The Laplac operator on a metric graph is explained. Dispersion relation for this operator is derived. Inthis paper, we proved that the operator has a pure discrete spectrum consisting of eigenvalues tending to .

في هذي اهاحث،يمتهمنيقهن مؤثيلاهني الهعثيعلهاحنميلطهاليرهاحنشتيتهاحنميللهشاثيمممثهاعؤيلاهاحمثيمقهحني الهعثيعل ه اثم يمه هت يطهذي اهاحن اله كش هنممتعه م هش ،مشلهالرهؤ مه ام لاهممملبهإحرهاحنمع هم لا هه

Differential operator --- Metric graph --- Discrete spectrum.

:Differential Operators (Gradient, Laplacian and Biharmonic) have been used todetermine anomaly characteristics using theoretical gravity field for prismatic bodies with differenttop depths, dimensions and density contrasts. The concepts of gradient and laplacian operator arewidely used in image processing. The intersection between the gravity field and the three differentialoperator's fields could be used to estimate the depth to the top of the prismatic bodies regardless oftheir differences in dimensions, depths and density contrasts. The Biharmonic Operator has anexcellent result, were two zero closed contour line produced. The outline of the internal closed zerocontour line define precisely the dimension of the prismatic bodies. The distance between this zerocontour and the maxima of the Laplacian Operator define the exact depth to the top of the prismaticbodies. The maxima of the Biharmonic amplitude could be used for density contrast approximation.This is the first attempt to use such technique for estimating body characteristics. Also, theBiharmonic Operator has high sensitivity to resolve hidden small anomaly due the effect of largeneighborhood anomaly, the 2nd derivative Laplacian Filter could reveal these small anomaly but theBiharmonic Operator could indicate the exact depth. The user for such technique should be very careto the accuracy of digitizing the data due to the high sensitivity of Biharmonic Operator. The validityof the method is tested using field example for salt dome in Gulf Coast basin

استخدمت المعاملات التفاضلیة ( الانحدار، لابلاس والتوافقیة المزدوجة) لتحدید خصائص الشواذ لاجسام موشوریة ثلاثیة الابعاد لها اعماق وابعادوتباین كثافي مختلف ومن خلال مجالها الجذبي المحسوب نظریا. مبدئي الانحدار ولابلاس یستخدمان وبشكل واسع ضمن مبادىء التحلیل الصوري. تقاطعالمجال الجذبي مع المجالات التفاضلیة المحسوبة یمكن ان تساعد في حساب العمق الى السطح العلوي للاجسام الموشوریة بغض النظر عن الاختلاف فيابعادها واعماقها وتباینها الكثافي. معامل التوافقیة المزدوجة اعطى نتائج ممتا زة ، حیث یعطي انغلاقین كنتوریین وبقیمة صفریة. حدود الانغلاق الداخلي ذيالقیمة الصفریة یحدد وبصورة مضبوطة ابعاد الاجسام الموشوریة. المسافة بین الكنتور الصفري والقیم العظمى لمعامل لابلاس تحدد العمق الحقیقي الى السطحالعلوي للاجسام الموشوریة. القیم العظمى للتوافقیة المزدوجة یمكن استخدامها لتقییم التباین الكثافي. هذه هي المحاولة الاولى لاستخدام هذه الطریقة فيحساب خصائص الجسم ، وللمعامل التوافقي المزدوج حساسیة كبیرة لاظهار الت ا ركب الصغیرة والمخفیة بتأثیر الاجسام القریبة والكبیرة ، مرشح لابلاسلحساب المشتقة الثانیة یمكنه ایضا اظهار هذه الت ا ركیب الصغیرة ولكن معامل التوافقیة المزدوجة یمكنه حساب العمق المضبوط. المستخدم لهذه الطریقة یجبان یكون حذ ا ر خلال حساب ق ا رءاته لكون معامل التوافیة المزدوجة حساس جدا لتغیر هذه الق ا رءات. صلاحیة الطریقة اختبرت على مثال لقبة ملحیة في حوضساحل الخلیج.

Gravity --- Depth Estimation --- Prismatic bodies --- Differential Operator --- Gradient --- Laplacian --- Biharmonic

In This paper the generalized spline method and Caputo differential operator are applied to solve linear fractional integro-differential equations of the second kind. Comparison of the applied method with exact solutions reveals that the method is tremendously effective

Caputo differential operator --- Fractional integro-differential equations --- generalized spline

In the present paper, we have studied a class of analytic and meromorphic univalent functions defined by differential operator in the punctured unit disk and obtain some sharp results including coefficient inequality, distortion theorem, radii of starlikeness and convexity, Hadamard product, closure theorems. We also obtain some results connected with neighborhoods on and integral operator

Meromorphic univalent function --- Differential operator --- Distortion theorem --- Radii of starlikeness --- Hadamard product --- Neighborhood --- Integral operator

In the submitted search ,by making use of Differential operator ,we drive coefficient bounds and some important properties of the subclass Ti(n,p,q,a, A) (P,j E N= {1,2,...}; q,n,E N0=N U {0};0<= a

Multivalent function --- Coefficient bounds --- Distortion inequality --- - neighbourhood --- Differential operator --- integral and fractional operators .