wave

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Authors: Sachin Kumar

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Fuzzy fractional diffusion equation is widely useful to depict different physical processes arising in physics, biology, and hydrology. The motive of this article is to deal with the fuzzy fractional diffusion equation. We study a mathematical model of fuzzy space-time fractional diffusion equation in which unknown function, coefficients, and initial-boundary conditions are fuzzy numbers. First, we find out a fuzzy operational matrix of Legendre polynomial of Caputo type fuzzy fractional derivative having a non-singular Mittag-Leffler kernel. The main advantages of this method are that it reduces the fuzzy fractional partial differential equation (FFPDE) to a system of fuzzy algebraic equations from which we can find the solution of the problem. The feasibility of our approach is shown by some numerical examples. Hence, our method is suitable to deal with FFPDE and has good accuracy.

Keywords: fractional PDE, fuzzy valued function, diffusion equation, Legendre polynomial, spectral method

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Authors: Chor Nejmeddine, Ines Ben Omrane, Mohamed Abdelwahed

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In this paper, we present a posteriori analysis of the discretization of the heat equation by spectral element method. We apply Euler's implicit scheme in time and spectral method in space. We propose two families of error indicators, both of which are built from the residual of the equation and we prove that they satisfy some optimal estimates. We present some numerical results which are coherent with the theoretical ones.

Keywords: heat equation, spectral elements discretization, error indicators, Euler

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Authors: Emad K. Jaradat, Ala’a Al-Faqih

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Nonlinear Schrödinger equations are regularly experienced in numerous parts of science and designing. Varieties of analytical methods have been proposed for solving these equations. In this work, we construct an approximate solution for the nonlinear Schrodinger equations, with harmonic oscillator potential, by Elzaki Decomposition Method (EDM). To illustrate the effects of harmonic oscillator on the behavior wave function, nonlinear Schrodinger equation in one and two dimensions is provided. The results show that, it is more perfectly convenient and easy to apply the EDM in one- and two-dimensional Schrodinger equation.

Keywords: non-linear Schrodinger equation, Elzaki decomposition method, harmonic oscillator, one and two-dimensional Schrodinger equation

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