Praveen Kumar Gupta

Homotopy perturbation method for fractional Fornberg-Whitham equation

This article presents the approximate analytical solutions to solve the nonlinear Fornberg-Whitham equation with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm like homotopy perturbation method. The fractional derivatives are taken in the Caputo sense. Numerical results show that the HPM is easy to implement and accurate when applied to time-fractional PDEs.

A mathematical model on fractional Lotka–Volterra equations

The article presents the solutions of Lotka-Volterra equations of fractional-order time derivatives with the help of analytical method of nonlinear problem called the homotopy perturbation method (HPM). By using initial values, the explicit solutions of predator and prey populations for different particular cases have been derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The method performs extremely well in terms of efficiency and simplicity to solve this historical biological model.

Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method

In this paper, the approximate analytical solutions of Benney-Lin equation with fractional time derivative are obtained with the help of a general framework of the reduced differential transform method (RDTM) and the homotopy perturbation method (HPM). RDTM technique does not require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. Comparing the methodology (RDTM) with some known technique (HPM) shows that the present approach is effective and powerful. The numerical calculations are carried out when the initial conditions in the form of periodic functions and the results are depicted through graphs. The eight different cases have studied and proved that the method is extremely effective due to its simplistic approach and performance.

A Fractional Predator-Prey Model and its Solution

A fractional Lotka-Volterra model is suggested, and its solution is obtained using the homotopy perturbation method. The effect of the fractional order on populations of the predator and the prey is discussed.

An approximate analytical solution of time-fractional telegraph equation

Abstract In this article, the powerful, easy-to-use and effective approximate analytical mathematical tool like homotopy analysis method (HAM) is used to solve the telegraph equation with fractional time derivative α (1

An Approximate Analytical Solution of the Fractional Diffusion Equation with Absorbent Term and External Force by Homotopy Perturbation Method

In the present paper, the approximate analytical solutions of a general diffusion equation with fractional time derivative in the presence of an absorbent term and a linear external force are obtained with the help of the powerful homotopy perturbation method (HPM). By using initial values, the approximate analytical solutions of the equation are derived. The results are deduced for different particular cases. The numerical results show that only a few iterations are needed to obtain accurate approximate solutions and these are presented graphically. The presented method is extremely simple, concise, and highly efficient as a mathematical tool in comparison with the other existing techniques.

Solution of the heat transfer problem in tissues during hyperthermia by finite difference-decomposition method

In this article, a mathematical model describing the process of heat transfer in biological tissues with blood perfusion having different values under different temperature range for various coordinate system and different boundary conditions during thermal therapy by electromagnetic radiation is studied. Using the method of finite differences, the boundary value problem governing this process has been converted to an initial value problem of ordinary differential equations, which is solved by the Adomian decomposition method. The effects of blood perfusion rate intended for a different temperature range of temperature at target point during thermal therapy for different boundary conditions are discussed in detail. And we have also checked our results with the result of exact solutions for one case and it shows a good agreement.

Homotopy analysis method for solving fractional hyperbolic partial differential equations

In the present paper, the solutions of the hyperbolic partial differential equation with fractional time derivative of order α (1<α≤2) are obtained with the help of approximate analytical method of nonlinear problems called the homotopy analysis method. By using initial values, the explicit solutions of the equations for different particular cases have been derived which demonstrate the effectiveness, validity, potentiality and reliability of the method in reality. Numerical results for different particular cases are presented graphically. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions.

Solution of fractional bioheat equations by finite difference method and HPM

Abstract In this paper, we present a mathematical model of space–time fractional bioheat equation governing the process of heat transfer in tissues during thermal therapy. Using fractional backward finite difference scheme, the problem is converted into an initial value problem of vector-matrix form and homotopy perturbation method is used to solve it. Results are interpreted in the form of standard case and anomalous cases for taking different orders of space and time fractional derivatives.

A numerical study on heat transfer in tissues during hyperthermia

A mathematical model describing the process of heat transfer in tissues during high temperature thermal therapy by electromagnetic radiation of organs in human body for different coordinate systems and under different boundary conditions is proposed. The heat transfer in tissues is examined using the modified Pennes bioheat transfer equation. The boundary value problem governing this process has been solved by Galerkins method using the Bernstein polynomial as a basis function. The system of ordinary differential equations in an unknown time variable, thus obtained, is solved by the variational iteration method. The whole analysis is presented in a dimensionless form. The dimensionless time to achieve the hyperthermia position is calculated. The effects of variability of Pf, Pm, Pr, Ki and Bi on dimensionless tissue temperature during steady state are shown graphically. It has been observed that during thermal therapy, probe shape, boundary conditions and internal heat source should not be the same and must be changed from organ to organ in the human body.