Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method

Nonlinear equations are of great importance to our contemporary world. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. Despite the importance of obtaining the exact solution of nonlinear partial differential equations in physics and applied mathematics, there is still the daunting problem of finding new methods to discover new exact or approximate solutions. The purpose of this chapter is to impart a safe strategy for solving some linear and nonlinear partial differential equations in applied science and physics fields, by combining Laplace transform and the modified variational iteration method (VIM). This method is founded on the variational iteration method, Laplace transforms and convolution integral, such that, we put in an alternative Laplace correction functional and express the integral as a convolution. Some examples in physical engineering are provided to illustrate the simplicity and reliability of this method. The solutions of these examples are contingent only on the initial conditions

Explicit Analytic Solution of Vibration Equation for large domain by mean of the Elzaki projected Differential Transform Method

The aim of this paper is to present a reliable and efficient algorithm Elzaki projected differential transform method (EPDTM) to obtain the explicit solution of vibration equation for a very large membrane with given initial conditions. By using initial conditions, explicit series solutions for six different cases have been derived for the fast convergence of the solution. Numerical results show the reliability, efficiency and accuracy of Elzaki projected differential transform method (EPDTM). Numerical results for the six different cases are presented graphically

A New Homotopy Perturbation Method for Solving Systems of Nonlinear Equations of Emden-Fowler Type

In this work, we apply the new homotopy perturbation method (NHPM) to get accurate results for solving systems of nonlinear equations of Emden–Fowler type, we indicate that our method (NHPM) is equivalent  to the variational iteration method (VIM) with a specific convex. Four examples  are given  to illustrate our proposed methods. The method is easy to carry out and gives very accurate solutions for solving linear and nonlinear differential equations

Solution of Linear and Nonlinear Partial Differential Equations Using Mixture of Elzaki Transform and the Projected Differential Transform Method

The aim of this study is to solve some linear and nonlinear partial differential equations using the new integral transform "Elzaki transform" and projected differential transform method. The nonlinear terms can be handled by using of projected differential transform method; this method is more efficient and easy to handle such partial differential equations in comparison to other methods. The results show the efficiency and validation of this method. Keywords: Elzaki transform, projected differential transform method, nonlinear partial differential equations

Solution of Telegraph Equation by Modified of Double Sumudu Transform "Elzaki Transform"

In this paper, we apply modified version of double Sumudu transform which is called double Elzaki transform to solve the general linear telegraph equation. The applicability of this new transform is demonstrated using some functions, which arise in the solution of general linear telegraph equation. Keywords: Double Elzaki Transform, modified of double Sumudu transforms, Double Laplace transform, Telegraph Equation

On the Generalized Solution for Composite Type Differential Equation

Abstract In this paper, we study a boundary-value problem for a class of composite equation of a mixed-type problem in the space. The existence and uniqueness of the generalized solution is proved, the proof is based on an energy inequality and the density of the range of the operator generated by the problem

Exact Solution of nonlinear time-fractional Biological population Equation

ABSTRACT: This paper proposes a modified version of the differential transform method, which is termed the projected differential transform method (PDTM) .This method involves less computational work and can, thus, be easily applied to initial value problems. (PDTM) is used to determine the exact solutions of some nonlinear timefractional partial differential equations. A number of illustrative examples are provided and compared with the other methods. The numerical results obtained by these examples are found to be the same

On existence and uniqueness of generalized solutions for a mixed-type differential equation.

In this paper, we study a boundary value problem for a mixed–type differential equation. The existence and uniqueness of generalized solution is proved. The proof is based on an energy inequality and the density of the range of the operator generated by this problem

Analytical Solution for Telegraph Equation by Modified of Sumudu Transform "Elzaki Transform"

In this work modified of Sumudu transform [10,11,12] which is called Elzaki transform method ( new integral transform) is considered to solve general linear telegraph equation, this method is a powerful tool for solving differential equations and integral equations [1, 2, 3, 4, 5]. Using modified of Sumudu transform or Elzaki transform, it is possible to find the exact solution of telegraph equation. This method is more efficient and easier to handle as compare to the Sumudu transform method and variational iteration method. To illustrate the ability of the method some examples are provided.   Keywords: modified of Sumudu transform- Elzaki transform - Telegraph equation - Partial Derivative

Biometric Data of Adults’ Aortic Knob Diameter in Posteroanterior Chest Radiograph, Correlation to Age and Normative Heart Diameter: A Cross-Sectional Study

Background: The aortic knob (AK) is an essential feature on a chest x-ray. It could be the first sign of a cardiovascular problem if there is any deformation or enlargement of the knob. This study aimed to measure the normal AK diameter (AKD) on a posteroanterior chest radiograph in healthy adult Sudanese. Methods and Results: The study was conducted in the Department of Radiology and Imaging in Ribat Hospital (Sudan) between Jun 2019 and Jan 2020. A total of 113 participants of both sexes (45.1% males and 54.9% females) with a normal chest x-ray and no history of diabetes, blood hypertension, cardiovascular disease, or skeletal abnormality were selected. Participants' age fluctuated from 18 to 75 years. The measurements (AK, heart diameter [HD], cardiothoracic ratio [CTR]) were carried out with the measuring tools available on the software of the computed radiography system. The mean AKD was 2.8±0.8 cm (2.94±0.8 cm in males and 2.51±0.77 cm in females, P=0.005)). The mean HD was 9.22±2.8 cm (9.8±3.3 cm in males and 8.7±0.2.1 cm in females, P=0.005). The mean CTR was estimated as 46.6±7.7% with a significant difference between males and females and significantly correlated with HD and BMI (P<0.05). The AKD increased by 0.019 cm with an increase of one year of age (AKD = 0.0199(age)+1.9469), and there was a strong positive correlation between age and AKD (P<0.001). Conclusion: The study found a significant positive correlation between age and AKD. Increased heart sizes increase AKD. The AKD value is greater in males than in females