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The purpose of this paper is to apply, for the first time, the authors' newly developed perturbation iteration method to heat transfer problems. The effectiveness of the new method in nonlinear heat transfer problems will be tested.

Nonlinear heat transfer problems are solved by perturbation iteration method. They are also solved by the well‐known technique variational iteration method in the literature.

It is found that perturbation iteration solutions converge faster to the numerical solutions. More accurate results can be achieved with this new method for nonlinear heat transfer problems.

A few iterations are actually sufficient. Further iterations need symbolic packages to calculate the solutions.

This new technique can practically be applied to many heat and flow problems.

The new perturbation iteration technique is successfully implemented to nonlinear heat transfer problems. Results show good agreement with the direct numerical simulations and the method performs better than the existing variational iteration method.

This study aims to propose the parametric-guiding algorithm, the complex-root (CR) tunneling algorithm and the method that integrates both algorithms for the heat and fluid flow (HFF) community, and apply them to nonlinear Bratu’s boundary-value problem (BVP) and Blasius BVP.

In the first algorithm, iterations are primarily guided by a diminishing parameter that is introduced to reduce magnitudes of fictitious source terms. In the second algorithm, when iteration-related barriers are encountered, CRs are generated to tunnel through the barrier. At the exit of the tunnel, imaginary parts of CRs are trimmed.

In terms of the robustness of convergence, the proposed method outperforms the traditional Newton–Raphson (NR) method. For most pulsed initial guesses that resemble pulsed initial conditions for the transient Bratu BVP, the proposed method has not failed to converge.

To the best of the authors’ knowledge, the parametric-guiding algorithm, the CR tunneling algorithm and the method that integrates both have not been reported in the computational-HFF-related literature.

To develop a subdomain perturbation technique to calculate skin and proximity effects in inductors within frequency and time domain finite element (FE) analyses.

A reference limit eddy current FE problem is first solved by considering perfect conductors via appropriate boundary conditions. Its solution gives the source for eddy current FE perturbation subproblems in each conductor with its actual conductivity. Each of these problems requires an appropriate mesh of the associated conductor and its surrounding region.

The skin and proximity effects in inductors can be accurately determined in a wide frequency range, allowing for a precise consideration of inductive phenomena as well as Joule losses calculations in thermal coupling.

The developed subdomain method allows to accurately determine the current density distributions and ensuing Joule losses in conductors of any shape, not only in the frequency domain but also in the time domain. It extends the domain of validity and applicability of impedance boundary condition techniques. It also allows the solution process to be lightened, as well as efficient parameterized analyses on signal forms and conductor characteristics.

The purpose of this paper is to investigate the inverse problem of determining a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions.

The variational iteration method (VIM) is employed as a numerical technique to develop numerical solution. A numerical example is presented to illustrate the advantages of the method.

Using this method, we obtain the exact solution of this problem. Whether or not there is a noisy overdetermination data, our numerical results are stable. Thus the VIM is suitable for finding the approximation solution of the problem.

This method is based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional and gives rapidly convergent successive approximations of the exact solution if such a solution exists.

The purpose of this paper is to apply an efficient hybrid computational numerical technique, namely, q-homotopy analysis Sumudu transform method (q-HASTM) and residual power series method (RPSM) for finding the analytical solution of the non-linear time-fractional Hirota–Satsuma coupled KdV (HS-cKdV) equations.

The proposed technique q-HASTM is the graceful amalgamations of q-homotopy analysis method with Sumudu transform via Caputo fractional derivative, whereas RPSM depend on generalized formula of Taylors series along with residual error function.

To illustrate and validate the efficiency of the proposed technique, the authors analyzed the projected non-linear coupled equations in terms of fractional order. Moreover, the physical behavior of the attained solution has been captured in terms of plots and by examining the L₂ and L_∞ error norm for diverse value of fractional order.

The authors implemented two technique, q-HASTM and RPSM to obtain the solution of non-linear time-fractional HS-cKdV equations. The obtained results and comparison between q-HASTM and RPSM, shows that the proposed methods provide the solution of non-linear models in form of a convergent series, without using any restrictive assumption. Also, the proposed algorithm is easy to implement and highly efficient to analyze the behavior of non-linear coupled fractional differential equation arisen in various area of science and engineering.

Investigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking model. Nonetheless, due to observational or experimental errors, or any other circumstance, it may contain some incomplete information. Fuzzy sets can be used to deal with uncertainty. Yet, there may be some inconsistency in the membership as well. As a result, the primary goal of this proposed work is to numerically solve the model in a type-2 fuzzy environment.

Triangular perfect quasi type-2 fuzzy numbers (TPQT2FNs) are used to deal with the uncertainty in the model. In this work, concepts of r2-cut at r1-plane are used to model the problem's uncertain parameter. The Legendre wavelet method (LWM) is then utilised to solve the giving up smoking model in a type-2 fuzzy environment.

LWM has been effectively employed in conjunction with the r2-cut at r1-plane notion of type-2 fuzzy sets to solve the model. The LWM has the advantage of converting the non-linear fractional order model into a set of non-linear algebraic equations. LWM scheme solutions are found to be well agreed with RK4 scheme solutions. The existence and uniqueness of the model's solution have also been demonstrated.

To deal with the uncertainty, type-2 fuzzy numbers are used. The use of LWM in a type-2 fuzzy uncertain environment to achieve the model's required solutions is quite fascinating, and this is the key focus of this work.