Search results for: Christiansen equation evapotranspiration

Authors: Ranjeet Sable, Mahotsavi Patil, Aadesh Nimbalkar, Prajakta Palaskar, Ritu Sagar

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The exact amount of water used by various crops in different climatic conditions is necessary to step for design, planning, and management of irrigation schemes, water resources, scheduling of irrigation systems. Evaporation and transpiration are combinable called as evapotranspiration. Water loss from trees during photosynthesis is called as transpiration and when water gets converted into gaseous state is called evaporation. For calculation of correct evapotranspiration, we have to choose the method in such way that is should be suitable and require minimum climatic data also it should be applicable for wide range of climatic conditions. In hydrology, there are multiple correlations and regression is generally used to develop relationships between three or more hydrological variables by knowing the dependence between them. This research work includes the study of various methods for calculation of evapotranspiration and selects reasonable and suitable one Pune region (Maharashtra state). As field methods are very costly, time-consuming and not give appropriate results if the suitable climate is not maintained. Observation recorded at Pune metrological stations are used to calculate evapotranspiration with the help of Radiation Method (RAD), Modified Penman Method (MPM), Thornthwaite Method (THW), Blaney-Criddle (BCL), Christiansen Equation (CNM), Hargreaves Method (HGM), from which Hargreaves and Thornthwaite are temperature based methods. Performance of all these methods are compared with Modified Penman method and method which showing less variation with standard Modified Penman method (MPM) is selected as the suitable one. Evapotranspiration values are estimated on a monthly basis. Comparative analysis in this research used for selection for raw data-dependent methods in case of missing data.

Keywords: Blaney-Criddle, Christiansen equation evapotranspiration, Hargreaves method, precipitations, Penman method, water use efficiency

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Authors: Ali Dehgan Moroozeh, B. Farhadi Bansouleh

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Evapotranspiration is one of the most important components of the hydrological cycle. Evapotranspiration (ETo) is an important variable in water and energy balances on the earth’s surface, and knowledge of the distribution of ET is a key factor in hydrology, climatology, agronomy and ecology studies. Many researchers have a valid relationship, which is a function of climate factors, to estimate the potential evapotranspiration presented to the plant water stress or water loss, prevent. The FAO-Penman method (PM) had been recommended as a standard method. This method requires many data and these data are not available in every area of world. So, other methods should be evaluated for these conditions. When sufficient or reliable data to solve the PM equation are not available then Hargreaves equation can be used. The Hargreaves equation (HG) requires only daily mean, maximum and minimum air temperature extraterrestrial radiation .In this study, Hargreaves method (HG) were evaluated in 12 stations in the North West region of Iran. Results of HG and M.HG methods were compared with results of PM method. Statistical analysis of this comparison showed that calibration process has had significant effect on efficiency of Hargreaves method.

Keywords: evapotranspiration, hargreaves, equation, FAO-Penman method

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Authors: Mishari AlNaim

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A field experiment was conducted for two seasons of 2011 and 2012 in The Agricultural Experiment Research Station in King Faisal University at Al-Hasa region, Saudi Arabia to estimate evapotranspiration (ETC) of Eggplant crop using Drainage Lysimeter with surface area of 2 x 2 m and depth of 1.5 m. The irrigation was applied daily. The amount of drainage was measured before each irrigation event. The results showed that there was almost no difference in the seasonal evapotranspiration of eggplant crop in the two seasons. The average evapotranspiration values for eggplant crop for the summer and winter seasons were 823.4 mm and 479.7 mm respectively. The highest and the lowest weekly measured values of (ETC) of eggplant crop during the two summer seasons were 8.6 mm/day and 3.9 mm/day respectively, while the highest and lowest weekly measured values of (ETC) of eggplant crop during the two winter seasons were 3.9 mm/day and 2.0 mm/day respectively. The measured values of ETc, in conjunction with the results of Penmen-Monteith equation for reference Evapotranspiration (ETR), were used to determine the crop coefficient (KC ini, KC mid and KC end) for eggplant crop. The average values were 0.50, 84 and 0.60 for KC ini, KC mid and KC end in Al-Hasa region, respectively. These estimated values for KC were used to approximate (ETc) for eggplant crop. High positive correlation coefficient (0.959) was detected between the approximated and measured values of eggplant crop evapotranspiration.

Keywords: evapotranspiration, eggpant, ETC, Al-Hasa

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Authors: Alper Serdar Anli

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Analysis of temporal variation of reference evapotranspiration (ET0) is important in arid and semi-arid regions where water resources are limited. In this study, temporal variation of reference evapotranspiration (ET0) and meteorological drought analysis through SPEI (Standardized Precipitation Evapotranspiration Index) method have been carried out in provinces of Central Anatolia Region, Turkey. Reference evapotranspiration of concerning provinces in the region has been estimated using Penman-Monteith method and one calendar year has been split up four periods as r1, r2, r3 and r4. Temporal variation of reference evapotranspiration according to four periods has been analyzed through parametric Dickey-Fuller test and non-parametric Mann-Whitney U test. As a result, significant increasing trends for reference evapotranspiration have been detected and according to SPEI method used for estimating meteorological drought in provinces, mild drought has been experienced in general, and however there have been also a significant amount of events where moderate and severely droughts occurred.

Keywords: central Anatolia region, drought index, Penman-Monteith, reference evapotranspiration, temporal variation

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Authors: J. Bahrawi, M. Elhag

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Estimation of evapotranspiration is always a major component in water resources management. Traditional techniques of calculating daily evapotranspiration based on field measurements are valid only for local scales. Earth observation satellite sensors are thus used to overcome difficulties in obtaining daily evapotranspiration measurements on regional scale. The Surface Energy Balance System (SEBS) model was adopted to estimate daily evapotranspiration and relative evaporation along with other land surface energy fluxes. The model requires agro-climatic data that improve the model outputs. Advance Along Track Scanning Radiometer (AATSR) and Medium Spectral Resolution Imaging Spectrometer (MERIS) imageries were used to estimate the daily evapotranspiration and relative evaporation over the entire Nile Delta region in Egypt supported by meteorological data collected from six different weather stations located within the study area. Daily evapotranspiration maps derived from SEBS model show a strong agreement with actual ground-truth data taken from 92 points uniformly distributed all over the study area. Moreover, daily evapotranspiration and relative evaporation are strongly correlated. The reliable estimation of daily evapotranspiration supports the decision makers to review the current land use practices in terms of water management, while enabling them to propose proper land use changes.

Keywords: daily evapotranspiration, relative evaporation, SEBS, AATSR, MERIS, Nile Delta

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Authors: M. Grifoni, G. Petruzzelli, M. Barbafieri, I. Rosellini, B. Pezzarossa, F. Pedron

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Evapotranspiration (ET) covers are an alternative cover system that utilizes water balance approach to maximize the ET process to reduce the contaminants leaching through the soil profile. Microcosm tests allow to identify in a short time the most suitable plant species to be used as alternative covers, their survival capacity, and simultaneously the transpiration and evaporation rate of the cover in a specific contaminated soil. This work shows the soil characterization and ET results of microcosm tests carried out on two contaminated soils by using Triticum durum and Helianthus annuus species. The data indicated that transpiration was higher than evaporation, supporting the use of plants as alternative cover at this contaminated site.

Keywords: contaminated sites, evapotranspiration cover, evapotranspiration, microcosm experiments

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Authors: Khosro Shafie Motlaghi, Amir Reza Salemian

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In arid and semi-arid regions like our country Evapotranspiration is the greatestportion of water resource. Therefor knowlege of its changing and other climate parameters plays an important role for planning, development, and management of water resource. In this search the Trend of long changing of Evapotranspiration (ET0), average temprature, monthly rainfall were tested. To dose, all synoptic station s in iran were divided according to the climate with Domarton climate. The present research was done in semi-arid climate of Iran, and in which 14 synoptic with 30 years period of statistics were investigated with 3 methods of minimum square error, Mann Kendoll, and Vald-Volfoytz Evapotranspiration was calculated by using the method of FAO-Penman. The results of investigation in periods of statistic has shown that the process Evapotranspiration parameter of 24 percent of stations is positive, and for 2 percent is negative, and for 47 percent. It was without any Trend. Similary for 22 percent of stations was positive the Trend of parameter of temperature for 19 percent , the trend was negative and for 64 percent, it was without any Trend. The results of rainfall trend has shown that the amount of rainfall in most stations was not considered as a meaningful trend. The result of Mann-kendoll method similar to minimum square error method. regarding the acquired result was can admit that in future years Some regions will face increase of temperature and Evapotranspiration.

Keywords: analysis, algorithm, SANN, ET0

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Authors: Muhammad Tayyab Afzal, Muhammad Arslan, Mirza Muhammad Waqar

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One of the key components of the water cycle is evapotranspiration (ET), which represents water consumption by vegetated and non-vegetated surfaces. Conventional techniques for measurements of ET are point based and representative of the local scale only. Satellite remote sensing data with large area coverage and high temporal frequency provide representative measurements of several relevant biophysical parameters required for estimation of ET at regional scales. The objective is of this research is to exploit satellite data in order to estimate evapotranspiration. This study uses Surface Energy Balance System (SEBS) model to calculate daily actual evapotranspiration (ETa) in Larkana District, Sindh Pakistan using Landsat TM data for clouds-free days. As there is no flux tower in the study area for direct measurement of latent heat flux or evapotranspiration and sensible heat flux, therefore, the model estimated values of ET were compared with reference evapotranspiration (ETo) computed by FAO-56 Penman Monteith Method using meteorological data. For a country like Pakistan, agriculture by irrigation in the river basins is the largest user of fresh water. For the better assessment and management of irrigation water requirement, the estimation of consumptive use of water for agriculture is very important because it is the main consumer of water. ET is yet an essential issue of water imbalance due to major loss of irrigation water and precipitation on cropland. As large amount of irrigated water is lost through ET, therefore its accurate estimation can be helpful for efficient management of irrigation water. Results of this study can be used to analyse surface conditions, i.e. temperature, energy budgets and relevant characteristics. Through this information we can monitor vegetation health and suitable agricultural conditions and can take controlling steps to increase agriculture production.

Keywords: SEBS, remote sensing, evapotranspiration, ETa

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Authors: Saeed Jahanbakhsh Asl, Yaghob Dinpazhoh, Masoumeh Foroughi

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Reference evapotranspiration (ET₀) is an important element in the water cycle that integrates atmospheric demands and surface conditions, and analysis of changes in ET₀ is of great significance for understanding climate change and its impacts on hydrology. As ET₀ is an integrated effect of climate variables, increases in air temperature should lead to increases in ET₀. ET₀ estimated by using the globally accepted Food and Agriculture Organization (FAO) Penman-Monteith (FAO-56 PM) method in 18 meteorological stations located in the West of Iran. The trends of ET₀ detected by using the Mann-Kendall (MK) test. The slopes of the trend lines were computed by using the Sen’s slope estimator. The results showed significant increasing as well as decreasing trends in the annual and monthly ET₀. However, ET₀ trends were increasing. In the monthly scale, the number of the increasing trends was more than the number of decreasing trends, in the majority of warm months of the year.

Keywords: climate change, Mann–Kendall, Penman-Monteith method (FAO-56 PM), reference crop evapotranspiration

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Authors: Enze Zhang

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One of the most challenging steps in implementing virtual water content (VWC) analysis of crops is to get properly the total volume of consumptive water use (CWU) and, therefore, the choice of a reliable crop CWU estimation method. In practice, lots of previous researches obtaining CWU of crops follow a classical procedure for calculating crop evapotranspiration which is determined by multiplying reference evapotranspiration by appropriate coefficient, such as crop coefficient and water stress coefficients. However, this manner of calculation requires lots of field experimental data at point scale and more seriously, when current growing conditions differ from the standard conditions, may easily produce deviation between the calculated CWU and the actual CWU. Since evapotranspiration caused by crop planting always plays a vital role in surface water-energy balance in an agricultural region, this study decided to alternatively estimates crop evapotranspiration by Budyko framework. After brief introduce the development process of Budyko framework. We choose a modified Budyko framework under unsteady-state to better evaluated the actual CWU and apply it in an agricultural irrigation area in North China Plain which rely on underground water for irrigation. With the agricultural statistic data, this calculated CWU was further converted into VWC and its subdivision of crops at the annual scale. Results show that all the average values of VWC, VWC_blue and VWC_green show a downward trend with increased agricultural production and improved acreage. By comparison with the previous research, VWC calculated by Budyko framework agree well with part of the previous research and for some other research the value is greater. Our research also suggests that this methodology and findings may be reliable and convenient for investigation of virtual water throughout various agriculture regions of the world.

Keywords: virtual water content, Budyko framework, consumptive water use, crop evapotranspiration

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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: Bensaid A., Mostephaoui T., Nedjai R.

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Climatic factors are the subject of considerable research, both methodologically and instrumentally. Under the effect of climate change, the approach to climate parameters with precision remains one of the main objectives of the scientific community. This is from the perspective of assessing climate change and its repercussions on humans and the environment. However, many regions of the world suffer from a severe lack of reliable instruments that can make up for this deficit. Alternatively, the use of empirical methods becomes the only way to assess certain parameters that can act as climate indicators. Several scientific methods are used for the evaluation of evapotranspiration which leads to its evaluation either directly at the level of the climatic stations or by empirical methods. All these methods make a point approach and, in no case, allow the spatial variation of this parameter. We, therefore, propose in this paper the use of three sources of information (network of weather stations of Meteo France, World Databases, and Moodis satellite images) to evaluate spatial evapotranspiration (ETP) using the Turc method. This first step will reflect the degree of relevance of the indirect (satellite) methods and their generalization to sites without stations. The spatial variation representation of this parameter using the geographical information system (GIS) accounts for the heterogeneity of the behaviour of this parameter. This heterogeneity is due to the influence of site morphological factors and will make it possible to appreciate the role of certain topographic and hydrological parameters. A phase of predicting the evolution over the medium and long term of evapotranspiration under the effect of climate change by the application of the Intergovernmental Panel on Climate Change (IPCC) scenarios gives a realistic overview as to the contribution of aquatic systems to the scale of the region.

Keywords: climate change, ETP, MODIS, GIEC scenarios

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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: 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: 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: Hamidreza Miri, Farahnaz Momtazi

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Wheat is one of the most important crops in Fars province, and because of water shortage, there is a great emphasis on its water use efficiency in the production field. A field experiment was conducted in 2021 and 2022 in order to evaluate wheat yield and its components in raised planting system in Arsanjan, Fars province. The experiment was conducted as a split plot with three irrigation treatments (irrigation equal to evapotranspiration, 80% of evapotranspiration irrigation (moderate drought stress), and 60% of evapotranspiration irrigation (severe drought stress)) as the main plot and three planting methods (conventional flat planting, 60 cm raised bed planting and 120 cm raised bed planting) as a subplot. The results indicated that drought stress significantly decreased traits such as plant height, grain yield, ear number, seed number, and biological yield while increasing seed protein. Raised bed planting significantly increased the traits in comparison with conventional flat planting. So that plating with a 120 cm raised bed increased grain yield by 22.1% and 25.9% in the first and second years, respectively. This increase was 17% for biological, 75 for ear number, and 21% for seed number. Planting in raised bed system reduced the adverse effect of drought stress on wheat traits. In conclusion, based on the observed results planting in raised bed system can be adopted as an appropriate planting pattern for improving yield and water productivity in experimental regions and similar climates.

Keywords: wheat, raised bed planting, drought stress, yield, water use

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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: Waqas Ahmed Khan Afridi, Subhas Chandra Mukhopadhyay, Bandita Mainali

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The research paper presents a unique approach to evapotranspiration (ET) prediction using a Support Vector Machine (SVM) learning algorithm. The study leverages real-time sensor node data to develop an accurate and adaptable prediction model, addressing the inherent challenges of traditional ET estimation methods. The integration of the SVM algorithm with real-time sensor node data offers great potential to improve spatial and temporal resolution in ET predictions. In the model development, key input features are measured and computed using mathematical equations such as Penman-Monteith (FAO56) and soil water balance (SWB), which include soil-environmental parameters such as; solar radiation (Rs), air temperature (T), atmospheric pressure (P), relative humidity (RH), wind speed (u2), rain (R), deep percolation (DP), soil temperature (ST), and change in soil moisture (∆SM). The one-year field data are split into combinations of three proportions i.e. train, test, and validation sets. While kernel functions with tuning hyperparameters have been used to train and improve the accuracy of the prediction model with multiple iterations. This paper also outlines the existing methods and the machine learning techniques to determine Evapotranspiration, data collection and preprocessing, model construction, and evaluation metrics, highlighting the significance of SVM in advancing the field of ET prediction. The results demonstrate the robustness and high predictability of the developed model on the basis of performance evaluation metrics (R2, RMSE, MAE). The effectiveness of the proposed model in capturing complex relationships within soil and environmental parameters provide insights into its potential applications for water resource management and hydrological ecosystem.

Keywords: evapotranspiration, FAO56, KNIME, machine learning, RStudio, SVM, sensors

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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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