In this paper, we developed a new family of self starting second derivative Simpson’s type block methods (SDSM) of uniform order for step number. The new block methods forwere seen to possess good stability property as they possessed good regions of absolute stability. They were also found to be consistent, zero stable and A-stable (Fig.4). This essential property made them suitable for the solution of stiff system of ordinary differential equations. Four numerical examples were considered and results obtained show improved accuracy in terms of their Maximum absolute errors when compared with the work of existing scholars. The newly developed block methods were seen to approximate well with the stiff Ode Solver (Fig. 5, 6, 7 and 8).

A Modified ThreeStep Block Hybrid Extended Trapezoidal Multistep Method of Second Kind (BHETR 2 s) with two off-grid points, one at interpolation and another at collocation point yielding uniform order six (6,6,6,6,6) T for the Numerical Integration of initial value problems of stiff Ordinary Differential Equations was developed. The main method and additional equations were obtained from the same continuous formulation through interpolation and collocation procedures. The stability properties of the method was discussed and from the stability region obtained, the method is suitable for the solution Stiff Ordinary Differential Equations. Three numerical examples were considered to illustrate the efficiency and accuracy.

Many problems in natural and engineering sciences such as heat transfer, elasticity, quantum mechanics, water flow, and others are modelled mathematically by partial differential equations. Some of these problems may be linear, nonlinear, homogeneous, non-homogeneous, and order greater or equal one. Finding the theoretical solution to these problems with less cumbersome techniques is an active area of research in the aforementioned field. In this research paper, we have developed a new application of the double Laplace transform method to solve homogeneous and non-homogeneous linear partial differential equations (pdes) with higher-order derivatives (i.e order n where n≥2) in science and engineering. We discussed a brief theory of double Laplace transforms that helped in its application. The main advantage of our method is the reduction of computational effort in finding solution to pdes. Another major benefit of our method is solving problems in the form of (21) directly by transforming to an algebraic equation where the inverse double Laplace transform is implemented for analytical solution, unlike other integral transform methods that would first transform to a system of ODEs before they are solved, is it also very effective in solving linear high-order partial differential equations and yield fast convergence. We present a well-simplified solution for easier comprehension by upcoming researchers.

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