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This study focuses on the flow of viscous, electrically conducting incompressible fluid over a stretching plate. The Falkner–Skan equation is a nonlinear, third-order boundary value problem. No closed-form solutions are available for this two-point boundary value problem. Here, we developed a new functional matrix of integration using the Bernoulli wavelet and also generated a new technique called Bernoulli wavelet collocation method (BWCM) to solve the nonlinear differential equation that arises in the fluid flow over a stretching plate. The boundary layer model is transformed to a nonlinear ordinary differential equation called the Falkner-Skan type equation using suitable transformation. Using BWCM, we have solved the unbounded governing equations of different types that arise in the MHD boundary-layer flow of a viscous fluid over a stretching plate. Several aspects of this problem are justified using the Haar wavelet and the previously obtained theoretical results. It is observed that the boundary-layer thickness decreases as the pressure gradient and magnetic field parameters increase. The overshoots and undershoots are observed for some particular parameters using BWCM. Furthermore, our research yields dual solutions for some physical parameters, which are investigated for the first time in the literature using the Bernoulli wavelet approach. The nature of the flow problem is discussed through the graphs by varying the physical parameters.

The key idea and contribution of this study are to present the innovative functional matrix approach for solving the two chemical, mathematical problems such as the absorption of $$C{O}_{2}$$ C O 2 into phenyl glycidyl ether (PGE) and the chemical kinetic problem. These chemical problems are characterized as a system of nonlinear ODEs with initial and boundary conditions. Numerical outcomes are obtained to establish the simplicity and efficacy of the developed scheme. Graphs and tables show how consistently and effectively the developed strategy works. Results obtained show that the newly adopted technique is more precise and effective than other methods such as Adam–Bashforth–Moultan method (ABM), Runge–Kutta method (RK4), Adomian Decomposition method (ADM), and residual method (RM). Mathematical software called Mathematica 11.3 has been used to perform all calculations. Theorems explain the convergence of this approach.

The steady two-dimensional flow of an incompressible electrically conducting fluid near the stagnation point on a stretching sheet is investigated. The governing highly nonlinear PDEs are altered into highly nonlinear ODE by utilizing non-similar and similar variables and then solved numerically using a Hermite wavelet collocation method (HWCM). It is interesting to note that the results of the previous authors are in good agreement with the results of this study tabulated which is evident from the tabular values. Further, the effects of the magnetic parameter, viscoelastic parameter, stagnation point flow, and stretching of the sheet parameters on the flow field characteristics are obtained and discussed. The Lorentz force opposes the flow. Further, the fluid flow velocity decreases as the viscoelastic parameter increases. This practically indicates that the characteristics of fluid flow in viscoelastic fluids may be influenced by modifying the change in the viscoelastic parameter.

In this article, the two-dimensional micropolar rotating nanofluid between two parallel plates under Hall's influence can be solved using the Hermite wavelet technique (HWT). The nanofluid flow between two parallel plates is taken under the influence of the Hall current. The flow of micropolar nanofluid has been assumed in a steady state. We obtain a coupled, highly nonlinear ordinary differential equations (ODEs) from the fundamental governing equations via appropriate similarity variables. The behavior of velocity, temperature, concentration, and micro rotation are computationally discussed concerning the various governing parameters. Our outcome illustrates that the HWT is immensely capable and accessible for finding solutions to this type of coupled nonlinear ODEs. The results are in outstanding agreement for coupled nonlinear ODEs in engineering applications.

This study proposed an efficient numerical technique for coupled differential equations (CDEs) using the clique polynomials of the Complete graph. Recently, Graph theory has dragged the attention of many mathematicians due to its wide applications. Here, some problems have been considered to examine the proposed scheme proficiency. Some theorems on convergence are discussed. Here, the CDEs are rehabilitated into an algebraic equation system using the clique polynomials and collocation technique. The proposed scheme results are compared with the undetermined coefficients (UDCM) and variation of parameters method (VPM) solutions through tables and graphs. Obtained results reveal that the current approach is more accurate.

In this article, we propose a novel technique called the Hermite wavelets collocation method (HWCM) and fractional derivatives of functions to solve multi-term fractional differential equations (MT...

Hermite wavelet method (HWM) is applied to approximate the solution of the integro-differential equations. The Hermite wavelet and operational matrix of integration are used to reduce the integro-differential equations into algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.

In this article, we present the Laguerre wavelet exact Parseval frame method (LWPM) for the two-dimensional flow of a rotating micropolar fluid in a porous channel with huge mass transfer. This flow is governed by highly nonlinear coupled partial differential equations (PDEs) are reduced to the nonlinear coupled ordinary differential equations (ODEs) using Berman's similarity transformation before being solved numerically by a Laguerre wavelet exact Parseval frame method. We also compared this work with the other methods in the literature available. Moreover, in the graphs of the velocity distribution and microrotation, we shown that the proposed scheme's solutions are more accurate and applicable than other existing methods in the literature. Numerical results explaining the effects of various physical parameters connected with the flow are discussed.

In this article, we generated a new operational matrix of integration using Clique polynomials of complete graphs and also introducing a new numerical technique to solve nonlinear Klein–Gordon equation. These equations describe a variety of physical phenomena such as ferroelectric and ferromagnetic domain walls, and DNA dynamics. We obtain an approximate solution for the nonlinear Klein–Gordon equation using the present method by transforming a system of nonlinear algebraic equations. The proposed scheme is applied to some examples and compared with another method in the literature that demonstrates the effectiveness of this method.

This paper has developed a novel functional matrix using the standard basis of $$(n+1)$$ dimensional polynomial linear space to solve second-order singular initial and boundary problems. The linearly independent polynomials properties are used to convert the differential equations into algebraic equations with suitable solvers that can efficiently solve. Seven numerical examples are considered to demonstrate this technique's applicability and efficiency. The obtained results are compared favorably with the exact solutions. Also, we proved some theorems on convergence, exact solutions, and uniform convergence.

This article proposed an efficient numerical technique for the solution of (2+1) dimensional Sobolev and regularized long wave equations that arise in fluid mechanics using the Laguerre wavelet collocation method. Five examples are illustrated to inspect the proposed technique efficiency, and convergence analysis is discussed in terms of a theorem. Here, the Sobolev and regularized long wave equations are converted into a system of algebraic equations using the properties of Laguerre wavelet, and solutions obtained by the proposed scheme are more accurate, and they are compared with the analytical solution and other methods in the literature by calculating L 2 and L ∞ Errors.

In this paper, we proposed a new operational matrix of differentiation using CAS wavelets. By aid of these matrices, CAS wavelets method (CASWM) is developed for the Integro-differential eq...

This study introduced a generalized operational matrix using Clique polynomials of a complete graph and proposed the latest approach to solve the non-linear Klein–Gordon (KG) equation. KG equations describe many real physical phenomena in fluid dynamics, electrical engineering, biogenetics, tribology. By using the properties of the operational-matrix, we transform-the non-linear KG equation into a system-of algebraic-equations. Unknown coefficients to be determined by Newton’s method. The present-technique is applied-to four problems, and the obtained-results are-compared with-another-method in the literature. Also, we discussed some theorems on convergence analysis and continuous property.

In this study, the Laguerre wavelets exact Parseval frame is introduced and proposed an effective numerical algorithm to get a numerical solution for the system of differential equations based on the Laguerre wavelets exact Parseval frame. This algorithm includes the collocation method and truncated Laguerre wavelet frames. Here, we reduce the system of differential equations into a set of algebraic equations which are having unknown Laguerre wavelet frame coefficients. Some numerical examples are given and compared to the numerical solution by the present method with the Adomian decomposition method. Moreover, the modeling of the spreading of a non-fatal disease in a population, which represents a system of an ordinary differential equation is numerically solved by the proposed technique and compared with the Adomina decomposed method. The obtained results reveal that the present algorithm provides a good approximation than existing methods.

In this paper, we present an approximate solution for solving the nonlinear Rosenau–Hyman equation. The method is based on adapting the wavelet technique accompanied with the Hermit polynomials. Convergence analysis for the proposed method is being investigated, proving that the Hermite wavelet expansion is uniformly convergent. A test example is presented for different values of the parameters, and the obtained results are compared to other relevant methods from the literature. The process proves to have the ability to produce accurate results than the other compared methods. Some graphical representations for the problem are drawn to illustrate the behavior of the solution.

In this article, Shannon wavelets are studied together with their properties and we focused on Shannon wavelets and scalar functions through linear algebraic concept such as linearly independent, nested properties and direct product etc. Also, we generated Laguerre wavelets frames and discussed some results on it.

In this paper, Cardinal B-spline wavelet numerical method is developed for the solution of Generalized Burgers–Huxley(GBH) Equation. It is a new approach for the accurate numerical solution of the GBH equation with the initial and boundary conditions using Cardinal B-spline wavelets. This method is based on the truncated Cardinal B-spline wavelet expansions is used to convert the initial and boundary value problems into system of algebraic equations which can be efficiently solved by suitable solvers. Numerical results of the GBH equation with initial and boundary conditions shows the efficiency and accuracy of the present method.