Complex Nonlinear Equations Research Articles

INTELLIGENT CONTROL SYSTEM OF MULTI-MOTION MOBILE ROBOT

Technical devices with irregular motion are described by complex non-linear differential equations, as they have an uncertain environment. Due to interactions and simplifications in these types of control objects, their mathematical models may have certain errors. For this reason, by known methods, for example, the linear matrix inequality method, Lyapunov's quadratic function, etc. The synthesized automatic control systems (ACS) for dynamic objects written by nonlinear mathematical models based on the quadratic matrix determined by The synthesis of regulators for the management of dynamic objects described in nonlinear and uncertain conditions is based on fuzzy logic theory. Since the controllers synthesized by fuzzy logic are based on knowledge, their application to other objects (robots) is limited. In such systems, it is difficult or not at all to determine the dependencies between ACS quality indicators, stability and object parameters. Taking into account the above, a method of analytically parametric synthesis of regulators that ensures the degree of stability and accuracy in ACS’s designed for fuzzy TS-type dynamic objects is proposed. In the article, the solution to the problem of the synthesis of the control system of the mobile robot was considered, and it was successfully applied in the multi-motion mobile robot system. Modeling and experimental results confirm that the operation and stability of the obtained system fully meet the technical requirements. The proposed method has led to an increase in both tracking and control accuracy during high-speed movements. Keywords: multi-motion mobile robot, mathematical model, mechatronic modules, fuzzy TS model, robust controller, fuzzy logic.

A health monitoring technique for spherical structures based on multi-acoustic source localization

Multi-acoustic source localization (MASL) technique has important applications in the early warning and maintenance of spherical structures. Without solving complex nonlinear equations and without knowing the wave velocity distribution a priori, this work demonstrates the feasibility of MASL on the surface of spherical structures using L-shaped sensor clusters. The positions of multiple acoustic sources can be localized using only time difference of arrival values. Relative location determination and relative probability density analysis have been presented and verified to eliminate two types of pseudo-sources. Simulations are performed for isotropic and anisotropic spherical shells. The proposed technique is validated experimentally for stainless steel spherical shells. Simulation and experimental results show that the proposed technique can enable MASL in spherical structures without knowing the wave velocity in the material.

Dynamics of optical solitons in the extended (3+1)‐dimensional nonlinear conformable Kudryashov equation with generalized anti‐cubic nonlinearity

The nonlinear Schrödinger equation (NLSE) is a fundamental equation in the field of nonlinear optics and plays an important role in the study of many physical phenomena. The present study introduces a new model that demonstrates the novelty of the paper and provides the advancement of knowledge in the area of nonlinear optics by solving a challenging problem known as the extended ‐dimensional nonlinear conformable Kudryashov's equation (CKE) with generalized anti‐cubic nonlinearity, which is a generalization of the NLSE to three spatial dimension and one temporal dimension for the first time. This work is significant because it advances our understanding of nonlinear optics and its applications to solve complex equations in physics and related disciplines. The extended hyperbolic function method (EHFM) and Nucci's reduction method are applied to the extended ‐dimensional nonlinear CKE with generalized anti‐cubic nonlinearity. The equation is solved by using the concept of conformable derivative, a recently developed operator in fractional calculus, which has advantages over other fractional derivatives in terms of accuracy and flexibility. The attained solutions include periodic singular, dark 1‐soliton, singular 1‐soliton, and bright 1‐soliton which are visualized using 3D and contour plots. This study highlights the potential of using conformable derivative and the applied techniques to solve complex nonlinear differential equations in various fields. The obtained solutions and analysis will be useful in the design and analysis of optical communication systems and other related fields. Overall, this study contributes for the understanding of the dynamics of the extended ‐dimensional nonlinear CKE and offers new insights into the use of mathematical techniques to tackle complex problems in physics and related fields.

An algorithm to determine the impact point on the target surface using ultrasonic sensors

The article addresses the issue of using ultrasonic sensors to determine the bullet's impact point on the target surface, serving to evaluate the results of firing tests. The method of determining the impact point is based on the time difference of the time of arrival of the sound wave to the ultrasonic sensors. The constructed equations are complex nonlinear equations. To be able to solve this system of equations, the construction of the equation is combined with the arrangement of sensors to isolate each unknown, helping to simplify the solution of the system of equations. Sensor layout diagram and impact point determination algorithm have been proposed.

Travelling Wave-like Solutions for Some Nonlinear Equations with Cubic Nonlinearities

We exercised the series-expansion method to extract solitary wave solutions for complex nonlinear evolution equations (NLEEs) such as the coupled Higgs equation (CHE), the (3+1)-dimensional dynamical system, the (3+1)-dimensional nonlinear Schrodinger(NLSE) equation which have many physical importance in different branches. Varieties of periodic and solitonic wave-like solutions are extracted. Computational work has been done and plots and counter graphs are plotted using Wolfram Mathematica 11.

Abundant solitons for highly dispersive nonlinear Schrödinger equation with sextic-power law refractive index using modified extended direct algebraic method

In this paper, we investigate soliton solutions and other exact solutions of the highly dispersive perturbed nonlinear Schrödinger equation having Kudryashov's arbitrary form with sextic-power law of refractive index and generalized non-local laws. Studying is conducted by applying the modified extended direct algebraic method, which offers several types of solutions. The extracted solutions including (combo bright-dark, singular, dark, bright) solitons, exponential solutions, rational solutions and singular periodic solutions. The extracted solutions confirmed the effectiveness and strength of the current technology. To illustrate the properties of some solutions, graphic representations of those solutions are given. This study contributes to the understanding of nonlinear wave phenomena and showcases the applicability of the modified extended direct algebraic method in obtaining exact solutions for complex nonlinear equations. The optical solitons produced in respect to this form have never been explored by the proposed technique before, and the results have never been published.

Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2

Abstract Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1, and ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g}, where g ( z ) g\left(z) is a nonconstant polynomial and a j , b j {a}_{j},{b}_{j} , and c j ( j = 1 , 2 ) {c}_{j}\left(j=1,2) are constants in C {\mathbb{C}} . The finite-order transcendental entire solution u u of the first equation is of the following forms: u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\pm \frac{1}{\sqrt{{a}_{1}{a}_{2}}}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, or u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e − Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\frac{1}{2{a}_{1}}{e}^{Q\left({z}_{1},{z}_{2})}+\frac{1}{2{a}_{2}}{e}^{-Q\left({z}_{1},{z}_{2})}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, where D = b 1 c 2 − b 2 c 1 D={b}_{1}{c}_{2}-{b}_{2}{c}_{1} , η 0 ∈ C − { 0 } {\eta }_{0}\in {\mathbb{C}}-\left\{0\right\} , and Q ( z 1 , z 2 ) = − 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C . Q\left({z}_{1},{z}_{2})=-\frac{1}{D}\left[\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\eta }_{1},\hspace{1em}{\eta }_{1}\in {\mathbb{C}}. The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.

Retrieving land surface temperatures from IASI hyperspectral thermal infrared data using an AFNO-transformer model.

An adaptive Fourier neural operator (AFNO)-transformer model was developed to retrieve land surface temperature (LST) data from infrared atmospheric sounding interferometer (IASI) observations. A weight selection scheme based on linearization of the radiative transfer equation was proposed to solve the hyperspectral data channel redundancy problem. The IASI brightness temperatures and Advanced Very High Resolution Radiometer onboard MetOp (AVHRR/MetOp) LST product were selected to construct the training and test datasets. The AFNO-transformer performed effective token mixing through self-attention and effectively solved the global convolution problem in the Fourier domain, which can better learn complex nonlinear equations and achieve time-series forecasting. The root mean square error indicated that the LST in Eastern Spain and North Africa could be retrieved with an error of less than 2.5 K compared with the AVHRR/MetOp LST product. Moreover, the validation results from other time period data showed that the retrieval accuracy of this model can be less than 3 K. The proposed model provides a novel approach for hyperspectral LST retrieval.

Analysis of bidirectional flow of Williamson micropolar fluid in porous medium with activation energy and thermal radiation

The prime objective of the proposed model is to explore the impacts of viscous dissipation and Ohmic heating on the bidirectional flow of micropolar and Williamson fluids in porous medium over a stretching surface. The velocity slips and heat convection are assessed at the surface as well. The concernment of chemical reaction, thermophoresis, and Brownian motion repercussions is also considered. More complex nonlinear ordinary differential equations arise when the variables are combined in a suitable manner to represent the mathematical model of the problem. The Galerkin weighted residual approach is used to solve non - linear differential equations’ system that has been developed. Flow patterns are presented by utilizing multiple plots and assessing the influence of various emerging factors. Tables are frequently utilized to characterize the rates of mass and heat transfer. The current findings have yielded several important crucial findings, including the fact that skin friction improves with improvements in the porosity, Williamson fluid, and micropolar fluid parameters. The Nusselt number decays with increments in Eckert numbers and heat source parameters. The velocity field speeds up in the x-direction as the slip parameter enhances, but it lessens in both directions with rise in the porosity parameter and the Williamson fluid parameter. The microrotation velocity profiles in both directions experience a significant reduction as the microrotation slip parameter increases. Also, it has been noted that the nano-liquid concentration profile becomes lesser while the temperature distribution becomes higher for higher values of the Eckert numbers. The thermal profile accelerates as the heat source parameter increases.

Offshore Wind Energy Assessment with a Clustering Approach to Mixture Model Parameter Estimation

In wind resource assessment research, mixture models are gaining importance due to the complex characteristics of wind data. The precision of parameter estimations for these models is paramount, as it directly affects the reliability of wind energy forecasts. Traditionally, the expectation–maximization (EM) algorithm has served as a primary tool for such estimations. However, challenges are often encountered with this method when handling complex probability distributions. Given these limitations, the objective of this study is to propose a new clustering algorithm, designed to transform mixture distribution models into simpler probability clusters. To validate its efficacy, a numerical experiment was conducted, and its outcomes were compared with those derived from the established EM algorithm. The results demonstrated a significant alignment between the new method and the traditional EM approach, indicating that comparable accuracy can be achieved without the need for solving complex nonlinear equations. Moreover, the new algorithm was utilized to examine the joint probabilistic structure of wind speed and air density in China’s coastal regions. Notably, the clustering algorithm demonstrated its robustness, with the root mean square error value being notably minimal and the coefficient of determination exceeding 0.9. The proposed approach is suggested as a compelling alternative for parameter estimation in mixture models, particularly when dealing with complex probability models.

SONets: Sub-operator learning enhanced neural networks for solving parametric partial differential equations

Complex nonlinear partial differential equations (PDEs) can be decomposed into subsystems comprising certain linear and nonlinear sub-operators. We usually have fundamental solutions to the linear subsystems and have a good knowledge of their uniqueness and regularity. The nonlinear subsystems have more straightforward formulas than the original complex nonlinear PDEs, even with reduced dimensions. Solutions of the complex nonlinear PDEs based on simpler subsystems are investigated in this study. We derive a proper subsystem decomposition of the nonlinear PDEs into linear and nonlinear subsystems to get unique solutions. We propose the sub-operator learning enhanced neural networks (SONets) based on this proper subsystem decomposition to solve nonlinear PDEs. The solutions of nonlinear subsystems with generated force terms are modeled via operator learning. In contrast, only the cumbersome integration and series parts of fundamental solutions of linear subsystems are modeled. The complex nonlinear PDEs are then solved by finding the force terms to minimize the physics-informed loss functions. It is also convenient to generalize SONets for a new PDE parameter, e.g., the diffusion coefficient. Next, we extend the SONets to a meta version (meta-SONets) for parametric PDEs by transfer learning. In meta-SONets, the initial conditions are explicitly encoded via operator learning neural networks (ENNs) to output force terms, followed by the whole SONets to solve PDEs. In the pre-training stage of meta-SONets, parametric PDEs given randomly generated initial conditions are solved by training ENNs to minimize the physics-informed loss functions. At the fine-tuning stage, the trainable parameters of ENNs are fine-tuned to solve the PDE given a new initial condition. Numerical experiments are conducted to solve the Burgers equation, the nonlinear diffusion-reaction system, and the Allen-Cahn equation. The numerical results indicate that SONets with neural network-parameterized force terms obtain solutions with good accuracy. With transfer learning, the computational cost of meta-SONets for solving the nonlinear PDEs with a new initial condition is dramatically reduced.

Coupled vibration of an annular plate-cylindrical drum structure with material-nonlinearity damping patches considering position-mistuned and stiffness-mistuned connections

Considering position-mistuned and stiffness-mistuned connections, the coupled vibration of an annular plate-cylindrical drum structure with material-nonlinearity damping patches is studied. These two mistuned connections are inevitable, but they are seldom concerned in published works. Moreover, the mistuned connections can cause local excessive vibrations, so vibration suppression is in demand. Spraying hard coatings with material nonlinearity is a good damping method. Their damping-related and stiffness-related parameters will change with strain, which will improve the vibration-reduction ability but bring obstacles to modelling. This paper focuses on the consideration of both mistuned connections and material-nonlinearity damping patches (MDP) and presents a general modeling method of connection-mistuned annular plate-cylindrical drum structures with MDP. Position-mistuned and stiffness-mistuned connections are emulated by developing the spring approach, and dynamic equations are created by combining Sanders’ shell theory, Kirchhoff’s plate theory, and Lagrange equation. Furthermore, the region decomposition approach is developed for considering the material nonlinearity, and an efficient solution is provided to solve the corresponding nonlinear complex equations. The study is adequately validated. Finally, numerical results indicate that mistuned connections cause the local-vibration behavior, frequency separation, and coupled vibration between substructures and modes. Moreover, mistuned connections also seriously affect the response curves and response cloud charts.

On the Analysis of a Fractional Tuberculosis Model with the Effect of an Imperfect Vaccine and Exogenous Factors under the Mittag–Leffler Kernel

This research study aims to investigate the effects of vaccination on reducing disease burden by analyzing a complex nonlinear ordinary differential equation system. The study focuses on five distinct sub-classes within the system to comprehensively explore the impact of vaccination. Specifically, the mathematical model employed in this investigation is a fractional representation of tuberculosis, utilizing the Atangana–Baleanu fractional derivative in the Caputo sense. The validity of the proposed model is established through a rigorous qualitative analysis. The existence and uniqueness of the solution are rigorously determined by applying the fundamental theorems of the fixed point approach. The stability analysis of the model is conducted using the Ulam–Hyers approach. Additionally, the study employs the widely recognized iterative Adams–Bashforth technique to obtain an approximate solution for the suggested model. The numerical simulation of the tuberculosis model is comprehensively discussed, with a particular focus on the assumptions made regarding vaccination. The model assumes that only a limited portion of the population is vaccinated at a steady rate, and the efficacy of the vaccine is a critical factor in reducing disease burden. The findings of the study indicate that the proposed model can effectively assess the impact of vaccination on mitigating the spread of tuberculosis. Furthermore, the numerical simulation underscores the significance of vaccination as an effective control measure against tuberculosis.

Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2

This article is devoted to exploring the existence and the form of finite order transcendental entire solutions of Fermat-type second order partial differential-difference equations $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$ and $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$ where \(\delta,\eta\in\mathbb{C}\) and \(g(z_1,z_2)\) is a polynomial in \(\mathbb{C}^2\). Our results improve the results of Liu and Dong [23] Liu et al. [24] and Liu and Yang [25] Several examples confirm that the form of tr

Generalized (G’ / G) – expansion method for the loaded shallow water wave equation

This article is devoted to finding solutions for the traveling wave of the loaded wave equation in shallow water. One of the approaches to finding solutions by the expansion method (G / G) is given, which is one of the most effective ways to obtain solutions. When parameters are taken as special values, solitary waves are also derived from traveling waves. Solutions for the traveling wave are expressed by hyperbolic and trigonometric functions. This method is easy to implement using well-known software packages that allow solving complex nonlinear evolutionary equations of mathematical physics.

Efficient (2N+1) selective harmonic elimination in modular multilevel converters using an evolutionary many-tasking approach with prior knowledge

(2N+1) selective harmonic elimination pulse with modulation (SHE-PWM) is an important technique for a modular multi-level converter (MMC) to enhance the control ability and improve the fundamental wave amplitude accuracy of the output waveform. While finding optimal switch angles by solving the complex non-linear equation system is one of its main challenges. A current popular solution is to run a metaheuristic or evolutionary algorithm repeatedly on various modulation indexes. These methods ignore the fact that some of the optimization tasks are similar, and the optimization experiences could be shared among these similar tasks. To fill this gap, this study proposes an evolutionary many-tasking approach with prior knowledge (EMT-PK). EMT-PK facilitates positive knowledge transfer between similar tasks and is capable to solve all the sub SHE-PWM problems simultaneously through a single run. Numerical results on 7-, 9- and 11-level MMCs show that EMT-PK provides superior results for the (2N+1) SHE-PWM in comparison with 8 other widely used metaheuristics/evolutionary algorithms in this field including differential evolution (DE), ant colony optimization (ACO), teaching and learning-based optimization (TLBO), bee algorithm (BA), genetic algorithm (GA), particle swarm optimization (PSO), generalized pattern search (GPS) and asynchronous particle swarm optimization-genetic algorithm(APSO-GA). The superior performance of EMT-PK is further validated by the MATLAB Simulink simulation and the laboratory experiment on a 7-level MMC.

On solutions for several systems of complex nonlinear partial differential equations with two variables

This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations $$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}$$ where $$a,b,c,d\in {\mathbb {C}}$$ , and g is a polynomial in $${\mathbb {C}}^2$$ . Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations.

A New High-Performance Photovoltaic Emulator Suitable for Simulating and Validating Maximum Power Point Tracking Controllers

Photovoltaic (PV) research is rapidly growing, and the need for controlled environments to validate new MPPT controllers is becoming increasingly important. Currently, researchers face several challenges in testing MPPT algorithms due to the unpredictable nature of solar PV power generation. In this paper, we propose a new photovoltaic emulator (PVE) that could replace solar panels and ensure a highly controllable environment suitable for testing photovoltaic (PV) systems. In this PVE, the complex nonlinear equations of the PV cell/module are fast computed and resolved by a new linearization technique which involves the systematic breakdown of the current-voltage ( I - V ) curve of the PV into twelve linear segments. Based on input environmental conditions, an artificial neural network (ANN) was constructed to assist the linearization process by predicting the current-voltage boundary coordinates of these segments. Using simple linear equations, with the segment boundary coordinates, a reference voltage was generated for the PVE. A nonlinear backstepping controller was designed to exploit the reference voltage and stabilize the power conversion stage (PCS). The PVE was optimized using particle swarm optimization (PSO). Several tests have shown that the proposed nonlinear controller provides better dynamic and robust performance than the PI controller, the most reputable and recurrent control method in the area of PVE. The PVE was coupled with a recently proposed integral backstepping MPPT controller and analyzed under several dynamic conditions, including the MPPT test specified by EN 50530. It was found that the accuracy of the proposed PVE measured by its relative error is less than 0.5%, with an MPPT efficiency of greater than 99.5%. The attractive results achieved by this PVE make it especially suitable for simulating and validating MPPT controllers.

Numerical Analysis of the Fractional-Order Belousov–Zhabotinsky System

This paper presents a new approach for finding analytic solutions to the Belousov–Zhabotinsky system by combining the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) with the Elzaki transform. The ADM and HPM are both powerful techniques for solving nonlinear differential equations, and their combination allows for a more efficient and accurate solution. The Elzaki transform, on the other hand, is a mathematical tool that transforms the system into a simpler form, making it easier to solve. The proposed method is applied to the Belousov–Zhabotinsky system, which is a well-known model for studying nonlinear chemical reactions. The results show that the combined method is capable of providing accurate analytic solutions to the system. Furthermore, the method is also able to capture the complex behavior of the system, such as the formation of oscillatory patterns. Overall, the proposed method offers a promising approach for solving complex nonlinear differential equations, such as those encountered in the field of chemical kinetics. The combination of ADM, HPM, and the Elzaki transform allows for a more efficient and accurate solution, which can provide valuable insights into the behavior of nonlinear systems.

Solar thermal energy on the convective transport in microchannel with exponential heat sink

In modern years, the conversion on solar radiation toward thermal energy had taken the important consideration by the appeal about sustainable power and heat raises. Nanofluid could show the significant role on developing the behavior on solar-thermal systems when that our capabilities about heat transfer improvement. That article analyzed numerically the flow on the hybrid nanofluid over the microchannel in existence of nonlinear solar radiation about different solar-thermal appliance. Effect of exponential heat source or sink and convective boundary condition are the focal concern of this study. The flow form was transferred with non-dimensionless model by convenient transformation, this numerical outcomes on nonlinear complex equations was formed to applying bvp4c technique. Subsequently, to influences on applicable parameters by the heat transfer of fluid had being revealed through the favor into tabular and graphical access. Outcomes communicate this solar radiation faster heat transport into division. The hybrid solution show massive increasing on heat transport 44.59% about division. Temperature decreased in the microchannel due to Biot number. The imitation outcomes are approved from the past issued work.